A Multi-Target Trajectory Optimization Method for Robotic Arms
By employing a multi-objective trajectory optimization method, utilizing clamped B-spline curves and an improved NSGA-II algorithm, the problem of balancing multiple performance indicators in robot trajectory optimization was solved, enabling efficient and stable movement of the robotic arm in different application scenarios.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- TSINGHUA SHENZHEN INTERNATIONAL GRADUATE SCHOOL
- Filing Date
- 2024-06-27
- Publication Date
- 2026-06-30
AI Technical Summary
Existing technologies struggle to balance various performance metrics, such as execution time, energy consumption, and joint impact, in robot trajectory optimization, resulting in insufficient adaptability in different application scenarios.
A multi-objective trajectory optimization method is adopted. By transforming the workspace path points to the joint space, the clamped B-spline curve is used for planning. Combined with Pareto dominance and the improved NSGA-II algorithm, the total running time, total energy consumption and average pulsation of the robotic arm are optimized. The optimal trajectory is selected using the SSM comprehensive performance metric method.
It achieves smooth and efficient robotic arm movement, reduces joint impact, and improves the robot's adaptability and stability in different application scenarios, with significant optimization results.
Smart Images
Figure CN118596150B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of industrial robot control technology, specifically to a method for optimizing the multi-target trajectory of a robotic arm. Background Technology
[0002] With the advancement of technology, robotics is also developing rapidly, leading to the widespread application of robotic arms in industry. For many practical robotic applications (such as palletizing, labeling, and spot welding), trajectory planning is crucial for task completion. The goal of trajectory planning is to find a geometric path with given motion laws in Cartesian space or joint space, enabling the manipulator to achieve fast, accurate, and stable movement while satisfying obstacle constraints, kinematic constraints, and dynamic constraints. Therefore, trajectory planning has significant theoretical and engineering implications.
[0003] Industrial robots perform basic operations by connecting the target points of a task through trajectory planning. However, different task applications have different performance requirements. Therefore, further optimization of the robot's trajectory based on trajectory planning is of great significance. Trajectory optimization aims to obtain trajectory parameters that better meet the needs of actual applications by optimizing the performance indicators related to the operation process of the industrial robot. The main performance indicators typically focused on include: execution time, energy consumption, maximum power, and driving torque. Depending on the optimization objective, trajectory optimization can be further divided into single-objective trajectory optimization and multi-objective trajectory optimization.
[0004] In trajectory optimization, considering only a single objective function may not be suitable for meeting the diverse requirements of real-world applications. For example, while execution time and joint stress are important for improving productivity and maintaining trajectory smoothness, solely pursuing shorter execution times inevitably leads to greater stress on the joints. Therefore, multi-objective trajectory optimization methods can balance different performance indicators of industrial robots, better meeting their performance needs in various application scenarios. Currently, multi-objective evolutionary algorithms utilize non-dominated sorting to provide decision-makers with Pareto solutions, offering considerable flexibility. Consequently, this optimization technique is increasingly popular among researchers. Summary of the Invention
[0005] To address the shortcomings of existing technologies, this invention provides a multi-target trajectory optimization method for robotic arms. It enables trajectory optimization of industrial robots in various application scenarios, offering advantages such as high practicality and excellent optimization results.
[0006] The technical problem of this invention is solved by the following technical solution:
[0007] This invention provides a method for optimizing the multi-target trajectory of a robotic arm, comprising the following steps:
[0008] This invention provides a method for optimizing the multi-target trajectory of a robotic arm, comprising the following steps:
[0009] S1: Transform the path points in the workspace into the joint space, and use the clamped B spline curve with virtual value points to plan the trajectory, and obtain the clamped B spline trajectory curve parameters;
[0010] S2: Perform multi-objective trajectory optimization, taking the total running time, total energy consumption and average pulse of the robotic arm to be optimized as the objective function of trajectory optimization, and transforming the position, velocity, acceleration and jerk constraints into control point constraints, and finding the optimal solution based on the Pareto dominance relationship;
[0011] S3: Select the optimal trajectory that meets the requirements using the SSM comprehensive performance metric method.
[0012] In some embodiments, step S1, converting path points in the workspace to joint space, specifically involves inputting the workspace path points into the inverse kinematics function, solving for the corresponding shape points in the joint space, and further solving for the control points of the trajectory to complete the conversion.
[0013] In some embodiments, in step S1, a clamped B-spline curve with virtual shape points is used to generate the trajectory between shape points, and the trajectory is planned using the following formula:
[0014]
[0015] Where p(u) represents the type value at time node u, and d j N represents the position coordinates of the j-th control point of the trajectory curve. j，k (u) represents the value of the k-th order spline basis function corresponding to the j-th control point at time node u, where m is the number of time nodes, obtained recursively using the de Boer formula:
[0016]
[0017] Among them, u j u j+k u j+k+1 u j+1 The normalized node vector is calculated by normalizing the time series corresponding to the shape points. For the n+1 shape points in the joint space, the clamped B spline curve specifies that the number of control points is n+k, corresponding to the number of time nodes m = n+2k+1, expressed as:
[0018] d = [d0, d1, ..., d n+k-1 ]
[0019] u = [u0, u1, ..., u]n+2k ]
[0020] The overlap between the first and last time nodes is k+1, thus ensuring that the positions of the initial and final control points coincide with the initial and final value points.
[0021]
[0022] Among them, i=k+1,...,n+k-1,t j Let j be the time corresponding to time j. Since the time series is represented by floating-point numbers, normalization is required to improve the numerical accuracy of floating-point calculations.
[0023] In some embodiments, step S1 solves for the coordinates of the corresponding control points using a system of linear equations, providing n+1 constraints for n+k control points d and n+1 shape points p:
[0024]
[0025] For the clamped B-spline curve, solving for the control points requires an additional k-1 constraints, which are constructed by adding boundary conditions where the velocity and acceleration of the initial and final value points are zero.
[0026] In some embodiments, in step S1, there are two virtual model points, located after the first model point and before the last model point, respectively. Two time nodes and two control points are added accordingly. The boundary condition that the jerk at the start and end positions is 0 is introduced into the control point solution equation for these two control points. All boundary conditions are as follows:
[0027]
[0028]
[0029]
[0030]
[0031]
[0032]
[0033] in It is the j-th control point of order l:
[0034]
[0035] The control points of the Clamped B-spline curve can be derived from the value points obtained through discrete programming using the above formula, and thus the value of any point on the curve can be obtained.
[0036] In some embodiments, in step S2, the optimization variable for multi-objective trajectory optimization is the time interval between target points, and the optimization evaluation indicators include the total running time of the robotic arm, average energy consumption, and average impact. The objective function for the above evaluation indicators is:
[0037]
[0038]
[0039]
[0040] Where F1, F2, and F3 represent the robot's efficiency, energy consumption, and stability, respectively, tj is the time at time j, and t0 and t... n These represent the times corresponding to the first and last model value points, respectively. These are the joint acceleration and jerk in joint space, respectively, and T is a marker for the total trajectory running time.
[0041] In some embodiments, in step S2, the constraints for multi-objective trajectory optimization include robotic arm kinematic constraints, which include constraints on joint position, velocity, acceleration, and jerk, as shown below:
[0042]
[0043] in P is the i-th derivative of the j-th joint of the robotic arm. j It is the joint position constraint, V j It is the joint velocity limit, W j It is the joint acceleration limit, J j This refers to the joint jerk constraint. Utilizing the properties of B-spline curves, the kinematic constraints are transformed into control point constraints, reducing the computational burden of constraint processing. The resulting control point constraints are shown below:
[0044]
[0045] in, It is the i-th l-th control point of the j-th joint of the robotic arm, and the scale coefficient c is estimated through numerical simulation.
[0046] In some embodiments, in step S2, multi-objective trajectory optimization is performed by improving the NSGA-II algorithm (Non-dominated Sorting Genetic Algorithm II, a classic multi-objective optimization algorithm). Specifically, the improved NSGA-II algorithm improves the genetic operation part, reducing the complexity of the algorithm and increasing the diversity and stability of the obtained solutions. The genetic operation part includes selection operation, crossover operation and mutation operation.
[0047] In some embodiments, in step S2,
[0048] For the selection operation, the roulette wheel selection method is replaced with a sorted group selection method, reducing the computational complexity from O(n^2) to O(n^2). 2 The value of logn is reduced to O(nlog n); P t Q t Let F1, F2, and F3 be the sets of individuals with the highest and lowest fitness rankings among the newly generated individuals. Then, a certain proportion of the top-ranked individuals are sorted and stratified according to fitness and crowding to obtain F3. This part corresponds to P. t+1 That is, the basic individuals for the next round; divide these individuals into x i Groups i = 1, 2, 3, 4 are matched and cross-matched according to the grouping method shown in the diagram;
[0049] For the matching crossover operation, the simulated binary crossover operator is replaced with a directed group crossover. For a matched group, the crossover operation is performed on the individuals within that group. The expression for the crossover operation is:
[0050]
[0051] in, To select the i-th individual from group A obtained during the operation, To select the j-th component of the i-th individual in groups A and B obtained during the selection operation, r represents the spatial distance on the corresponding components. ij A random number between [-1, 1] The new individual obtained from the crossover operation.
[0052] In some embodiments, step S3 includes the following steps:
[0053] S3-1: Evaluate the diversity of the obtained solutions using the SSM performance metric method, specifically:
[0054]
[0055] Where M is the number of solutions on the Pareto front, s i The distance between adjacent solutions. For s i The mean, s f s l These are the Euclidean distances between the boundary solutions and the extrema, respectively. A smaller SSM metric indicates better diversity in the distribution of solutions.
[0056] S3-2: The fitness value of the Pareto solution is evaluated using the fuzzy membership function, specifically as follows:
[0057]
[0058] Among them, S i (j) represents the i-th objective function value of the solution on the j-th Pareto front, S imax S imin Let represent the maximum and minimum values of the i-th objective function on the Pareto front;
[0059] S3-3: Evaluation is conducted using fitness combination evaluation metrics, specifically:
[0060]
[0061] S3-4: Will u syn The solution corresponding to (j) = 1 is the optimal solution, and the trajectory corresponding to the optimal solution is the best trajectory.
[0062] The present invention has the following beneficial effects:
[0063] This invention uses clamped B-spline curves with added virtual value points to construct continuous path motion trajectories. This results in good local support for the trajectory while ensuring that the robot's velocity, acceleration, and jerk are all zero at the beginning and end of the motion. This effectively reduces joint impact during robot movement and extends the robot's lifespan. By transforming robot kinematic constraints into control point constraints, the computational load for constraint handling in optimization problems is effectively reduced, leading to better trajectory planning results. Furthermore, this invention's SSM (Simultaneous Performance Measurement) method selects the optimal solution based on the actual application scenario, yielding results with good optimization performance and environmental adaptability. This method significantly improves the robot's operational efficiency and stability.
[0064] In some embodiments, this invention achieves multi-objective trajectory optimization using an improved NSGA-II algorithm. The robot's time, energy consumption, and impact are used as objective functions for trajectory optimization, and the optimal solution is found based on Pareto dominance. A sorting and grouping selection method replaces the traditional genetic selection method, reducing computational complexity while maintaining population diversity. Directed grouping crossover replaces the traditional genetic crossover method, improving solution diversity while ensuring convergence. This method yields the optimal trajectory solution set in a shorter time and is highly efficient.
[0065] Other beneficial effects of the embodiments of the present invention will be further described below. Attached Figure Description
[0066] Figure 1 This is a flowchart of the method in an embodiment of the present invention;
[0067] Figure 2 This is a schematic diagram of the control points of the clamped B-spline curve used in the embodiments of the present invention;
[0068] Figure 3 This is a schematic diagram illustrating the constraint transformation principle of the clamped B-spline curve used in the embodiments of the present invention;
[0069] Figure 4 This is a schematic diagram illustrating the principle of the sorting and grouping selection method used in this embodiment of the invention;
[0070] Figure 5 This is a schematic diagram illustrating the principle of the directional grouping crossover method used in this embodiment of the invention. Detailed Implementation
[0071] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.
[0072] This invention provides a multi-target trajectory optimization method for robotic arms based on an improved NSGA-II algorithm. The main workflow is as follows: Figure 1 As shown, based on the workspace path points planned by the upstream task, and combined with kinematic constraints, the problem is transformed into a trajectory optimization problem in joint space. A multi-objective trajectory optimization task is performed, with time, energy consumption, and impact as optimization objectives. The optimal trajectory parameter set is output, and the multi-objective optimal trajectory parameters for the current task are obtained by combining the evaluation method. Robot kinematic constraints refer to the restrictions imposed on the robot during movement, usually defined in joint space; control point constraints refer to the constraints set on the positions of control points in trajectory optimization. The above process includes the following steps:
[0073] S1. Transform the path points in the workspace to the joint space, and plan the trajectory using a 5th-order non-uniform B-spline curve with virtual value points to obtain a 5th-order non-uniform B-spline trajectory curve under velocity, acceleration, and jerk constraints. Here, the workspace is a three-dimensional space, and the joint space refers to the vector space formed by the joint angles of the robotic arm. First, based on the task requirements, determine a series of path points for the robotic arm in the workspace. Then, using forward and inverse kinematics algorithms, transform these path points from the workspace to the joint space to obtain the corresponding joint angle sequence.
[0074] Specifically, the conversion of path points in the workspace to joint space involves inputting the path points in the workspace into the inverse kinematics function, solving for the corresponding shape points in the joint space, and further solving for the control points of the trajectory to complete the conversion.
[0075] In some embodiments of the present invention, step S1 uses a clad B-spline curve with virtual shape points added to generate the trajectory between shape points, and the formula used is:
[0076]
[0077] Where p(u) represents the type value at time node u, d j N represents the position coordinates of the j-th control point of the trajectory curve. j，k (u) represents the value of the k-th order spline basis function corresponding to the j-th control point at time node u, which is obtained recursively using the De Boer formula:
[0078]
[0079] Where u j The normalized node vector is calculated by normalizing the time series corresponding to the shape points. For the n+1 shape points in the joint space, the clampe d B spline curve specifies that the number of control points is n+k, corresponding to the number of time nodes m = n+2k+1, expressed as:
[0080] d = [d0, d1, ..., d n+k-1 ]
[0081] u = [u0, u1, ..., u] n+2k ]
[0082]
[0083] Where i = k+1, ..., n+k-1, the overlap between the first and last time nodes is k+1, thus ensuring that the positions of the initial and final control points coincide with the initial and final value points. Since the time series is represented by floating-point numbers, normalization is required to improve the numerical accuracy of floating-point calculations.
[0084] The coordinates of the corresponding control points are solved by a system of linear equations. For n+k control points d and n+1 shape point p, n+1 constraints are provided:
[0085]
[0086] Solving for the control points still requires k-1 constraints. For the clamped B-spline curve used in this invention, boundary conditions with zero velocity and acceleration at the initial and final points need to be added, resulting in a total of 4 constraints. In actual production, we also need to ensure that the jerk of the industrial robot is zero at the initial and final positions to avoid damage to mechanical parts caused by sudden stops. Therefore, we add two virtual points, one after the first and one before the last, requiring two time nodes and two control points. To solve for the two newly introduced control points, the boundary condition of zero jerk at the initial and final positions is introduced into the control point solution equation. All boundary conditions are shown below:
[0087]
[0088]
[0089]
[0090]
[0091]
[0092]
[0093] in It is the j-th control point of order l:
[0094]
[0095] Rearranging it into matrix form, we get:
[0096]
[0097] Where C is a (n+k+2)×(n+k+2) matrix, we can use the value points obtained by discrete programming to inversely calculate the control points of the clamped B spline curve based on the above formula, and then obtain the value of any point on the curve.
[0098] S2. Multi-objective trajectory optimization is achieved by improving the NSGA-II algorithm. The total running time, total energy consumption and average pulse of the robot to be optimized are taken as the objective function of trajectory optimization, and the optimal solution is found based on the Pareto dominance relationship.
[0099] More specifically, the optimization variables in the multi-objective trajectory optimization problem are the time intervals between target points, and the optimization evaluation indicators include the robot's total running time, average energy consumption, and average impact. The objective function for these evaluation indicators is:
[0100]
[0101]
[0102]
[0103] Where F1, F2, and F3 represent the robot's efficiency, energy consumption, and stability, respectively, tj is the time at time j, and t0 and t... n These represent the times corresponding to the first and last model value points, respectively. These are the joint acceleration and jerk in joint space, respectively, and T is a marker for the total trajectory running time.
[0104] During trajectory optimization, it is also essential to ensure that the robot remains in a safe state throughout the entire operation, ensuring that the position, velocity, acceleration, jerk, and torque of all robot joints simultaneously satisfy the constraints. The constraints of the optimization problem are primarily robot kinematic constraints, including constraints on joint position, velocity, acceleration, and jerk, as shown below:
[0105]
[0106] in P is the i-th derivative of the j-th joint of the robotic arm. j It is the joint position constraint, V j It is the joint velocity limit, W j It is the joint acceleration limit, J j It is the joint acceleration limit, because Difficult to express analytically, and numerical methods require enormous computation, this invention utilizes the properties of B-spline curves. Specifically, the obtained B-spline curve is completely enclosed by the convex hull formed by its control points, such as... Figure 2 As shown, this invention transforms kinematic constraints into control point constraints, effectively reducing the computational load of constraint processing. The resulting control point constraints are shown below:
[0107]
[0108] in It is the i-th l-th control point of the j-th joint of the robotic arm, and the scale coefficient c can be estimated through numerical simulation.
[0109] The improved NSGA-II algorithm is used to solve multi-objective trajectory optimization problems, and includes fast non-dominated sorting, crowding calculation, genetic operations, and elite strategies. This method uses Pareto dominance relations to evaluate the superiority or inferiority among different individuals, and finds a set of Pareto optimal solutions based on the dominance relations to satisfy the sub-objective functions F. i Approaching the optimal state. The total population size is N, and the state of an individual in the population is marked by the time interval parameter Δt = {t1-t0, t2-t1, ..., t...} between adjacent path points. n -t n-1}
[0110] For each individual i in the population, the fast non-dominated sorting defines the following two parameters: n i (The number of individuals that dominate individual i in the population) and S i (The set of individuals in the population dominated by individual i). The specific implementation steps of fast non-dominated sorting hierarchical structure are as follows:
[0111] ① Find all n in the population i Individuals with a value of 0 are stored in set F1;
[0112] ② For each individual i in set F1, the set of individuals it governs is S. i traverse S i Each of them
[0113] Individual l, n l =n l -1, if n l =0, store individual l in set F2 (second level);
[0114] ③ Using F2 as the current set, repeat the above operation until all individuals in the population are stratified.
[0115] Crowding ordering uses crowding distance to represent the density of other individuals around a given individual in a population. The crowding distance of an individual i in the population is... d This is equal to the perimeter of the largest rectangle surrounding the individual in the population that includes that individual but excludes all other individuals. Using this method ensures a uniform distribution of optimal solutions on the Pareto optimal front, guaranteeing good population diversity. After fast non-dominated sorting and crowding distance sorting, each individual in the population has two attribute values: non-dominated index i. rank and crowded distance i d .
[0116] The elite strategy refers to merging the new population generated in a certain generation with its parents to create a new population of size 2N. Then, the best N individuals are selected from this new population to form a new population for genetic manipulation. The specific steps are as follows:
[0117] ①Recombine all individuals of a certain generation of the population and its parent population to form a new population of size 2N;
[0118] ② Perform non-dominated ranking and crowding distance calculations on all individuals in the new population, and select the top N individuals to form the new population;
[0119] ③ Perform genetic operations on the new population to obtain the next generation population;
[0120] ④ Repeat the above steps to allow the population to continuously evolve.
[0121] Compared to the traditional NSGA-II algorithm, the improved NSGA-II algorithm has improved the genetic operations (selection, crossover, mutation), which reduces the complexity of the algorithm and increases the diversity and stability of the obtained solutions.
[0122] For the selection operation, this invention replaces the commonly used roulette wheel selection method with a sorting and grouping selection method, reducing the computational complexity from O(n^2) to O(n^2). 2 The solution reduces the time complexity from O(log n) to O(nlog n), as shown in the following example. Figure 4 As shown: P t Q t These are the sets of individuals with the highest and lowest fitness rankings among the newly generated individuals. A certain proportion (usually 50%) of the top-ranked individuals are then sorted and stratified according to fitness and crowding to obtain F1, F2, and F3. This part corresponds to P. t+1 These individuals are the foundational individuals for the next round. Divide these individuals into x groups. i Groups i = 1, 2, 3, 4 are matched and cross-matched according to the grouping method shown in the figure.
[0123] For crossover operations, this invention replaces the commonly used analog binary crossover operator with directional group crossover, such as... Figure 5 As shown, for Figure 4 Given the matched groups A and B, perform a crossover operation on the individuals within each group. The expression for the crossover operation is shown below:
[0124]
[0125] in To select the i-th individual from group A obtained during the operation, To select the j-th component of the i-th individual in groups A and B obtained during the selection operation, r represents the spatial distance on the corresponding components. ij A random number between [-1, 1] The new individual obtained from the crossover operation.
[0126] S3. Select the optimal trajectory that meets the requirements using the SSM comprehensive performance measurement method;
[0127] The diversity of the obtained solutions is evaluated using the SSM performance metric method, and the calculation method is as follows:
[0128]
[0129] Where M is the number of solutions on the Pareto front, s i The distance between adjacent solutions. For s i The mean, s f s l , representing the Euclidean distance between the boundary solutions and the extrema, respectively; a smaller SSM metric indicates better diversity in solution distribution. The fitness value of the Pareto solution is evaluated using a fuzzy membership function, calculated using the following formula:
[0130]
[0131] Where S i (j) represents the i-th objective function value of the solution on the j-th Pareto front, indicating that S imax S imin To comprehensively consider all performance metrics, we use the fitness combinatorial evaluation index to evaluate the maximum and minimum values of the i-th objective function on the Pareto front. The calculation method is as follows:
[0132]
[0133] The best solution is u syn The solution corresponding to (j)=1.
[0134] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, several equivalent substitutions or obvious modifications can be made without departing from the concept of the present invention, and all such modifications, achieving the same performance or purpose, should be considered within the scope of protection of the present invention.
Claims
1. A method for optimizing the multi-target trajectory of a robotic arm, characterized in that, Includes the following steps: S1: Transform the path points in the workspace into the joint space, and use the clamped B-spline curve with virtual value points to plan the trajectory, and obtain the clamped B-spline trajectory curve parameters; In step S1, there are two virtual model points, located after the first model point and before the last model point, respectively. Two time nodes and two control points are added accordingly. For these two control points, the boundary condition of zero jerk at the start and end positions is introduced into the control point solution equation. All boundary conditions are shown below: in It is the j-th control point of order l: The control points of the clamped B spline curve can be derived from the model points obtained by discrete programming using the above formula, and then the value of any point on the curve can be obtained. S2: Perform multi-objective trajectory optimization, taking the total running time, total energy consumption and average pulse of the robotic arm to be optimized as the objective function of trajectory optimization, and transforming the position, velocity, acceleration and jerk constraints into control point constraints, and finding the optimal solution based on the Pareto dominance relationship; S3: Select the optimal trajectory that meets the requirements using the SSM comprehensive performance measurement method; Step S3 includes the following steps: S3-1: Evaluate the diversity of the obtained solutions using the SSM performance metric method, specifically: in Let be the number of solutions on the Pareto front. The distance between adjacent solutions. for The mean, These are the Euclidean distances between the boundary solutions and the extrema, respectively. A smaller SSM metric indicates better diversity in the distribution of solutions. S3-2: The fitness value of the Pareto solution is evaluated using the fuzzy membership function, specifically as follows: in, For the first The solution on the Pareto front The objective function value, Let represent the maximum and minimum values of the i-th objective function on the Pareto front; S3-3: Evaluation is conducted using fitness combination evaluation metrics, specifically: S3-4: Will The corresponding solution is the optimal solution, and the trajectory corresponding to the optimal solution is the best trajectory.
2. The method according to claim 1, characterized in that, In step S1, the conversion of path points in the workspace to joint space is specifically achieved by inputting the path points in the workspace into the inverse kinematics function, solving for the corresponding shape points in the joint space, and further solving for the control points of the trajectory to complete the conversion.
3. The method according to claim 2, characterized in that, In step S1, a clamped B-spline curve with virtual shape points is used to generate the trajectory between shape points, and the trajectory is planned using the following formula: in, The type value represents the time node u. This represents the position coordinates of the j-th control point on the trajectory curve. This represents the value of the k-th order spline basis function corresponding to the j-th control point at time node u. The number of time points is obtained recursively using the de Boer formula: in, , , , The normalized node vector is obtained by normalizing the time series corresponding to the type value points. The number of control points in the joint space, as specified by the clamped B-spline curve, is [number]. Number of corresponding time nodes One, represented as: The overlap between the first and last time points is: This ensures that the positions of the initial and final control points coincide with the initial and final data points. in, Let j be the time corresponding to time j. Since the time series is represented by floating-point numbers, normalization is required to improve the numerical accuracy of floating-point calculations.
4. The method according to claim 3, characterized in that, Step S1 solves for the coordinates of the corresponding control points using a system of linear equations. Control points , Individual value points supply One constraint: For the clampedB spline curve, solving for the control points requires additional steps. Each constraint is constructed by adding boundary conditions where the velocity and acceleration of the initial and final value points are zero.
5. The method according to claim 1, characterized in that, In step S2, the optimization variable for the multi-objective trajectory optimization is the time interval between the target points. The optimization evaluation indicators include the total running time of the robotic arm, average energy consumption, and average impact. The objective function for the above evaluation indicators is: in, , , These represent the robot's efficiency, energy consumption, and stability, respectively, with tj being the time corresponding to moment j. and These represent the times corresponding to the first and last model value points, respectively. These are the joint acceleration and jerk in joint space, respectively, and T is a marker for the total trajectory running time.
6. The method according to claim 5, characterized in that, In step S2, the constraints for multi-objective trajectory optimization include robotic arm kinematic constraints, which include constraints on joint position, velocity, acceleration, and jerk, as shown below: in It is the first robotic arm Each joint First derivative, It is the joint position restriction amount. It is the joint speed limit. It is the joint acceleration limit. This refers to the joint jerk constraint. Utilizing the properties of B-spline curves, the kinematic constraints are transformed into control point constraints, reducing the computational burden of constraint processing. The resulting control point constraints are shown below: in, It is the first robotic arm The first joint indivual Level control points, scaling factor Estimation is performed through numerical simulation.
7. The method according to claim 5, characterized in that, In step S2, multi-objective trajectory optimization is performed by improving the NSGA-II algorithm. Specifically, the improved NSGA-II algorithm improves the genetic operation part, reducing the complexity of the algorithm and increasing the diversity and stability of the obtained solutions. The genetic operation part includes selection operation, crossover operation and mutation operation.
8. The method according to claim 7, characterized in that, In step S2, For the selection operation, the roulette wheel selection method is replaced with a sorted grouping selection method, reducing the computational complexity from... Reduced to ; These are the sets of individuals with the highest and lowest fitness rankings among the newly generated individuals. A certain proportion of the top-ranked individuals are then sorted and stratified according to fitness and crowding. This part corresponds to That is, the basic individuals for the next round; these individuals are divided into Group them and match them according to the grouping method shown in the diagram; For the matching crossover operation, the simulated binary crossover operator is replaced with a directed group crossover. For a matched group, the crossover operation is performed on the individuals within that group. The expression for the crossover operation is: in, To select the first item in group A obtained during the selection operation Individual, To select the first from groups A and B obtained during the operation The first individual Quantity, This represents the spatial distance on the corresponding components. for Random numbers between, The new individual obtained from the crossover operation.