Abstract
Aiming at the problem of alignment deviation between shaft and hole caused by accumulated error in the assembly process of brake disc shaft hole, a trajectory planning method for compensating accumulated error is proposed. First, the path before the error is planned, and then the second compensation path is planned between the error position and the actual target point, so as to realize the accurate assembly of the brake disc. In this paper, the IRB 1410 robot is taken as the research object, and its kinematic model is established by using the improved D-H parameter method. The joint space quintic B-spline interpolation method was used to carry out trajectory planning, and the improved particle swarm optimization algorithm was introduced to solve the optimization. The optimal time and smooth trajectory were expected to be obtained. Then, using MATLAB software to simulate, compared with the nonoptimized trajectory, not only is the trajectory running time reduced from 8.6 s to 6.7 s but also the maximum joint change angle of each joint is reduced, which proves that the algorithm has optimization effect on trajectory running time and stability. Finally, the accuracy verification experiment of the algorithm is carried out, and the error between simulation and experiment is less than 6%, which shows the effectiveness of the method. This research provides a theoretical basis for improving the responsiveness and stability of brake disc assembly.
1. Introduction
6-Dof Robot is widely used in daily production and life, greatly reducing the amount of human labor. In order to pursue a more efficient, stable, and long-lived robots, a large number of researchers have made in-depth research on robot structure, robot control [1], robot trajectory planning [2–4], and so on. There are many methods for the research of robot control. For example, Roveda et al. [5] put forward a sensorless optimal switching impact/force (OSIF) controller, designed the adaptive mechanism of impedance control parameters, realized the smooth and stable transition between optimal impact control and optimal force control, so as to realize the stable control and operation of the robot. In terms of robot trajectory planning, trajectory planning, as the basic research of robots, has always been a hot spot in robot research. Trajectory planning is the basis of robot control, which is of great significance to the work efficiency, motion stability, and energy consumption of robots. In the complete research process of trajectory planning, first, interpolation operation should be performed on the trajectory points to obtain smooth trajectory, then the optimization algorithm should be used to optimize the obtained trajectory, and finally the optimized trajectory is obtained to ensure that the robot can move smoothly and efficiently.
In the research of trajectory interpolation, the interpolation algorithm is generally used to establish the time trajectory sequence of each axis of the robot, including the trajectory position, speed, and acceleration information [6, 7]. Early researchers proposed the polynomial interpolation method [8, 9], which has a small amount of calculation, but is prone to distortion and has the disadvantages of high order and no convex hull. With the improvement of technology, researchers began to use spline curves for robot trajectory planning. Zhang [10] proposed a method for planning robot trajectories by using cubic spline curves. Taking the approaching degree of the path points and the parameter continuity of the trajectory curve as a reference, a smooth trajectory model of the robot was established. The algorithm has fewer constraints and faster calculation speed, but there is jitter in the acceleration curve, resulting in greater wear and tear of the robot. In order to obtain the joint trajectory with continuous speed and acceleration, Jin et al. [11] uses a cubic uniform B-spline curve to construct the joint space trajectory of the manipulator first. The trajectory curve is relatively smooth, but initial acceleration and final acceleration cannot be specified by themselves. He et al. [12] get the relationship between joint variables and time by inverse solution in joint space trajectory planning and establishes the interpolated trajectory of fifth-order polynomial joint variables and time to obtain the time-optimal trajectory. Shen et al. [13] use five-degree nonuniform B-spline interpolation in joint space to ensure the continuity of the pulsation curve. It can be seen from the above that the polynomial B-spline curve can be processed in sections and has the characteristics of local support, convex hull, and derivative continuity [14]. It is widely used and its effect is obviously better than other interpolation methods.
In the research of trajectory optimization [15, 16], researchers mainly focus on using various algorithms to optimize performance indicators such as time, energy consumption, and flatness [17–19]. Most of them use genetic algorithm (GA), ant colony algorithm (ACO), particle swarm algorithm (PSO) [20–22], etc., among intelligent algorithms in the group. Yang et al. [23] build a system dynamics model based on the Lagrangian principle and uses the linear iteration method (ILP) to plan the optimal trajectory of the movement time. Peng et al. [24] used the nondominated neighborhood immune genetic algorithm to optimize the multiobjective trajectory function of the robot and obtained the optimal position, velocity, acceleration, and acceleration planning curve of each joint. Ma et al. [25] use genetic algorithm to construct quintic B-spline interpolation trajectory. With the goal of time optimization, the simulation results show that the operating efficiency of Robai Cyton Gamma 300 robot is significantly improved. Baghli et al. [26] use ant colony algorithm as an optimization tool to solve the optimal path of five rectangular obstacles from the initial position to the final position, which reflects the robustness of ant colony algorithm to search and optimize problems. In domestic and foreign research, the research on particle swarm optimization algorithm is obviously less than the research on genetic and ant colony optimization algorithm. However, because the particle swarm optimization algorithm is simple and effective, it is not only suitable for scientific research but also particularly suitable for engineering applications [27], and it has gradually attracted the attention of many scholars. The study of Feng et al. [28] is based on particle swarm algorithms and uses high-order polynomial interpolation to fit the motion trajectory of the robot joint space. Liu et al. [29] established a closed-chain vector kinematics model, constructed the trajectory by using the 5th B-spline interpolation method of nonshaft drive space, and optimized the trajectory by using an improved particle swarm optimization algorithm to obtain the time-optimal motion trajectory curve. Jin and Geng [30] proposed a Cartesian trajectory planning method based on combined functions and transformed the trajectory planning problem into a multiobjective optimization problem, which was solved by the chaotic particle swarm optimization (CPSO) algorithm of mixed integer programming.
In this paper, the goal is to complete the task of automatic assembly of brake disc shaft hole. Assembly technology [31] has always been the weak link in modern production. Realizing automatic assembly can effectively improve production efficiency and product quality and is a symbol of realizing national industrial automation. Therefore, in order to complete the task of brake disc shaft hole assembly more accurately, it is of great research significance to plan the assembly trajectory. But the above research is aimed at the planning and optimization of robot trajectory with the same starting point and end point, without considering the cumulative error of the structure caused by the robot’s long-term work. For the task of assembling the axle hole of the automobile brake disc, the centering requirement of the axle and the hole is high. This paper proposes a trajectory planning method that considers shaft hole alignment compensation and improves shaft hole alignment accuracy through secondary trajectory planning. The second section establishes the robot kinematics model, the third section plans the trajectory of the brake disc assembly path and uses the particle swarm algorithm to optimize the trajectory, the fourth section uses MATLAB software for simulation, the fifth section conduct algorithm verification experiments, and finally the sixth section concludes.
2. Kinematics Modeling
2.1. Improved D-H Method Modeling
The D-H method describes the position and angle relationship between the adjacent links of the robot through four parameters: link length, link angle, link offset, and joint angle. The traditional D-H method has computational advantages in dealing with tandem structure robots, but there is ambiguity in dealing with robots with tree structure or closed-loop structures. In this paper, the improved D-H method is used for kinematic modeling, and a unified definition is used to deal with robots with series structure, tree structure, and closed-loop structure.
The coordinate system is established sequentially at the head end of the connecting rod ( = 1∼6). Since the motion axes of the connecting rod 4 and the connecting rod 5 intersect at one point, the origin of the coordinate system corresponding to these three connecting rods can be set at the intersection point, and the coordinate system shown in Figure 1 can be established for the robot based on the above settings.
The transformation of the coordinate system {} relative to the coordinate system { − 1} can be obtained through four D-H parameters. For a given robot, usually only one parameter changes during the transformation process, and the other three parameters are determined by the dimensions of the robot body. From this, it can be concluded that the D-H parameters of the robot are shown in Table 1.
2.2. Kinematics Equation Solving
Based on the aforementioned spatial coordinate system and D-H parameter table, the kinematic characteristics of the robot can be analyzed. Kinematic analysis is used to study the relationship between the joint variable space and the position and posture of the end effector. The robot is analyzed through the solutions of positive and inverse, respectively.
2.2.1. Solving Positive Kinematics
The purpose of positive kinematics of the robot is to obtain the position and posture of the end of the robot according to the motion of each joint. Based on the coordinate system established on each connecting rod, the positive kinematics problem can be divided into multiple subproblems to solve. The transformation of the vector defined in the coordinate system {} to the vector in the coordinate system { − 1} can be obtained by the following transformation:
Which is
Among them:
In the above formula, is the conversion matrix between the coordinate system {} and the coordinate system { − 1}, which can be obtained by the following formula:
Among them, represents the homogeneous transformation matrix obtained by rotating α around the Q axis, and represents the homogeneous transformation matrix obtained by translating d along the Q axis. The general expression of can be obtained from the above formula:
Substituting the data in Table 1 into the formula, you can get the conversion matrix between the coordinate systems:
The positive kinematic equation of the robot obtained by multiplying the above transformation matrix expressions is
2.2.2. Solving Inverse Kinematics
The purpose of inverse kinematics is to know the pose coordinates of the end of the robot in Cartesian space, and based on the results of the positive solution of the robot kinematics, the angle of each joint of the robot is calculated by geometric methods, numerical methods, analytical methods, and other methods. In this paper, the Pieper method is used to solve the inverse kinematic equations. From equation (7), we can get:
Which is
Calculated by the above formula:
From equation (10), the value of can be obtained:
The same can be obtained:
Among them:
In the identical transformation of , we can get:
By solving the trigonometric function relationship, we can get:
3. Trajectory Planning
In order to obtain a robot trajectory with optimal time and stable operation, an optimal trajectory planning based on improved particle swarm optimization algorithm is proposed, and the flow chart is shown in Figure 2. The first step is to carry out trajectory planning in the joint space. First, we select several important trajectory nodes and use the quintic B-spline interpolation method to obtain the coordinates of the intermediate points of the nodes. Then, the position and angle of each robot joint are calculated from the coordinates of these points through the robot inverse kinematics algorithm. Finally, a point on the required trajectory is obtained by the angular position closed-loop control system, and the interpolation is continued and the above process is repeated, so as to obtain a trajectory that meets the requirements. The second step is the establishment of optimization objective function. In order to get the trajectory with optimal time and stable operation, an optimization function with time and stationary performance as the optimization objectives is defined. The last step is the trajectory optimization based on the improved particle swarm optimization algorithm, which is used to optimize trajectory and get the optimal trajectory with optimal time and stable operation.
3.1. Trajectory Planning Based on Quintic B-Spline Interpolation
The process of manipulator trajectory planning first is to establish a preliminary trajectory in the joint space according to the operational task of the end effector of the robotic arm, then select multiple path points between the start position and the end position of the trajectory, and interpolate the obtained path points. Finally, the trajectory is described as a function of the joint angle. This paper takes the IRB1410 six-degree-of-freedom industrial robot as the research object to carry out the trajectory planning and optimization of the assembly process of the axle hole of the automobile brake disc. Each joint of the robot is a rotary joint.
In the process of trajectory planning, the given path is discretized, and m points are selected as the path points of the manipulator trajectory. Through inverse kinematics solution, the TCP spatial position sequence and the corresponding time node sequence ti are obtained and then a set of spatial position-time sequences are obtained:
Based on the continuity requirements of the trajectory, this paper uses the quintic B-spline to interpolate the joint path points. All the B-spline trajectory curves can be described as
In the formula, is the control vertex of the joint; t is the time to pass the control vertex; is the th gauge B-spline basis function; and the formula can be obtained through the De Boor recurrence relation:
In the formula, k is the number of B-spline; i represents the th segment of B-spline curve; and the condition of 0/0 = 0 must be satisfied. It can be seen from equation (18) that is not equal to 0 only in the support interval , and the others are all 0, so the th B-spline trajectory curve can also be expressed for:
Equation (19) shows that the th B-spline function is only related to − +1 control vertices in the support interval . In order to further analyze the velocity, acceleration, and jerk trajectory curves of the trajectory, it is only necessary to derive the both sides of equation (19) separately and then obtain the l-order derivative of the th B-spline joint trajectory:
Derived from the De Boor formula:
From equations (20) and (21), it can be concluded that the nodal velocity trajectory curve , the acceleration trajectory curve , and the pulse trajectory curve are
In this paper, we need to obtain the trajectory of the brake disc shaft hole assembly, which requires smooth motion, high precision, and continuous pulsation. Therefore, the continuous quintic B-spline interpolation method of is used to obtain the continuous trajectory of . According to the properties of the B-spline curve, it can be obtained that control vertices need to be solved, and the equations can be listed by known path points:
It is obviously not enough to solve control vertices with equations, so 4 additional boundary conditions need to be added: set the initial point velocity and acceleration as and , respectively, and the end point velocity and acceleration as and , respectively. Combine formula (22) to get:
Combining equations equations (17)–(24), the matrix equation trajectory curve of the th space joint can be obtained asIn the formula, = 1, …, 6; the coefficient of the matrix equation ; and the control vertex .
The control vertex vector can be obtained by the above formula and then the motion trajectory of each joint and the expression of the speed, acceleration, and jerk of each joint can be obtained according to formulas (17) and (22).
3.2. Establish Optimization Objective Function and Constraint Conditions
The assembly movement of the shaft hole of the brake disc requires relatively high accuracy, so it is generally hoped that the robot has the highest working efficiency and the most stable running track. In order to achieve the comprehensive optimization of time and stationarity and considering the constraint conditions of position, velocity, and acceleration in kinematics, the following optimization objective functions are defined:In the formula, is the time required for the robot to complete the specified task trajectory from the initial position to the end position; are the duration, start time, and end time of the motion track of segment , respectively; is the motion acceleration of the th joint of the manipulator; aAmong the above two optimization objectives, is the motion time, which is the sum of the motion time between each type value point adjacent to the joint trajectory of the manipulator, which is used to measure the motion efficiency of the manipulator; is the average acceleration of the joint, which is used to measure the smoothness performance index of the trajectory.
In the process of trajectory planning, the joint displacement, velocity, and acceleration constraints of the manipulator are considered as the constraints of trajectory planning:In the formula, are the displacement, velocity, and acceleration of the segment trajectory of the joint of the manipulator and represent the maximum displacement, maximum velocity, and maximum acceleration of joint, respectively.
3.3. Trajectory Optimization Based on Improved Particle Swarm Algorithm
Particle Swarm Optimization (PSO) algorithm is an evolutionary computing technology based on iteration. The system is initialized as a set of random solutions, and the optimal value is searched through iteration. Suppose the position of the -th particle in the -dimensional search space is denoted as , and the velocity is denoted as . In each iteration, the particle updates its position and velocity by tracking the two optimal solutions. One is the optimal solution of the individual particle in the optimization process, that is, the individual extreme value , and the other optimal solution is the optimal solution currently found by the entire particle swarm, that is, the global optimal solution . After determining these two optimal solutions, use the following formula to update your own speed and position.In the formula, is the inertia weighting factor, , are acceleration factors, , are any numbers within the range of (0, 1), is the individual optimal solution of the th individual, and is the optimal solution of the particle swarm group.
The traditional particle swarm algorithm lacks dynamic adjustment of speed and is easy to fall into the local optimum, resulting in low convergence accuracy and difficult convergence. In order to solve this problem, this paper adopts an improved particle swarm optimization algorithm with a constraint factor to optimize the trajectory. The factor can control the weight of the speed in the optimization process, which makes the convergence of the algorithm easier and enhances the performance of the algorithm. The formula for calculating is as follows:
The constraint factor method controls the final convergence of the system behavior and can effectively search different regions. This method can obtain high-quality solutions. The value of is between (0, 1), and the particle swarm size = 30, = 2.7, = 1.4. At this time, , . The speed update equation of this method is
According to Figure 2, it can be seen that the basic process of the improved particle swarm optimization algorithm for the optimization of the brake disc axle hole assembly trajectory: first, set the particle swarm size , the number of iterations , and initialize the individual optimal value of each particle, the global optimal solution , the initial velocity and the initial position. Then calculate the fitness value of each particle’s initial position according to the optimization objective function, construct the hierarchical structure of the population, and update the individual optimal value and the global optimal solution according to the fitness, and then update the particle position and velocity in the optimization process. If the maximum number of iterations is reached or the minimum limit of the global optimal position is met, the optimization is ended, and the final desired trajectory, as well as the velocity curve, acceleration curve, and pulsation curve are obtained. Otherwise, return to the step of calculating fitness and continue to run the process.
4. Simulation and Analysis of Trajectory Planning
First, the quintic B-spline interpolation method is introduced to construct the first trajectory plan before error compensation. At this time, there is a certain centering position error between the hole of the brake disc and the shaft. In order to solve this problem, the vision system is used to locate the center point of the axis, and the compensation planning under the second stage of accumulated error is carried out. And the particle swarm optimization algorithm is used to iteratively seek to optimize the two trajectories. The two optimized trajectories are synthesized to obtain the final alignment trajectory with optimal time and smooth operation, which realizes the precise assembly of the brake disc shaft hole.
In order to verify the effectiveness of the trajectory planning scheme of the quintic B-spline interpolation curve based on PSO, MATLAB software is used to establish the IRB1410 robot kinematics model, as shown in Figure 3, and the robot trajectory planning method was simulated and verified. It is expected to get a track with short running time and stable operation.
The trajectory planning based on the particle swarm optimization algorithm needs to set some relevant parameters. The particle swarm size is 30, the acceleration factors , , the constraint factor and the angular velocity of the initial point and the end point, the sum angular acceleration is 0, and the maximum number of iterations is 100. After MATLAB simulation, the optimal global fitness curve can be obtained, as shown in Figure 4.
The population size and the number of iterations should not be too large or too small. These two values affect the search ability and calculation amount of the algorithm. If the population size is too small, the algorithm converges quickly, but it is easy to fall into local optimum, which makes it difficult to iterate to the optimal fitness, while increasing the running time, the increase in the number of evolution will reduce the diversity of the population, so it is not easy to be too large or too small. Therefore, this experiment takes 100 iterations and the population size is 30. It can be seen from the figure that the fitness value optimization process takes 22 iterations to obtain the optimal fitness degree of 9.7277.
Through simulation, the change curve of each joint position can be obtained, as shown in Figures 5 and 6.
From Figures 5 and 6, it can be seen that the trajectory motion time before compensation is reduced from 6 s to 4.8 s after being optimized by improved particle swarm optimization algorithm, and the trajectory motion time after compensation is reduced from 2.6 s to 1.9 s after being optimized by improved particle swarm optimization algorithm. It can be seen that the improved PSO algorithm has achieved obvious results in time optimization.
It can also be seen from Figures 5 and 6 that the maximum joint change angle of joint 2 is reduced from 19.2° to 18.1° after optimization, the maximum joint angle of joint 2 is reduced from 58.2° to 53.7° after optimization, the maximum joint angle of joint 6 is reduced from 161.2° to 159.4° after optimization, and the trajectory curve of each joint obtained after optimization is smoother, continuous, and no sudden change compared with the nonoptimized curve, which makes the trajectory running more stable and reduces the possibility of running shock.
In order to further test the effect of the improved strategy in this paper and verify the superiority of the algorithm in this paper, the improved particle swarm optimization algorithm adopted in this paper is compared with the traditional particle swarm optimization algorithm by simulation:
Figures 7–9 are the comparative diagrams of movement position change, angular velocity change, and angular acceleration change of each joint of the robot obtained by using the improved particle swarm optimization algorithm and the traditional particle swarm optimization algorithm, respectively. The solid line in the figure is the optimization result of the improved particle swarm optimization algorithm, and the dotted line is the optimization result of the traditional particle swarm optimization algorithm.
It can be seen that the curves of displacement, velocity, and acceleration are smooth and continuous, and the variation range of each curve is within the kinematic constraint, which meets the requirements of work and design. It can be seen from the figure that the overall running time has not changed, indicating that the improved particle swarm optimization algorithm is not superior to the traditional particle swarm optimization algorithm in time optimization.
However, in Figures 8 and 9, the dotted lines are the velocity and acceleration curves optimized by the traditional particle swarm optimization algorithm. The velocity variation amplitudes of each joint are 35°/s, 40°/s, 67°/s, 101°/s, 78°/s, and 213°/s, XD, respectively, and the acceleration variation amplitudes are 94°/s2, 112°/s2, 232°/s2, 124°/s2, 159°/s2, and 384°/s2, respectively; the solid line is the velocity and acceleration curve optimized by the improved particle swarm optimization algorithm. The velocity variation amplitude of each joint is 23°/s, 26°/s, 41°/s, 72°/s, 57°/s, and 155°/s, respectively, and the acceleration variation amplitude is 66°/s2, 71°/s2, 124°/s2, 98°/s2, 136°/s2, and 271°/s2, respectively. It can be clearly seen that the trajectory optimized by the improved particle swarm optimization algorithm is significantly smaller than that of the traditional particle swarm optimization algorithm in terms of speed and acceleration variation amplitude. In addition, it can be clearly seen from the figure that the trajectory optimized by the improved particle swarm optimization algorithm is smoother and has no obvious mutation than the trajectory optimized by the traditional particle swarm optimization algorithm. To sum up, it shows that the improved particle swarm optimization algorithm has a high degree of optimization ability in displacement, velocity, and acceleration under the same movement time and can effectively reduce the mechanical wear and excessive vibration of the robot. It is proved that compared with the traditional particle swarm optimization algorithm, the improved particle swarm optimization algorithm has obvious optimization effect in terms of running stability.
In summary, the quintic B-spline trajectory planning based on PSO algorithm can get the motion trajectory with short running time, stable running, and no mutation, which ensures the high working efficiency and stability of the robot. The validity of the trajectory planning of quintic B-spline interpolation based on PSO algorithm is verified.
5. Algorithm Verification Experiment
5.1. Prototype Development
The experimental IRB1410 robot and related equipment are shown in Figure 10. The parameters of the IRB1410 industrial robot are as follows: load capacity: 5 kg; TCP maximum speed: 2.1 m/s; power supply rated voltage: 200–600 V, 50/60 Hz; robot weight: 225 kg; operating radius: 1440 mm; and positioning accuracy: 0.05 mm. The motion range of each axis is shown in Table 2.
5.2. Experimental Results and Analysis
First, let us set the robot TCP starting point , grab the middle point of the brake disc , and the target end point of the brake disc assembly , the actual position of each point is shown in Figure 11.
In order to observe the effect of trajectory planning, conventional PID control is used in this experiment. Based on the expected trajectory in the simulation part of chapter 4, trajectory tracking control is performed on the trajectory of each driving node to obtain the position of each point on the desired trajectory. And we use the teach pendant to program the trajectory, get the change diagram of each joint position in the actual trajectory process, and compare it with the change diagram of each joint position in the expected trajectory. The comparison diagram is shown in Figures 12 and 13.
As can be seen from Figures 12 and 13, since the simulation model is the motion trajectory of the robot in an ideal state, and there is an error between the motion trajectory of the robot in the actual working environment and the motion trajectory of the simulation model, so there is a certain deviation between the two lines in the figure. But the absolute error between the actual trajectory and the simulated trajectory is less than 6%, which shows that the improved particle swarm optimization algorithm based on the quintic B-spline interpolation curve can obtain the trajectory with the optimal time and running smoothly without sudden change, which verifies the correctness of the algorithm in this paper.
6. Conclusions
(1)In this paper, an error compensation trajectory planning method is proposed. First, the first stage of trajectory planning before error compensation is constructed. At this time, there is a certain centering position error between the hole and the shaft of the brake disc. Then, the center point of the axis is located by the vision system, and then the compensation planning under the second stage of accumulated error is carried out. The quadratic trajectory planning method is used to eliminate errors and improves the positioning accuracy of the robot.(2)The kinematics model of IRB1410 robot was established by the improved D-H parameter method, and its positive and inverse kinematics analytical solutions were obtained. The trajectory planning method of the quintic B-spline interpolation method is proposed, and the continuous characteristic of is used to deduce the trajectory planning formula of the driving node of the manipulator, which converts the kinematics constraints of the manipulator into the control vertex constraints of the trajectory curve, and improves the versatility of the manipulator trajectory planning method.(3)The quintic B-spline interpolation trajectory is optimized based on the improved particle swarm optimization algorithm. The simulation results show that the proposed trajectory planning method reduces the trajectory running time from 8.6 s before optimization to 6.7 s after optimization, which reduces the running time of the robot and improves the working efficiency of the robot. In addition, the maximum joint change angle of each joint has been reduced, and the trajectory curve of each joint is smoother and continuous without sudden change, which improves the stability of the trajectory operation.(4)The algorithm verification experiment is carried out, and the absolute error between the actual trajectory and the simulated trajectory is less than 6%, which improves the work efficiency and stability of the assembly task of the automobile brake disc shaft hole.Abbreviations
| : | Connect rod’s rotation angle (°) |
| : | Connecting rod length (mm) |
| : | Connecting rod offset length (mm) |
| : | Joint angle (°) |
| : | TCP spatial location sequence |
| : | TCP time node sequence |
| : | The control vertex of the joint |
| : | Time spent by controlling vertices (s) |
| : | The number of B-splines |
| : | Trajectory equation of nodal velocity |
| : | Trajectory equation of nodal acceleration |
| : | Trajectory equation of pulse |
| : | Velocity of initial point (m/s) |
| : | Acceleration of initial point (m/s2) |
| : | Velocity at the termination point (m/s) |
| : | Acceleration at termination point (m/s2) |
| : | Coefficient of matrix equation |
| : | Track running time (s) |
| : | Index of trajectory stationarity |
| : | Time for robot to complete task trajectory (s) |
| : | Individual optimal solution of particle swarm optimization |
| : | Group optimal solution of particle swarm optimization |
| : | Inertia weight factor |
| : | The size of particle swarm |
| : | Acceleration factor |
| : | Constraint factor. |
Data Availability
The data underlying the results presented in the study are available within the manuscript.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
Thanks are due to Dr. Ni for assistance with the experiments and to Dr. Zhao for valuable discussion. This work was supported by National Natural Science Foundation of China (Grant no. 51405419), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant no. 18KJB460029), and Yancheng Institute of Technology Training Program of Innovation and Entrepreneurship for Undergraduates (Grant no. 2020105).