2. Kinematic modeling and analysis

In this paper, we use the six-degree-of-freedom robot PUMA 560 as the research subject and establish the link coordinate system according to the MDH method. Figure 1 and Table I demonstrate the obtained DH parameters of the robot.

Table I. Robot DH parameter table.

Figure 1. The linkage coordinate system.

Then, establish the transformation relationship between axes by rotating the x axis, translating the x axis, and then rotating the z axis and translating the z axis:

(1) \begin{equation} {}^{i-1}{\boldsymbol{T}}{_{i}^{}}=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \cos \theta _{i} & {-}\sin \theta _{i} & 0 & a_{i}\\[5pt] \sin \theta _{i}\cos \alpha _{i-1} & \cos \theta _{i}\cos \alpha _{i-1} & {-}\sin \alpha _{i-1} & -d_{i}\sin \alpha _{i-1}\\[5pt] \sin \theta _{i}\sin \alpha _{i-1} & \cos \theta _{i}\sin \alpha _{i-1} & \cos \alpha _{i-1} & d_{i}\cos \alpha _{i-1}\\[5pt] 0 & 0 & 0 & 1 \end{array}\right) \end{equation}

Then, the transformation relationship between the robot end and the base is established according to the formula (1):

(2) \begin{equation} {}^{0}{\boldsymbol{T}}{_{6}^{}}={}^{0}{\boldsymbol{T}}{_{1}^{}}{}^{1}{\boldsymbol{T}}{_{2}^{}}{}^{2}{\boldsymbol{T}}{_{3}^{}}{}^{3}{\boldsymbol{T}}{_{4}^{}}{}^{4}{\boldsymbol{T}}{_{5}^{}}{}^{5}{\boldsymbol{T}}{_{6}^{}}=\left(\begin{array}{c@{\quad}c} \boldsymbol{R} & \boldsymbol{P}\\[5pt] 000 & 1 \end{array}\right) \end{equation}

In the equation, $\boldsymbol{R}$ is a 3 × 3 matrix representing the orientation of the robot, and $\boldsymbol{P}$ is a 3 × 1 vector representing the position of the robot.

The scenario of robot assembly operation can be simplified as shown in Fig. 2. Assuming that the end of the robot and the installation axis are regarded as one, it is necessary to ensure that the position and direction of the end and the hole overlap highly, so as to avoid collision and affect the quality of the workpiece.

Figure 2. Assembly operation scenario.

Corresponding the specific information of the hole $(x,y,z,\alpha,\beta,\gamma )$ to the end attitude of the robot, the assembly problem can be transformed into the inverse solution problem of the robot.

(3) \begin{equation} f^{-1}\left(\begin{array}{c@{\quad}c} \boldsymbol{R} & \boldsymbol{P}\\[5pt] 000 & 1 \end{array}\right)=\boldsymbol{q}\left(\theta _{1},\theta _{2},\theta _{3},\theta _{4},\theta _{5},\theta _{6}\right) \end{equation}

According to literature [Reference Liu, Xiao, Tong, Tao, Xu, Jiang, Chen, Cao and Sun17], the inverse solution exists only when the robot satisfies the Piper principle. In order to enable the general machine to find the inverse solution of the robot, the optimization algorithm is used to solve the inverse solution.

3. Fitness function design

The mounting hole’s positional data are transformed into the robot’s base coordinate system using MATLAB’s transl and rpy2tr functions, which are part of Professor Peter Corke’s robotic toolbox. By applying these functions to combine the position and rotation angles, we can obtain the desired pose matrix $\boldsymbol{T}_{g}$ . Among them, the transl function transforms the X, Y and Z information of holes in Cartesian space into a homogeneous transformation matrix. The rpy2tr function is usually used to convert Euler angles (pitch angle, roll angle and yaw angle) into elements in the rotation matrix or the direction cosine matrix in the homogeneous transformation matrix. The jtraj function generates a set of joint angles according to the initial joint position and the target joint position to ensure that the trajectory is smooth. Please note that Professor Peter Corke’s robotic toolbox is not a standard component of MATLAB and needs to be separately installed, as mentioned earlier.

In order to realize the shaft hole assembly, the current pose of the robot $\boldsymbol{T}_{p}$ is required not to exceed the limit of the joint Angle, which is simplified to a mathematical model:

(4) \begin{equation} f_{1}=\left\{\begin{array}{l} \boldsymbol{q}_{\min }\leq \boldsymbol{q}\leq \boldsymbol{q}_{\max }\\[11pt] \left\| \boldsymbol{T}_{g}-\boldsymbol{T}_{p}\right\| =0 \end{array}\right. \end{equation}

When the optimization algorithm is used to solve the inverse, the formula (4) can be regarded as the objective function and the minimum value. Since the last row of the target matrix $\boldsymbol{T}_{g}$ remains constant, the objective function can be organized as:

(5) \begin{equation} f_{1}=\left\{\begin{array}{l} \min \left(\left\| \boldsymbol{R}_{g}-\boldsymbol{R}_{p}\right\| +\left\| \boldsymbol{P}_{g}-\boldsymbol{P}_{p}\right\| \right)\\[5pt] \boldsymbol{q}_{\min }\leq \boldsymbol{q}\leq \boldsymbol{q}_{\max } \end{array}\right. \end{equation}

At the same time, in the assembly operation, it is necessary to ensure that the robot joint Angle change is gentle, and there is no joint direction change phenomenon, which affects the quality of the workpiece. Therefore, it is also necessary to introduce the joint angle variation as a fitness function:

(6) \begin{equation} f_{2}=\sum _{i=1}^{6}\left(\left\| \theta _{i}\left(t\right)-\theta _{i}\left(t-1\right)\right\| \right) \end{equation}

Among them, $f_{2}$ represents the sum of joint angle changes.

In conclusion, we design the fitness function as follows when utilizing the intelligent optimization algorithm to solve the kinematic inverse solution of the assembly robot:

(7) \begin{equation} f=f_{1}+f_{2} \end{equation}

4. PSO algorithm and its improvement

PSO is a stochastic optimization algorithm based on population. Since its proposal, it has undergone numerous formal changes and has been applied in various target optimization scenarios [Reference Wang, Tan and Liu18]. The principle behind PSO lies in the interaction of information among particles within the population, enhancing their ability to search for optimal solutions. As depicted in Fig. 3, each particle possesses its own position and speed, influenced by both the group and individual interactions, which collectively guide their search toward the optimal location.

Figure 3. The principle of PSO.

You can organize this into a mathematical model:

(8) \begin{equation} \left\{\begin{array}{l} \boldsymbol{v}_{iD}^{t+1}=\omega^{\ast} \boldsymbol{v}_{iD}^{t}+c_{1}r_{1}\left(\boldsymbol{p}_{iD}^{t}-\boldsymbol{x}_{iD}^{t}\right)+c_{2}r_{2}\left(\boldsymbol{p}_{gD}^{t}-\boldsymbol{x}_{iD}^{t}\right)\\[9pt] \boldsymbol{x}_{iD}^{t+1}=\boldsymbol{x}_{iD}^{t}+\boldsymbol{v}_{iD}^{t+1} \end{array}\right. \end{equation}

Among them, $\omega$ is the inertia weight, $c_{1}$ and $c_{2}$ are the learning factors that determine the influence of the current velocity, the particle-to-particle interaction, and the swarm-to-swarm interaction, respectively, and $r_{1}$ and $r_{2}$ are random numbers between 0 and 1.

The larger the inertial weight value, the stronger the global search ability; otherwise, the search ability is weakened and easy to converge near the optimal solution. However, fixed inertial weights can result in poor search performance, leading to local optima and premature convergence phenomena. Therefore, several variants of the PSO algorithm are proposed, such as the adaptive PSO (APSO) and the linear weight particle swarm optimization (LWPSO).

Based on the basis of the weight strategy, the nonlinear weight PSO (NWPSO) is proposed, the model is (9):

(9) \begin{equation} \omega ^{\ast }=\left\{\begin{array}{l@{\quad}l} \omega _{z}-\dfrac{\omega _{z}-\omega _{\min }}{t_{z}}^{\ast}t & t\lt t_{z}\enspace \\[9pt] \omega _{\min }+\dfrac{\omega _{\max }-\omega _{\min }}{t_{\max }-t_{z}}^{\ast}\left(t-t_{z}\right) & t\geq t_{z} \end{array}\right. \end{equation}

Among them, $\omega _{z}$ is the intermediate value of the inertia weight, which is set based on empirical experience, and $t_{z}$ is the intermediate value of the iteration number. By introducing two parameters, the advantage of linear weight change can be maintained before and after the weight change. At the beginning of iteration, the weight quickly becomes smaller, the particle speed update is accelerated, and the convergence speed is improved. When the optimum is about to be reached, change the weight change mode to improve the convergence accuracy. The flow of the PSO algorithm for inverse kinematics is shown in Fig. 4.

However, when the particle is updated, the particle will exceed the joint angle limit of the robot, so the penalty function is introduced to eliminate the unqualified particles.

(10) \begin{equation} f=\text{fitness}+1000 \end{equation}

Among them, $\text{fitness}$ is the fitness value of the particle.

5. Simulation validation

In this paper, the improved PSO algorithm is applied to solve the inverse kinematics of the PUMA 560 robot, and the optimization results are compared with several other algorithms. The parameter settings for the algorithm are shown in Table II. PSO refers to the standard PSO algorithm, APSO refers to the adaptive PSO algorithm, Starling-PSO refers to the algorithm used in reference [Reference Liang, Li and Huang12], LWPSO refers to the linear weight PSO algorithm, NWPSO refers to the nonlinear weight PSO algorithm, MPPSO refers to the multi-population PSO algorithm, and GA refers to the genetic algorithm. The population size and iteration number for the genetic algorithm are the same as those for the PSO algorithm, with a crossover rate of 0.8 and a mutation rate of 0.05.

Table II. Algorithm parameters.

Figure 4. Flow chart of the PSO.

In Table II, the value of $\omega _{1}$ ranges from 0.2 to 1.0 in increments of 0.2. $t_{z}$ is a parameter introduced to improve the PSO, which modifies the way the weight changes near the iteration where the optimal solution is reached to avoid the algorithm being trapped in a local optimum state.

(11) \begin{equation} t_{z}=10+\text{round}\left(10^{\ast}\text{rand}\left(1,1\right)\right) \end{equation}

Among them, round is a circular function and rand is a random function.

Then, MATLAB R2021b is used to model the robot and multiple optimization algorithms on a Dell desktop processor configured as i5-10500, 3.10 GHZ and 16G memory. To verify the performance of the improved PSO, multiple sets of experiments are set in this paper.

5.1. Analysis of the algorithm performance at the random points

In the experiment, the position information of the installation hole is set as $[0.5,0.2,0.3,0,\pi /2,\pi /3]$ , which is transformed into a homogeneous transformation matrix, that is, the target matrix $\boldsymbol{T}_{g}$ :

(12) \begin{equation} \boldsymbol{T}_{g}=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0.9995 & -0.0137 & 0.0274 & 0.5000\\[5pt] 0.0137 & 0.9999 & 0.0004 & 0.2000\\[5pt] -0.0274 & 0 & 0.9996 & 0.3000\\[5pt] 0 & 0 & 0 & 1.0000 \end{array}\right) \end{equation}

Afterward, the number of particles was set to 150, with 100 iterations, and the remaining parameters are detailed in Table III. Utilizing $f_{1}$ as the fitness function, each algorithm was executed 10 times in MATLAB, capturing both the optimal and mean values of the results. As shown in Table IV, the NWPSO algorithm proposed in this paper proves to be feasible for solving the inverse robot solution. While its optimal optimization effect may not be as impressive as the APSO algorithm, in comparison to other algorithms, it significantly reduces the robot’s attitude error. Furthermore, the algorithm exhibits the smallest average and variance values, increasing by two orders of magnitude. This demonstrates that the NWPSO algorithm is well-suited for solving inverse problems, providing the best optimal stability while maintaining solution accuracy.

Table III. The fitness function values corresponding to the algorithm.

Figure 5. Average fitness function curve.

Combined with Figs. 5 and 6, the NWPSO algorithm obviously accelerates the convergence rate of the algorithm, and the average fitness function of the algorithm is the smallest, that is, the error value of the robot from the target attitude.

Table IV. Optimization effects of the algorithm at the singularity.

Figure 6. The mean fitness function curves at the singular points.

5.2. Algorithm analysis at the singularity points

To further validate the optimization effectiveness of the algorithm, a specific target point in an unconventional position is selected. The algorithm parameters and fitness function remain unchanged. Initially, all the joint points of the robot are obtained through traversal, and then, the corresponding Jacobian matrix is derived. Subsequently, the Jacobian matrix’s full-rank condition is checked to identify singular position points. Using the target pose matrix with positive kinematics: $[\pi /4,\pi /6,\pi /3,\pi /5,$ $0,\pi /2]$ :

(13) \begin{equation} \boldsymbol{T}_{g}=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0.5721 & -0.4156 & -0.7071 & -0.1473\\[5pt] -0.5721 & 0.4156 & 0.7071 & 0.0636\\[5pt] 0.5878 & 0.8090 & 0 & -0.2362\\[5pt] 0 & 0 & 0 & 1.0000 \end{array}\right) \end{equation}

Then, the algorithm parameters and other information are set to be consistent with 5.1, and the optimal value and average value of the running results are also taken.

The NWPSO algorithm reduces the error at the singularity to 0.0046 m, which is less effective than the APSO and LWPSO algorithms. However, the overall effect of NWPSO algorithm is the same as that of the general point. The algorithm converges in approximately 45 generations, making it the fastest. Meanwhile, the optimization result of the algorithm has the least volatility, and the average result can constrain the attitude error to 0.0171 m.

5.3. Analysis of the algorithm performance in the assembly process

In the assembly process, not only the end attitude of the robot is required to coincide with the installation hole but also the joint Angle is constrained from large fluctuations. Therefore, the above universal point as the termination point introduces the amount of position change to simulate the actual position of the hole and set the starting point of the hole.

(14) \begin{equation} \left\{\begin{array}{l} x_{\text{start}}=x_{\mathrm{end}}+\Delta x\\[5pt] y_{\text{start}}=y_{\mathrm{end}}+\Delta y\\[5pt] z_{\text{start}}=z_{\mathrm{end}}+\Delta z \end{array}\right. \end{equation}

In the equation: $\Delta x$ : 0.1 m; $\Delta y$ : 0.05 m; $\Delta z$ : 0.1 m.

Figure 7. Comparison of iteration numbers graph.

Figure 8. Fitness function.

Selected $f$ as the objective function, the number of iterations was set to 500 and the population individuals to 150, and 10 intermediate points were set using the jtraj function of MATLAB to minimize the joint angles between continuities.

From Fig. 7, it can be seen that the NWPSO algorithm proposed in the paper has the fastest convergence speed, all converging around 50 iterations. Compared with other algorithms, it significantly improves the convergence speed of the algorithm. Combined with Fig. 8, it can be concluded that the convergence accuracy of the NWPSO algorithm has not decreased, and its fitness function value is close to that of the APSO algorithm and the LWPSO algorithm, remaining around 0.19. At this point, the fitness function value includes attitude error and joint angle change, and the smaller the value, the better the assembly accuracy of the robot.

5.4. Algorithm performance analysis of redundant robotic arm

To further verify the generality of the proposed algorithm, we used a seven-degrees-of-freedom redundant robotic arm for model validation [Reference Alebooyeh and Urbanic19]. The algorithm parameters remained consistent with experiment 5.1, changing the number of iterations to 500. We used $f_{1}$ as a function of fitness, and the experimental results are shown in Table V and Fig. 9.

Table V. Algorithm optimization effects of redundant robotic arms.

Figure 9. Average fitness function curves of the algorithm on the redundant robotic arms.

From Table V, it can be seen that the NWPSO algorithm has the smallest fitness value and achieves a convergence accuracy of m. At this point, the posture of the robot is closer to the target posture, and the error is minimized. This indicates that the algorithm has a good inverse kinematics optimization effect on redundant manipulators and verifies the universality of the proposed algorithm. However, the average and variance values of the NWPSO algorithm are only slightly larger than those of the LWPSO algorithm, with a difference of about 0.004. This indicates that the optimization effect and data fluctuation of the algorithm are still relatively small. Combined with Fig. 9, it can be seen that the NWPSO algorithm converges around 50 iterations, while the LWPSO algorithm converges around 400 iterations, indicating a slower convergence speed. The other algorithms do not perform as well as the NWPSO algorithm in terms of convergence speed and accuracy.

In order to verify the inverse solution performance of redundant manipulator during assembly, the algorithm parameters are consistent with experiment 5.3, the number of iterations is set to 500, the population is set to 150, and the $f$ function is used as the objective function. The experimental results are shown in Table VI and Figs. 10 and 11. Compared with other algorithms, NWPSO algorithm has better convergence speed and accuracy.

Table VI. Optimization effect of arithmetic of F function for redundant manipulators.

Figure 10. Comparison of iteration numbers graph.

Figure 11. Fitness function.

Finally, to further compare the performance of the algorithm, the optimization time of each algorithm is compared.

In Fig. 12, it is assumed that the running time of the PSO algorithm in the first experiment is denoted as T. In the first, second and fourth experiments, the running time of the APSO algorithm is approximately 2T, while other algorithms have a running time of about T. In the third and fifth experiments, the total time is divided by the number of interpolation points, which is regarded as the optimization time of a point. The running time of the APSO algorithm exceeds that of the PSO algorithm, whereas the GA algorithm’s running time is shorter than that of the PSO algorithm. The running time of other algorithms is comparable to that of the PSO algorithm. However, the accuracy of the inverse solution obtained by the GA algorithm is inferior to that of the NWPSO algorithm. Considering the optimization performance of the aforementioned algorithms, the NWPSO algorithm proposed in this paper outperforms other algorithms and is suitable for solving robot inverse problems. Importantly, it maintains convergence accuracy without increasing optimization time. Additionally, the algorithm is applicable for the inverse solution optimization of redundant manipulators.

Figure 12. Runtime comparison of the algorithm.