## 6 Simulation Based on Inverse Kinematics  
One of the tracks was randomly selected for the simulation experiment, and the three algorithms mentioned above were used in turn to set the evaluation criteria as the position allowable error and axis allowable error. When the closed-form solution was used, the [PERSON] rotation formula was utilized to obtain one of the possible poses of the target point, and \(T_{\text{robot}}\), which equals \(T\)-(\(T_{\text{tool}}\cdot T_{\text{dis}}\))\({}^{-1}\) is taken as the actual pose of the UR10 robot's end effector. Equation (33) describes the [PERSON] formula, Where n\({}_{0}\) equals [0,0,1]\({}^{T}\), and \(r\) is the vector product of n and n\({}_{0}\). \[R=E\text{cos}\theta+(1-\text{cos}\theta)r^{T}\cdot r+\text{sin}\theta\cdot \begin{bmatrix}0&-r_{z}&r_{y}\\ r_{z}&0&-r_{x}\\ -r_{y}&r_{x}&0\end{bmatrix}  \tag{33}\]  
In the BP-SQP and SQP algorithms, the permitted error value for the position is usually set to 0.01-0.03 mm, according to engineering experience, we take the maximum of 0.03 mm here. Likewise, for the axis set to 0.11\({}^{\circ}\). The selected track includes 500 points, and the selection step was set to 0.6 mm. Figure 7 is the trajectory diagram. The target trajectory was analyzed with the closed-form solution. The accuracy was 100%. The position and axis-based errors of all points were recorded. The result is shown in Figure 8. The observation showed that both errors were far lower than the allowable errors. Then, inverse kinematics obtained by this the closed-form solution can be regarded as the real inverse kinematic solution. The total time taken to find the solution was 78.8 s.  
Figure 8: Inverse kinematics via the closed-form solution. Figure 7: The trajectory diagram. The initial value of the SQP algorithm was set as the joint angle value of the previous point of the closed-form solution, and half of the position and axis based errors were introduced into Equation (2). The results are shown in Figure 9, according to which most of the points from the UR10 robot's kinematics via the SQP algorithm meet the kinematic accuracy requirements, and only two are outside the allowed range. The accuracy of the SQP algorithm is 99.5%. The total time taken for the SQP algorithm to find the solution was about 37 s, which is less than half of the time taken by the closed-form solution. When the BP neural network was utilized, the top 400 points which met the accuracy requirement in the SQP algorithm were extracted as the training set, and the remaining 98 points were extracted as the prediction set. In the P neural network of the simulation experiment, \(\lambda\) equaled 0.001, \(\lambda\)_inc equaled 10, and \(\lambda\)_dec equaled 0.1, which were the default initial values of the LM algorithm. The highest number of failed iterations was 10. The total time taken by the BP-SQP algorithm to resolve the UR10 robot's inverse kinematic was 13.1 s. The simulation results are shown in Figure 10, according to which the BP neural network during the training process divided tested samples into three sets, the training set, the verification set, which is the data set of the BP neural network's initial verification model, and testing set, which is the data set that tested the fitting result of the BP neural network. The accuracy of the BP neural network's testing set and verification set was 0.003. The accuracy of the BP neural network's training set was 0.002 and met the actual requirement. The data from the training set was input into the BP neural network. The output values obtained from the neural network were chosen as the initial value of SQP. Figure 11 shows the experimental results of the BP-SQP algorithm. The time taken by the BP-SQP algorithm to resolve the UR10 robot's inverse kinematics was only 4.2 s. The success rate of the simulation reached 100%. The results obtained from the three methods are shown in Table 1.  
Figure 10: Training results of the BP neural network. Figure 9: The inverse solution via the SQP algorithm.