# Inverse Kinematics of Large Hydraulic Manipulator Arm Based on ASWO Optimized BP Neural Network
## 5 Simulation and Experimental Validation
### Experimental Validation of Robotic Arm Inverse Solution  
In order to verify the validity of using actual positions as inputs to the BP neural network data set in this study, a comparison is made in Experiment 1, where theoretical positions are used as inputs to the data set, and the same algorithmic model as in this study is used for training. Fifty-five sets of data are randomly tested during the actual use of the robotic arm (the target positions are all within the operational space of the robotic arm), and the joint values and the actual positions of the robotic arm's end are recorded. The position errors of the two methods in the three XYZ directions under the same target positions are compared, and the experimental results are shown in Figure 11.  
Each vertical axis in Figure 11 represents the positioning error in the XYZ directions, while the horizontal axis represents the test points. Using the theoretical position as the data set input does not effectively compensate for the forward solution error. Additionally, the inaccuracy of the inverse solution model itself further increases the localization error, resulting in average localization errors of (104.48, 94.07, 128.47) mm in the XYZ directions (indicated by black dots). In this study, using the actual position as the data set input can compensate for the aforementioned issues and significantly reduce the positional error. This approach reduces the average positioning error in the XYZ directions to (18.16, 24.67, 27.21) mm (indicated by red pentagrams), respectively. The experimental results confirm the validity and effectiveness of the data set settings in this paper. Figure 11: Comparison of errors for the two positions as inputs to the data set. In order to avoid the influence of neural network model errors on the experiment, the Jacobi iterative method, which is commonly used in the industrial field, is employed in Experiment II to compare with the research method of this paper. The experimental steps are similar to those in Experiment I. The spatial position errors of the two methods in the XYZ directions under 55 sets of target positions are calculated, and the test results are shown in Figure 12.  
The Jacobi iterative method is more accurate and can make the theoretical position of the end reach the target position accurately, but it cannot compensate for the existing forward solution error, leading to a larger error in the final position. As can be seen from the green dots in the error distribution in Figure 12, these points are relatively far from the zero-error line. In contrast, the inverse solution error (indicated by the red pentagrams) obtained using the ASWO-optimized BP neural network algorithm is significantly lower, and its distribution is close to the zero-error line. Compared to the Jacobi iterative method, the maximum positioning errors in the XYZ directions are reduced from (181.26, 284.74, 305.81) mm to (42.59, 63.71, 67.08) mm, and the average positioning errors from (91.3, 87.38, 117.31) mm to (18.16, 24.67, 27.21) mm, thus greatly improving the positioning accuracy of the robotic arm by 80.11%, 71.78%, and 76.81% in the respective directions. As can be seen from Table 3, the inverse solution position errors reported in the literature [11; 19; 31] are significantly lower than those in this paper and in the literature [14; 28], primarily due to the smaller size of the electro-mechanical arm and its more stable operation. The average spatial position error of the method in this paper is 32.21 mm. Compared to the  
Figure 12: Comparison of end position error between ASWO-BP method and Jacobi iterative method. hydraulic robotic arm described in the literature [14; 28], the position error of the proposed method in this paper is larger, mainly because of the higher degree of freedom (DOF) and larger size of the robotic arm studied in this paper. An increase in the size of the robotic arm exponentially increases the position error of the end-effector.