## 4 Simulations and Experiments  
### Identification Simulations  
In order to verify the effectiveness of the proposed kinematic parameter identification method of the parallel robot, identification simulations of the parallel robot have been carried out. The original nominal kinematic parameters of the parallel robot are given in Table 1. According to the objective function \(L\) of the parallel robot, there are forty-two kinematic parameters to be identified in the identification simulations. Therefore, three actual kinematic parameter sets with the different normal distribution errors are listed in Tables 2-4. The five Gaussian measurement noises with variances of 0.000001 mm, 0.00001 mm, 0.0001 mm, 0.001 mm, and 0.01 mm were added to the actual poses of the parallel robots. Meanwhile, the influence of different numbers and configurations of the measurement poses on the identification accuracy was considered. The number of the measurement poses was set as 7, 14, 21, and 42, respectively. The configuration of the measurement poses was implemented by using the random selection method (Method I), the boundary exploration method [57] (Method II), and the optimal experiment design method [58] (Method III). Thus, the twelve pose sets were obtained and recorded in order, as shown in Table 5.  
The reduction in the identification error is not the ultimate goal of the kinematic parameter identification method of the parallel robot. The aim of the proposed kinematic parameter identification method is to enhance the positioning and orientation accuracy of the parallel robot. In order to describe the influence of the identification error on enhancing the positioning and orientation accuracy of the parallel robot, two accuracy indices (RMSPE and RMSOE) are introduced and expressed as follows:  
\[RMSPE =\sqrt{\frac{1}{n}\underset{i=1}{\overset{M}{\underset{i=1}{\sum}}} \left(\boldsymbol{e}_{i(1)}^{2}+\boldsymbol{e}_{i(2)}^{2}+\boldsymbol{e}_{i(3) }^{2}\right)} \tag{27}\] \[RMSOE =\sqrt{\frac{1}{n}\underset{i=1}{\overset{M}{\underset{i=1}{\sum}}} \left(\boldsymbol{e}_{i(4)}^{2}+\boldsymbol{e}_{i(5)}^{2}+\boldsymbol{e}_{i(6) }^{2}\right)}\]  
The relationship between positioning error reduction and identification error is shown in Figure 6, and the relationship between orientation error reduction and identification error is shown in Figure 7. Note that the positioning error of the parallel robot decreases as the percentage of identification error reduction increases. Similarly, the decrease of orientation error depends on the percentage of identification error reduction in a linear way. It is indicated that the proposed kinematic parameter identification method can improve the positioning and orientation accuracy of the parallel robot. Figure 5: Correlation between identification error and pose set for parameter set 3. Figure 6: Correlation between identification error and position error. The improvements of positioning and orientation accuracy affected by different measurement noise are shown in Figures 8-13. The positioning and orientation accuracy of the parallel robot can be improved with the reduction in measurement noise. Regardless of the magnitude of the initial kinematic parameter errors, the proposed kinematic parameter identification method can be perfectly performed while the measurement noise is small enough. However, the ultimate positioning and orientation accuracy of the parallel robot are not determined by the measurement noise alone. For different kinematic parameter sets, the curves in Figures 8-13 are significantly not identical. In addition, the larger the initial kinematic parameter errors of the parallel robot, the larger the identified kinematic parameter errors obtained by the proposed kinematic parameter identification method while the measurement noise is held constant.