## 7 Numerical Examples
Based on the Equations (34)-(37), the calculation results of reliability sensitivity are determined:  
\[\frac{\partial R}{\partial\mu^{T}}=(\frac{\partial R}{\partial\mu_{1}}\,\frac {\partial R}{\partial\mu_{2}}\,\frac{\partial R}{\partial\mu_{3}}\,\frac{ \partial R}{\partial\mu_{4}}\,\frac{\partial R}{\partial\mu_{5}}\,\frac{ \partial R}{\partial\mu_{6}}\,\frac{\partial R}{\partial\mu_{7}}\,\frac{ \partial R}{\partial\mu_{8}}\,\frac{\partial R}{\partial\mu_{9}}\,\frac{ \partial R}{\partial\mu_{10}}\,\frac{\partial R}{\partial\mu_{11}}\,\frac{ \partial R}{\partial\mu_{12}}\,)\]  
\(=(0.4114\ -\ 0.1889\ -\ 1.4998\ 0.7636\ 1.6337\ -\ 0.63636\ 0.0015\ 0.1467\)  
\(=-0.1288\ -\ 0.0492\ 0.01084\ 0.0796)  
\(\frac{\partial R}{\partial D^{T}}=(\frac{\partial R}{\partial D_{1}}\,\frac {\partial R}{\partial D_{2}}\,\frac{\partial R}{\partial D_{3}}\,\frac{ \partial R}{\partial D_{4}}\,\frac{\partial R}{\partial D_{5}}\,\frac{ \partial R}{\partial D_{6}}\,\frac{\partial R}{\partial D_{7}}\,\frac{ \partial R}{\partial D_{8}}\,\frac{\partial R}{\partial D_{9}}\,\frac{ \partial R}{\partial D_{10}}\,\frac{\partial R}{\partial D_{11}}\,\frac{ \partial R}{\partial D_{12}}\,)\)  
\(=(-6.0553\ -\ 6.9927\ -\ 4.5702\ -\ 58.7828\ -\ 107.8470\ -\ 21.2619\ -\ 12.0916\)  
\(-2.3411\ -\ 24.7880\ -\ 0.0350\ -\ 0.01708\ -\ 0.04111)  
It can be obtained from the sensitivity matrix \(\partial R/\partial\mu^{T}\) of kinematic reliability to the mean of the random parameter vector:  
1. Increasing the means of \(\Delta r_{22}\), \(\Delta r_{32}\), \(\Delta a_{3}\), \(\Delta\sigma_{3}\) and \(\Delta\theta_{11}\) will decrease the kinematic reliability of the mechanism, while the increase of the mean of the other errors will increase the kinematic reliability. The reliability of the improved Delta parallel mechanism with respect to the mean of the input errors is more sensitive to the changes in \(\mu_{\Delta a2}\), \(\mu_{\Delta a32}\), and \(\mu_{\Delta a1}\) than to the changes in \(\mu_{\Delta a3}\), with the mean sensitivities of the other errors being smaller. The mean sensitivity of the length error of the second driving arm \(|\partial R/\partial\mu_{\Delta a2}\,|=1.6337\) is the largest, meaning the reliability could be improved to some extent by controlling the mean of \(\Delta a_{2}\). 2. The increase of the mean of the input errors does not wholly decrease the kinematic reliability of the mechanism, which indicates that the output position errors of the end-effector of the improved Delta parallel mechanism are caused by the coupling  
Figure 7: Normal distribution test of \(g(x)\). errors of each branch, and even the influences of the same kind of errors on kinematic reliability are also different because of the arrangements of the three branches. It can be obtained from the variance sensitivity matrix \(\partial R/\partial D^{T}\) of kinematic reliability to the variances of the random parameter vector:  
1. The increase in the variances of all the random parameters will decrease the kinematic reliability of the parallel mechanism. Therefore, in order to increase kinematic reliability of the improved Delta parallel mechanism, some effective measures could be taken to minimize the variances of the input errors of all the design variables at a reasonable cost controlled during processing and manufacturing. 2. We can also learn that the variance reliability of kinematic accuracy of the improved Delta parallel mechanism is more sensitive to \(\Delta a_{2}\), \(\Delta a_{1}\), \(\Delta\varphi_{3}\) and \(\Delta a_{3}\) and followed by \(\Delta\varphi_{1}\), \(\Delta r_{22}\), \(\Delta r_{12}\), \(\Delta r_{32}\) and \(\Delta\varphi_{2}\). The variance sensitivity of the length errors of the second driving arm \(|\partial R/\partial D_{\Delta a_{2}}|=107.847\) is the largest, and the average of variance sensitivity of \(\Delta a_{i}\) (\(i=1,2,3\)) is equal to 62.6326 which is much larger than any other variance sensitivity. The increase in variances of driving arms have the most serious influence on the kinematic reliability.