## 5 Reliability Optimization Design Method Oriented to the Stability of Configuration Transformation
### Determination of Optimization Design Parameters Based on Reliability Sensitivity Analysis  
The discreteness of random variables of each structural parameter has different influence on the reliability of the configuration transformation of the metamorphic mechanism. If the variation in some structural parameter tolerance has a great influence on the reliability, it should be strictly controlled. Otherwise, the restrictions on them can be relaxed to reduce processing costs. Therefore, before the reliability optimization design of the configuration transformation ability of the metamorphic mechanism is carried out, the variance sensitivity of each structural parameter needs to be determined. Moreover, the random parameters whose tolerance values need to be strictly controlled should be determined, and the tolerance values should be selected as the optimization design variables.  
The variance sensitivity [51] of the configuration transformation reliability of the constrained metamorphic mechanism to the structural parameter vector \(X=[x_{1},x_{2},\cdots,x_{n}]^{T}\) is as follows:  
\[\frac{\partial R}{\partial Var(X)}=\frac{\partial R}{\partial\beta}\frac{ \partial\beta}{\partial\sigma_{Z}}\frac{\partial\sigma_{Z}}{\partial Var(X)} \tag{40}\]  
It can be seen from Equations (35) and (36):  
\[\frac{\partial R}{\partial\beta}=\varphi(\beta),\ \frac{\partial\beta}{ \partial\sigma_{Z}}=-\frac{\mu_{Z}}{\sigma_{Z}^{2}},\ \frac{\partial\sigma_{Z}}{\partial Var(X)}=\frac{1}{2\sigma_{z}}\left[\frac{ \partial\overline{z}}{\partial X}\otimes\frac{\partial\overline{z}}{\partial X }\right]\]  
\(Z\) is the limit state function. At the moment of working configuration I, configuration transformation and working configuration II, its expressions are Equations (28), (30) and (32), respectively. \(\varphi(\bullet)\) is the probability density function of the standard normal distribution. \(Var(X)\) represents the variance of the structural parameter vector. The symbol \(\otimes\) represents the Kronecker product, which is defined as \((A)_{p\times q}\otimes(B)_{s\times t}=\left(a_{ij}B\right)_{ps\times qt}\).  
Due to the inconsistency of units among the structural parameters that affect the reliability of the configuration transformation of the metamorphic mechanism, the reliability sensitivity of different parameters cannot be compared. Therefore, it is necessary to conduct dimensionless processing for reliability sensitivity [52]. After the dimensionless processing, the reliability to each variance sensitivity of structural parameters is as follows:  
\[s_{x_{i}}=\frac{\partial R}{\partial Var(X)}\frac{\sigma_{x_{i}}^{2}}{R} \tag{41}\]  
In order to better analyze the influence of the tolerance of each parameter on reliability, the variance sensitivity value is normalized to obtain the variance sensitivity coefficient \(S_{x_{i}}\) of each parameter:  
\[S_{x_{i}}=\left|\frac{s_{x_{i}}}{\sum s_{x_{i}}}\right| \tag{42}\]  
According to the calculation result of variance sensitivity coefficient of each parameter, tolerance of the parameter with higher sensitivity is taken as the optimization design variable.