## 1 Introduction
### Reliability Evaluation Model of Configuration Transformation Ability
\begin{array}{rl}\sigma_{Z_{CT}}^{2}=&\Big{(}\frac{\partial f}{\partial l_{ 12}}\Big{)}^{2}\sigma_{I_{12}}^{2}+\Big{(}\frac{\partial f}{\partial l_{23}} \Big{)}^{2}\sigma_{I_{23}}^{2}+\Big{(}\frac{\partial f}{\partial l_{34}} \Big{)}^{2}\sigma_{I_{34}}^{2}+\\ &\Big{(}\frac{\partial f}{\partial\kappa}\Big{)}^{2}\sigma_{K}^{2}+\Big{(} \frac{\partial f}{\partial m_{23}}\Big{)}^{2}\sigma_{d_{1}}^{2}+\Big{(}\frac {\partial f}{\partial m_{34}}\Big{)}^{2}\sigma_{d_{2}}^{2}+\\ &\Big{(}\frac{\partial f}{\partial\kappa}\Big{)}^{2}\sigma_{\alpha}^{2}+\Big{(} \frac{\partial f}{\partial m_{23}}\Big{)}^{2}\sigma_{m_{23}}^{2}+\Big{(} \frac{\partial f}{\partial m_{34}}\Big{)}^{2}\sigma_{n_{34}}^{2}+\\ &\Big{(}\frac{\partial f}{\partial\omega_{1}}\Big{)}^{2}\sigma_{\omega_{1}}^{ 2}+\Big{(}\frac{\partial f}{\partial\theta_{1}}\Big{)}^{2}\sigma_{\theta_{1} }^{2}\end{array}  \tag{34}\]  
Since the limit state functions of the three working stages of the constrained metamorphic mechanism (Figure 1) are known, and both the driving torque and the resistance torque accord with normal distribution, by applying the stress strength interference model, the reliability of configuration transformation/keeping ability of constrained metamorphic mechanisms is as follows:  
\[R=P(Z>0)=\int_{0}^{\infty}f(Z)\mathrm{d}Z=\int_{0}^{\infty}\frac{1}{\sqrt{2 \pi}\sigma_{Z}}\exp\left[-\frac{1}{2}\Big{(}\frac{Z-\mu_{Z}}{\sigma_{Z}} \Big{)}^{2}\right]\mathrm{d}Z \tag{35}\]  
Equation (36) is obtained by converting Equation (35) into standard normal distribution and sets \(z=\frac{Z-\mu_{Z}}{\sigma_{Z}}\)  
\[R=\int_{z_{0}}^{\infty}\frac{1}{\sqrt{2\pi}}\exp\left[-\frac{1}{2}z^{2}\right] \mathrm{d}z=1-\Phi(z_{0})=\Phi(\beta) \tag{36}\]  
\[z_{0}=-\frac{\mu_{Z}}{\sigma_{Z}};\ \beta=\frac{\mu_{Z}}{\sigma_{Z}}\]  
\(f(Z)\) is the limit state function whose mathematical expressions are Equations (28), (30) and (32) corresponding to configuration I, configuration II and the moment of configuration transformation, respectively. Take the configuration transformation moment for example, the actual expression of the limit state function \(f(Z_{CT})\) could be solved by Equations (27), (29) and (30). Moreover, the partial derivative \(\frac{\partial f}{\partial\pi_{i}}\) for each input parameter \(x_{i}\) (\(i=1,2,3,\ldots\), \(n\)) can be solved. Therefore, \(\mu_{Z}\) and \(\sigma_{Z}\) could be obtained. Due to page limitations, we will not go into details about the actual expression of \(f(Z_{CT})\), \(\mu_{Z}\) and \(\sigma_{Z}\). The constrained metamorphic mechanism is more likely to produce an undesired working sequence at the adjacent moment of configuration transformation. Therefore, in working configurations I and II of the equivalent resistance gradient curve, \(a\) and \(b\) adjacent moments of configuration transformation whose equivalent resistance coefficients are close to 1 are respectively selected to calculate the reliability of the corresponding moments. Referring to the reliability calculation of the series system, multiply the reliabilities of \((a+b)\) adjacent moments with the reliability \(R_{c}\) of the successful configuration transformation at the moment of configuration transformation. The interval evaluation method of reliability \(R_{j}\) for a single configuration transformation of the constrained metamorphic mechanism with two working configurations is as follows:  
\[R_{j}=\underset{i=1}{\overset{a}{\Pi}}R_{1i}R_{c}\underset{i=1}{\overset{b}{\Pi}}R _{2i} \tag{37}\]  
\(R_{1i}\) is the reliability of the \(i\)th adjacent moment near configuration transformation in working configuration I, and \(R_{2i}\) is the reliability of the \(i\)th adjacent moment near configuration transformation in working configuration II.