## 1 Introduction
### Reliability Evaluation Model of Configuration Transformation Ability
 \tag{26}\]  
Bring Equations (4), (6) into Equation (7), respectively, to calculate the resistance torque of metamorphic revolute joint \(R_{3}\):  
\[ \begin{array}{ll}T_{k}&=F_{k}d_{1}\sin\theta\\ &=k(a-d_{k})d_{1}\sin\theta\\ &=f(k,d_{1},d_{2},\alpha)\end{array}  \tag{27}\]  
Therefore, the limit state function of the ability of the metamorphic revolute joint \(R_{3}\) to keep static in working configuration I is expressed as follows:  
\[Z_{1}=T_{k}-\Delta T_{1} \tag{28}\]  
At the moment of configuration transformation, the metamorphic prismatic joint \(P_{7}\) begins to switch from moving to static, and the metamorphic revolute joint \(R_{3}\) changes from static to moving. Since the geometric constraint force is theoretically infinite, the probability of joint \(P_{7}\) to complete the configuration transformation is regarded to be nearly 1. Therefore, whether the mechanism shown in Figure 1 can complete configuration transformation at the moment is mainly determined by whether joint \(R_{3}\) can switch form static to moving accurately. Take Equations (11), (13), (14) into Equation (12), respectively, to obtain the driving torque of joint \(R_{3}\) at the moment of configuration transformation:  
\[ \begin{array}{ll}\Delta T_{CT}=&T_{23}-T_{43}+T_{53}-T_{63}=\\ &P_{23x}R_{2y}-P_{23y}R_{2x}-P_{43x}R_{4y}+\\ &P_{43y}R_{4x}+P_{53x}m_{23\delta}-P_{63x}m_{34\delta}=\\ &f(t_{12},t_{23},t_{34},k,d_{1},d_{2},\alpha,m_{23},m_{34},\omega_{1},\theta_{ 1})\end{array}  \tag{29}\]  
The spring is not compressed at the moment of configuration transformation. Consequently, the consistent torque of joint \(R_{3}\) keeps constant as Equation (27). Therefore, the limit state function of the kinematic status transformation ability of joint \(R_{3}\) at the moment of configuration transformation is expressed as Equation (30) similarly:  
\[Z_{CT}=\Delta T_{CT}-T_{k} \tag{30}\]  
In order to ensure the desired working sequence in working configuration II, joint \(P_{7}\) should always keep static. To keep joint \(P_{7}\) static, the slider requires sufficient driving force. In other words, the resultant force of the reaction force and the external force along the guide path is always to the right. The component of the reaction force on the slider along the moving direction is as follows:  
\[ \begin{array}{ll}R_{\beta}&=R_{4x}\cos\beta+R_{4y}\sin\beta\\ &=f(t_{12},t_{23},t_{34},k,d_{1},d_{2},m_{23},m_{34},\omega_{1},\theta_{1}) \end{array}  \tag{31}\]  
The calculation formula of \(R_{4x}\) and \(R_{4y}\) is shown in Equation (13). The component along the moving direction of the external force \(F_{\beta}\) acted on the slider is shown in Equation (10). In most cases, the direction of the reaction force on the slider is to the right, and the direction of the external force is to the left. Therefore, the limit state function of the ability of joint \(P_{7}\) to keep static in working configuration II can be expressed as follows:  
\[Z_{2}=R_{\beta}-F_{\beta} \tag{32}\]  
By applying the method of First-Order Second Moment, the first-order Taylor expressions of the limit state function of the above three working stages are expanded, and the error models of the limit state function are established. Taking the moment of configuration transformation as an example, the mean \(\mu_{T_{1}}\) and variance \(\sigma_{T_{1}}^{2}\) are calculated, respectively:  
\[\mu_{Z_{CT}}=f\big{(}\mu_{I_{12}},\mu_{I_{23}},\mu_{I_{34}},\mu_{K},\mu_{d_{1} },\mu_{d_{2}},\mu_{\alpha},\mu_{m_{23}},\mu_{m_{34}},\mu_{\omega_{1}},\mu_{ \phi_{1}}\big{)} \tag{33}\]  
\[