## 1 Introduction
### Reliability Evaluation Model of Configuration Transformation Ability  
Given the randomness of the driving force \(F\) and the resistance \(F_{c}\) of a metamorphic joint \(k\), when the kinematic status of the metamorphic joint changes from static to moving at the moment of configuration transformation, the reliability of kinematic status transformation is as follows:  
\[ \begin{array}{ll}R_{k}&=P(f_{e}<1)=P(F>F_{c})\\ &=\int_{-\infty}^{\infty}f_{s}(x_{s})\Big{[}\int_{-\infty}^{x_{s}}f_{1}(x_{1}) \mathrm{d}x_{1}\Big{]}\mathrm{d}x_{s}\end{array}  \tag{23}\]  
When the metamorphic joint switches from moving to static at the moment of configuration transformation, the reliability is as follows:  
\[ \begin{array}{ll}R_{k}&=P(f_{e}>1)=P(F<F_{c})\\ &=\int_{-\infty}^{\infty}f_{s}(x_{s})\Big{[}\int_{x_{s}}^{\infty}f_{1}(x_{1}) \mathrm{d}x_{1}\Big{]}\mathrm{d}x_{s}\end{array}  \tag{24}\]  
\(f_{s}(x_{s})\) is the probability density function of the driving force, and \(f_{1}(x_{1})\) is the probability density function of the resistance force. Obviously, the configuration transformation of the metamorphic mechanism is mainly realized by switching the motion state of metamorphic joints. Therefore, the probability of the successful transformation moment of the constrained metamorphic mechanism with \(n\) working configurations at a single configuration transformation is defined as the product of the probabilities \(R_{k}\) of the successful kinematic status transformations of \(n\) metamorphic joints. The joints participate in configuration transformation of constrained metamorphic mechanisms:  
\[R_{c}=\prod_{i=1}^{n}R_{ki}=P(f_{e1}(\theta_{CT})<1)\prod_{i=2}^{n}P(f_{ei}( \theta_{CT})>1) \tag{25}\]  
\(\theta_{CT}\) is the position angle of driving link at the moment of configuration transformation. According to the equivalent resistance gradient curves shown in Figures 8 and 9, the equivalent resistance coefficient is close to 1 at the adjacent moments of the configuration transformation. At the moment, the fluctuation in equivalent resistance coefficient is more likely to cause the undesired working sequence. Therefore, it is necessary to give the interval reliability calculation model of the configuration transformation. Planar metamorphic joints are divided into metamorphic revolute joints and metamorphic prismatic joints, and both of them exit in augmented Assur group RRRP, as shown in Figure 3. Therefore, taking the constrained metamorphic mechanism (Figure 1) with the RRRP augmented Assur group as an example, the interval reliability calculation model of the configuration transformation is established, which can provide reference to analyze the configuration transformation reliability of constrained metamorphic mechanisms containing other augmented Assur groups. Moreover, the force analysis method of the RRRP augmented Assur group can be referred to in Section 3.1. Some factors need to be considered, such as the uncertainties in the link dimensions, masses, angular velocity and angular displacement of driving link, spring stiffness, installation position of spring and constrained angular of geometric constraint. Meanwhile, it is assumed that all the uncertainty parameters are random variables, and each of them is expressed by an independent and normal, i.e., Gaussian distribution. As described in Section 4.1, whether the mechanism shown in Figure 1 can work well in working configuration I is mainly determined by whether the metamorphic revolute joint \(R_{3}\) can keep static all the time. Take Equations (8), (9), (11) into Equation (12), respectively, to calculate the driving torque of metamorphic revolute joint \(R_{3}\):  
\[ \begin{array}{ll}\Delta T_{1}=&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},\alpha,m_{23},m_{34},m_{slider},\omega_{1},\theta_{1}) \end{array}