## 1 Introduction
### Evaluation of Configuration Transformation Ability Based on Equivalent Resistance Gradient Model  
In order to enable the constrained metamorphic mechanism to accurately complete the configuration transformation and realize the desired movements, the types and constraint architectures of metamorphic joints should be determined. Moreover, the constraint forces of the corresponding metamorphic joints should be calculated. Therefore, the equivalent resistance gradient model is established in ref. [7]. First, a dimensionless coefficient \(f_{e}(\theta_{i})\) is defined:  
\[f_{em}(\theta_{i})=\left|\frac{F_{c}(\theta_{i})}{F_{\min}}\right|=\left|\frac{ T_{c}(\theta_{i})}{T_{\min}}\right| \tag{17}\]  
\[f_{es}(\theta_{i})=\left|\frac{F_{c}(\theta_{i})}{F_{\max}}\right|=\left|\frac {T_{c}(\theta_{i})}{T_{\max}}\right| \tag{18}\]  
\(f_{em}(\theta_{i})\) is the equivalent resistance coefficient of moving metamorphic joint in the working stages, \(f_{es}(\theta_{i})\) is the equivalent resistance coefficient of static metamorphic joint in the working stages. \(F_{\min}\) and \(T_{\min}\) are the minimum driving force and torque acted on moving metamorphic joint in the corresponding working stages, respectively. \(F_{\max}\) and \(T_{\max}\) are the maximum driving force and torque acted on static metamorphic joint in the corresponding working stages, respectively. Figure 8: Flowchart for probabilistic evaluation procedure of configuration transformation ability of constrained metamorphic mechanisms. The moving sequences of metamorphic joints should be proportional to the equivalent resistant forces coefficient according to the law of minimum resistance of kinematics. Therefore, the equivalent resistance gradient of the metamorphic joints in the working stages of constrained metamorphic process should be as follows:  
\[f_{\text{es}}(\theta_{i})\geq 1\text{ and }f_{\text{em}}(\theta_{i})\leq 1 \tag{19}\]  
Taking the constrained metamorphic mechanism shown in Figure 1 as an example, the force analysis method of the RRRP augmented Assur group can be referred to in Section 3.1. According to Equations (17)-(19), the equivalent resistance coefficients of metamorphic joints \(R_{3}\) and \(P_{7}\) must meet the following conditions in the two configurations:  
In working configuration I, the metamorphic revolute joint \(R_{3}\) should always keep static, and the metamorphic prismatic joint \(P_{7}\) should always keep moving to ensure the desired working sequence. The driving torque of joint \(R_{3}\) is provided by \(\Delta T\) in Equation (12), and the constraint torque is provided by the spring force constraint and geometric constraint together. When \(\Delta T<0\), the angle of the metamorphic revolute joint \(R_{3}\) tends to increase, but remains unchanged due to geometric constraint. When \(\Delta T>0\), the angle of joint \(R_{3}\) tends to decrease and the spring is compressed. In order to still keep joint \(R_{3}\) static during working configuration I, the constraint torque provided by the spring should be always greater than the driving torque:  
\[f_{\text{e3}}=\frac{F_{k}d_{1}\sin\theta}{\Delta T}\geq 1 \tag{20}\]  
The beginning of working configuration II is regarded as the moment of configuration transformation. At that moment, the metamorphic prismatic joint \(P_{7}\) begins to switch from moving to static because of geometric constraint, and the metamorphic revolute joint \(R_{3}\) begins to change from static to moving. Since the geometric constraint force is theoretically infinite, it is obvious that the probability of the metamorphic prismatic joint \(P_{7}\) to finish the configuration transformation is nearly 100%.