## 1 Introduction
### Modularized Force Analysis of AGGs
Modularized force analysis procedures are as follows:  
\[\left\{ \begin{array}{l}R_{4x}=\frac{(A_{2}P_{43x}-B_{2}P_{43x})}{P_{43x}P_{4 3x}-P_{43y}P_{43x}}\\ R_{4y}=\frac{(A_{2}P_{43y}-B_{2}P_{43x})}{P_{43x}P_{42y}-P_{43y}P_{42x}}\\ A_{2}=-(T_{51}+T_{61}+T_{1}+T_{2})\\ B_{2}=-(T_{63}+T_{2}+T_{k})\\ \end{array} \right. \tag{13}\]  
\[\left\{ \begin{array}{l}R_{2x}=-(R_{4x}+F_{5x}+F_{6x})\\ R_{2y}=-\big{(}R_{4y}+F_{5y}+F_{6y}\big{)}\\ \end{array} \right. \tag{14}\]  
\[\left\{ \begin{array}{l}R_{3x}=-(R_{4x}+F_{6x}+F_{6x})\\ R_{3y}=-\Big{(}R_{4y}+F_{6y}+F_{ky}\Big{)}\\ F_{kx}=F_{k}\cos(\theta_{34}+\theta-\pi)\\ F_{ky}=F_{k}\sin(\theta_{34}+\theta-\pi)\\ \end{array} \right. \tag{15}\]  
In the above equations, \(F_{kx}\) and \(F_{ky}\) denote the components of the spring force in \(x\) and \(y\) directions, respectively. The calculation method of the variables \(d_{k}\), \(\theta\), \(F_{k}\), \(T_{k}\), \(\Delta T\) and the meaning of the other variables can refer to the force analysis in working configuration I above. Figure 6: Force diagram of RRR group degraded by RRRP group. Structure Design of Metamorphic Revolute Joint According to Constraint Parameters Calculated in Modularized Force Analysis  
When the metamorphic revolute joint keeps static in a working configuration, the angle of its two links must be constrained, neither increasing nor decreasing. The common way to keep static is to use spring force constraints and/or geometric constraints in the structure design of metamorphic joints. In previous studies, metamorphic joints are often drawn in kinematic diagrams without structure design. For practical application, a simple and practical structure of the metamorphic revolute joint is designed in the paper as shown in Figure 7a,b, and the physical prototype is shown in Figure 7c. The link (1), the end cover (3) and the component (6) are connected as a united one, as shown in Figure 7b. The link (2) and the component (4) are connected as a united one by bolts. The balls at both ends of the component (6) can move along the grooves which are located on the internal surface of the component (4), so that the two links (1) and (2) can rotate relative to each other. As shown in Figure 7a, the pin (5) serves as an actual geometric constraint, and the angle of the metamorphic revolute joint cannot increase after the component (6) contacts the pin (5). The spring is compressed with a preload and installs on the links. When the driving torque of the metamorphic revolute joint is large enough to overcome the force of the compressed spring, link (2) rotates clockwise relative to link (1) due to the release of both geometrical constraint and spring force constraint. At the limiting position, geometric constraints of the metamorphic revolute joints theoretically provide infinite constrained force to ensure that the angle cannot increase. At the same time, if the metamorphic revolute joint plans to keep static in a working configuration, the constraint torque provided by the spring force from preload should be greater than the maximum value of driving torque during the entire process of configuration keeping to limit the reduction of the angle between links (1) and (2), that is, as follows:  
\[F_{k}\geq\frac{\Delta T}{d_{1}\sin\theta} \tag{16}\]  
\(F_{k}\) is the spring force from preload, \(d_{1}\) and \(\theta\) are the corresponding structural dimensions as shown in Figure 5. Furthermore, we can design the structural parameters of spring by using the range of spring preload calculated by Equation (16). Probabilistic Evaluation Model for the Configuration Transformation Ability of Constrained Metamorphic Mechanisms  
In general, the probabilistic evaluation procedure for the configuration transformation ability of constrained metamorphic mechanisms is shown in Figure 8, and the detailed steps to perform the probabilistic evaluation will be introduced in the following sections. Figure 7: Structure of metamorphic revolute joint.