## 1 Introduction
### Modularized Force Analysis of AGGs  
In working configuration I, the AAG RRRP degenerates into the Assur group RRP. The force analysis of the corresponding equivalent Assur group RRP is shown in Figure 5,  
Figure 4: Metamorphic process of RRRP group. where points \(N_{5}\), \(N_{6}\) and \(N_{7}\) are the centroids of the links \(N_{2}N_{3}\), \(N_{3}N_{4}\) and the slider, respectively, and \(F_{\beta}\) is the component of the external force acting on the slide along the moving direction. Supposing that the external forces \(F_{5}\), \(F_{6}\), \(F_{7}\) and torques \(T_{1}\), \(T_{2}\), \(T_{3}\) acting on the links \(N_{2}N_{3}\), \(N_{3}N_{4}\) and the slider, respectively, are known, the reaction force of each revolute joint and the driving torque and constraint torque of the metamorphic revolute joint are solved. Modularized force analysis procedures are as follows:  
\[\cos\alpha=\frac{l_{23}^{2}+l_{34}^{2}-l_{24}^{2}}{2l_{23}l_{34}}=\frac{d_{1}^ {2}+d_{2}^{2}-d_{k}^{2}}{2d_{1}d_{2}} \tag{2}\]  
\[d_{k}=\sqrt{d_{1}^{2}+d_{2}^{2}-\frac{d_{1}d_{2}\big{(}l_{23}^{2}+l_{34}^{2}- l_{24}^{2}\big{)}}{l_{23}l_{34}}} \tag{3}\]  
\[F_{k}=k(a-d_{k}) \tag{4}\]  
\[\cos\theta=\frac{d_{k}^{2}+d_{1}^{2}-d_{2}^{2}}{2d_{k}d_{1}} \tag{5}\]  
\[\theta=\arccos\left[\frac{2d_{1}^{2}-\frac{d_{1}d_{2}\big{(}l_{23}^{2}+l_{34} ^{2}-l_{34}^{2}\big{)}}{l_{23}l_{34}}}{2\sqrt{d_{1}^{2}+d_{2}^{2}-\frac{d_{1}d _{2}\big{(}l_{23}^{2}+l_{34}^{2}-l_{34}^{2}\big{)}}{l_{23}l_{34}}}}d_{1}\right] \tag{6}\]  
\[T_{k}=F_{k}d_{1}\sin\theta \tag{7}\]  
\[\left\{ \begin{array}{l}R_{2x}=\frac{(-A_{1}\sin\beta+B_{1}P_{2x_{1}})}{P_{2 x_{1}}\cos\beta+P_{2x_{1}}\sin\beta}\\ R_{2y}=\frac{\big{(}A_{1}\cos\beta+B_{1}P_{2y_{1}}\big{)}}{P_{2x_{1}}\cos\beta +P_{2y_{1}}\sin\beta}\\ A_{1}=-(T_{54}+T_{64}+T_{1}+T_{2})\\ B_{1}=-(F_{5x}+F_{6x}+F_{7x})\cos\beta-\\ \big{(}F_{5y}+F_{6y}+F_{7y})\sin\beta\end{array} \right. \tag{8}\]  
\[\left\{ \begin{array}{l}R_{4x}=-\big{(}R_{2x}+F_{5x}+F_{6x})\\ R_{4y}=-\big{(}R_{2y}+F_{5y}+F_{6y}\big{)}\end{array} \right. \tag{9}\]  
\[F_{\beta}=F_{7x}\cos\beta+F_{7y}\sin\beta \tag{10}\]  
In the above equations, \(k\) denotes the spring stiffness, \(a\) denotes the original length of the spring, \(P_{ijk}\) and \(P_{ijy}\) denote the distance in \(x\) and \(y\) directions, respectively, \(R_{ix}\) and \(R_{iy}\) denote the reaction forces of joint \(i\) in the \(x\) and \(y\) directions, respectively, \(F_{ix}\) and \(F_{iy}\) denote the forces acting on point \(i\) in \(x\) and \(y\) directions, respectively, \(T_{ij}\) denotes the torque of the  
Figure 5: Force diagram of RRP group degraded by RRRP group. force acting on point \(i\) to point \(j\), \(T_{i}\) denotes the torque acting on point \(i\), and \(T_{k}\) denotes the torque of spring force to point \(N_{3}\). Supposing that the counterclockwise direction is positive, \(T_{53}\) and \(T_{63}\) denote the torque of the gravities of the two links \(N_{2}N_{3}\) and \(N_{3}N_{4}\) to point \(N_{3}\), respectively, \(\Delta T\) denotes the driving torque of the metamorphic revolute joint:  
\[\left\{ \begin{array}{l}T_{23}=P_{23x}R_{2y}-P_{23y}R_{2x}\\ T_{43}=P_{43x}R_{4y}-P_{43y}R_{4x}\\ T_{53}=P_{53x}m_{23}g\\ T_{63}=P_{63x}m_{43}g\\ \end{array} \right. \tag{11}\]  
\[\Delta T=T_{23}-T_{43}+T_{53}-T_{63} \tag{12}\]  
In working configuration II, the AAG RRRP degenerates into the RRR group. The force analysis of the corresponding equivalent Assur group RRR is shown in Figure 6, where points \(N_{5}\) and \(N_{6}\) are the centroids of the links \(N_{2}N_{3}\) and \(N_{3}N_{4}\), respectively. Given that forces \(F_{5}\), \(F_{6}\) and torques \(T_{1}\), \(T_{2}\) act on the links \(N_{2}N_{3}\) and \(N_{3}N_{4}\), respectively, solve the reaction force of each revolute joint and the driving torque and constraint torque of the metamorphic revolute joint. According to the above conditions, the reaction force of each revolute joint and the driving torque and constraint torque of the metamorphic revolute joint are solved.