## 2 Constraints of the Reconfigurable (Rc)PU Limb  
The reconfigurable (Rc)PU limb is formed by the reconfigurable circular groove cross joint (Rc joint), a P pair (prismatic pair), and a U pair (universal joint). The limb switches between different motion modes based on the evolutionary rule (4) of the metamorphic mechanism [8]. As shown in Figure 1, the Rc joint has three axes of rotation at the same central point: the outer axis (Axis 1), the inner axis (Axis 2), and the side axis (Axis 3). The motion characteristics of the Rc joint are changed by altering the angle and configuration between the outer axis and the inner axis. The outer axis (Axis 1) is a fixed axis; its direction is unchanged, and the inner ends of the outer axis are connected to a circular groove through a revolute pair. The inner axis (Axis 2) is placed inside the circular groove, where it can rotate, and then the angle between the inner axis and the outer axis is changed; the position of the inner axis in the circular groove can be determined with the positioning hole. The angle change of the outer axis (Axis 1) and the inner axis (Axis 2) leads to different types of kinematic pairs of Rc joints. For the convenience of analysis, special configurations with two collinear axes and two perpendicular axes are selected as the two types of Rc joints. Figure 2a shows the configuration of the (Rc)PU limb when the two axes are collinear; in this case, the (Rc)PU limb is equivalent to the motion characteristics of the UPU limb and is called the (Rc)\({}_{1}\)PU limb. Figure 2b shows the configuration of the (Rc)PU limb when the two axes are perpendicular; in this case, the (Rc)PU limb is equivalent to the motion characteristics of the UPS limb and is called the (Rc)\({}_{2}\)PU limb. Figure 1: Structural diagram of Rc joint. 1. Cross joint. 2. U-shaped connector. 3. Joint bracket. 4. Circular groove. 5. Positioning hole. In the evolution of mechanisms, constraints are the decisive factor. Taking the (Rc)PU limb moving to a certain angle as the initial state of the limb, the geometric constraints of two different isomorphic states in the initial state of the limb are analyzed. As shown in Figure 2a, a local coordinate system \(A\)-\(x_{1}y_{1}z_{1}\) is established at the center \(A\) of the circular groove of the Rc joint. The outer axis of the Rc joint is in the direction of the \(x_{1}\) axis, the perpendicular direction of the joint support base is in the direction of the \(y_{1}\) axis, the vertical direction of the joint support side wall is in the direction of the \(z_{1}\) axis, and the distance between the center of the Rc joint circular groove and the U pair is \(a\) (constant). Then, the kinematic screw system of the (Rc)\({}_{1}\)PU limb can be expressed as follows:  
\[{}^{1}\mathbb{S}=\left\{ \begin{array}{l}\begin{array}{l}S_{11}=(1,\;0,\;0 ;\;0,\;0,\;0)^{\mathrm{T}}\\ S_{12}=(0,\;\alpha_{1},\;s\alpha_{1};\;0,\;0,\;0)^{\mathrm{T}}\\ S_{13}=(0,\;0,\;0;-s\alpha_{2},\;\alpha\alpha_{2},-s\alpha\alpha_{2})^{ \mathrm{T}}\\ S_{14}=(1,\;0,\;0;\;0,-as\alpha_{1}\alpha_{2},-ac\alpha_{1}\alpha_{2})^{ \mathrm{T}}\\ S_{15}=(0,\;\alpha_{1},\;s\alpha_{1};\;2 as\alpha_{1}\alpha_{1}\alpha_{2},\;as \alpha_{1}s\alpha_{2},-ac\alpha_{1}s\alpha_{2})^{\mathrm{T}}\\ \end{array}\right.\end{array} \right. \tag{1}\]  
where c is cos and s is sin;  
\(\alpha_{1}\) is the angle between the axis of screw \(S_{13}\) and plane \(A\)-\(x_{1}y_{1}\);  
\(\alpha_{2}\) is the angle between the axis of screw \(S_{13}\) and plane \(A\)-\(y_{1}z_{1}\). The kinematic screw of the (Rc)\({}_{1}\)PU limb can form a fifth-order screw system.