## 2 The Loop Equation of the Single-DOF Spherical Kinematic Chain with Only Revolute Pairs  
The loop equation is an effective method to study a spherical kinematic chain's motion. The loop equation considers each spherical link as a spatial vector. With the loop equation, the relationship between the input and output joints of the spherical kinematic chain can be established. To demonstrate how the loop equation is established, the four-bar spherical mechanism is utilized. Three steps are involved to establish the loop equation of a single DOF spherical kinematic chain with only revolute pairs. In step one, a three-dimensional Cartesian coordinate system at every vertex (or at every spherical revolute pair), where two spherical links meet, is established. As is shown in Figure 1, a spherical four-bar kinematic chain, with four spherical revolute pairs, has four three-dimensional Cartesian coordinate systems, each of which has a \(Z_{i}\)-axis (\(i\) = 1, 2, 3, 4) that goes from the center of the kinematic chain's sphere to one of the four vertices. The yellow spherical link in Figure 1 is fixed. Each of the three remaining blue spherical links in the spherical four-bar kinematic chain has two types of rotation, axis rotation \(\mathbf{Z}_{i}\) (\(i\) = 1, 2, 3, 4), which is represented by a matrix and side rotation \(\mathbf{S}_{ij}\) (\(i\) = 1, 2, 3, \(j\) = 2, 3, 4), which is also a matrix. Since the \(Z_{i}\)-axis of the three-dimensional coordinate system on each revolute pair \(i\) (\(i\) = 1, 2, 3, 4) is designated as a rotation axis around which the two spherical links connected by revolute pair \(i\) rotate, the \(\mathbf{X}_{i}\)- and \(\mathbf{Y}_{i}\)-axes of the same three-dimensional coordinate system can be obtained with the right-hand rule, which makes the right thumb aim towards the positive direction of the \(Z_{i}\)-axis, and it makes the curling motion of the other four fingers of the same right hand move from the positive direction of the \(\mathbf{X}_{i}\)-axis to that of the \(\mathbf{Y}_{i}\)-axis. The axis rotation \(\mathbf{Z}_{i}\) is the rotation of spherical link \(i(i+1)\) (\(i\) = 1, 2, 3). In Figure 1, each of the four axis rotations, \(\mathbf{Z}_{1}\), \(\mathbf{Z}_{2}\), \(\mathbf{Z}_{3}\), and \(\mathbf{Z}_{4}\), is a red dotted line with an arrow. Axis rotation \(\mathbf{Z}_{i}\) is depicted by Equation (1). \[\mathbf{Z}_{i}= \begin{bmatrix}\cos\theta_{i}&-\sin\theta_{i}&0\\ \sin\theta_{i}&\cos\theta_{i}&0\\ 0&0&1\end{bmatrix}  \tag{1}\]  
The value of \(\mathbf{Z}_{i}\) depends on the rotation angle \(\theta_{i}\) (\(i\) = 1, \(\cdots\) 4), which is the angular displacement of a particular spherical link relative to the adjacent spherical link. The connecting rotation between any two adjacent axis rotations is a side rotation \(\mathbf{S}_{ij}\) which is described in Equation (2). \[\mathbf{S}_{ij}= \begin{bmatrix}1&0&0\\ 0&\cos\alpha_{ij}&-\sin\alpha_{ij}\\ 0&\sin\alpha_{ij}&\cos\alpha_{ij}\end{bmatrix}  \tag{2}\]  
As the radius of the four-bar kinematic chain's sphere is predetermined to have one unit, \(\mathbf{S}_{ij}\) is determined by the arc length of a particular link in the kinematic chain, which is the central angle \(\alpha_{ij}\) (\(i\) = 1, 2, 3, \(j\) = 2, 3, 4), whose two sides are two adjacent rotation axes shown in Figure 1. The side rotation \(\mathbf{S}_{ij}\) is normally constant for a spherical mechanism, with the exception that the mechanism has spherical prismatic pairs, which is further discussed in Section 3. Therefore, the movement of a spherical link can be divided into axis and side rotations, which appear alternately. Figure 1: The schematic diagram of the single-DOF four-bar spherical mechanism. In step two, the loop equation is established and is shown Equation (3),  
\[\mathbf{Z}_{1}\mathbf{S}_{12}\mathbf{Z}_{2}\mathbf{S}_{23}\dots\mathbf{Z}_{k-1} \mathbf{S}_{(k-1)k}\mathbf{Z}_{k}\mathbf{S}_{k(k+1)}=\mathbf{I}\;(k=1,2,3\cdots n), \tag{3}\]  
where the identity matrix, which is named \(\mathbf{I}\), equals \(