## 2 Kinematics of a Generalized Mechanism for The Hybrid Robots Class III
\right]^{\mathrm{T}}\) as the posture of the EEF in \(O_{0}\), where \(\left[ \begin{array}{cccc}x_{E}&y_{E}&z_{E}\end{array} \right]^{\mathrm{T}}\) is the position and \(\left[ \begin{array}{cccc}\alpha&\beta&\gamma\end{array} \right]^{\mathrm{T}}\) the orientation of the EEF, the kinematic equation can be written as follows:  
\[\mathbf{p}_{E}=\mathbf{f}(\mathbf{q}) \tag{3}\]  
Let us consider all the local-closed loops Type A, B, C or D, which can be added to the main kinematic chain, as shown in Figure 3. The presence of a closed chain Type A does not change Equation (1). However, Equation (1) must be recalculated if a closed loop Type B, C or D is designed for the robot. For a local linkage Type B, since a link \(l_{k_{2}}\) is inserted in between \(l_{i}\) and \(l_{i+1}\), the transformation matrix \(\mathbf{H}_{i(i+1)}\big{(}q_{i+1}\big{)}\) in Equation (1) must be recalculated as follows:  
\[\mathbf{H}_{i(i+1)}\big{(}q_{i+1}\big{)}=\mathbf{H}_{i\mathbb{R}_{1}}\big{(} q_{k_{1}}\big{)}\mathbf{H}_{k_{1}k_{2}}\big{(}q_{k_{2}}\big{)}\mathbf{H}_{k_{2}(i+1)} \big{(}q_{i+1}\big{)} \tag{4}\]  
Figure 3: Four types of the local closed-loops. For a local linkage Type C and D, the matrix \(\mathbf{H}_{(i-1)i}\big{(}q_{i}\big{)}\) is replaced by the following matrix:  
\[\mathbf{H}_{(i-1)i}\big{(}q_{i}\big{)}=\mathbf{H}_{(i-1)k_{1}}\big{(}q_{k_{1}} \big{)}\mathbf{H}_{k_{1}i}\big{(}q_{i}\big{)} \tag{5}\]  
In particular, for a linkage Type D, not only \(\mathbf{H}_{(i-1)i}\big{(}q_{i}\big{)}\) is re-calculated with Equation (5), but also the matrix \(\mathbf{H}_{i(i+1)}\big{(}q_{i+1}\big{)}\) must be re-calculated as follows:  
\[\mathbf{H}_{i(i+1)}\big{(}q_{i+1}\big{)}=\mathbf{H}_{(i-1)k_{2}}\big{(}q_{k_{2 }}\big{)}\mathbf{H}_{k_{2}(i+1)}\big{(}q_{i+1}\big{)} \tag{6}\]  
Note that the variables \(q_{k_{1}}\) and \(q_{k_{2}}\) in Equations (4)-(6) can be determined with respect to \(q_{i}\) via the two constraint equations of a corresponding closed-loops \(\Theta_{k}\), which will be presented later in the next section. _Constraint equations_  
Different from the formulation of the kinematics model for a serial robot, the formulation of the hybrid kinematics models of the hybrid robots Class III must take into account the constraints due to the presence of the local closed-loops. As discussed before, each linkage \(\Theta_{k}\) includes two sub-kinematic chains that move relative to the base link. For example, as shown in Figure 3(a), the first sub-chain of a linkage Type A is the link \(l_{i}\), and the second sub-chain includes the two link \(l_{k_{1}}\) and \(l_{k_{2}}\). Obviously, the two sub-chains of a linkage are closed at an intersection point M (see Figure 4). In order to derive the constraint equation for a loop \(\Theta_{k}\), the position of the point M are determined along the two sub-chains. On one hand, the point M, \(\mathbf{r}_{M}^{i}=\left[ \begin{array}{cc}x_{M}^{i}&y_{M}^{i}&z_{M}^{i}&1 \end{array} \right]^{T}\) represented in the coordinate \(O_{i}\) of the link \(l_{i}\), can be determined via the position and orientation of the link \(l_{i}\) that can be calculated relative to the base link \(l_{i-1}\). On the other hand, the point M, \(\mathbf{r}_{M}^{k_{2}}=\left[ \begin{array}{cc}x_{M}^{k_{2}}&y_{M}^{k_{2}}&z_ {M}^{k_{2}}&1\end{array} \right]^{T}\) expressed in \(O_{k_{2}}\), can be calculated through the position and orientation of the link \(l_{k_{2}}\). Consequently, the constraint equation for a loop Type A and B can be yielded as follows:  
\[\mathbf{H}_{(i-1)i}(q_{i})\cdot\mathbf{r}_{M}^{i}=\mathbf{H}_{(i-1)k_{1}} \big{(}q_{k_{1}}\big{)}\cdot\mathbf{H}_{k_{1}k_{2}}\big{(}q_{k_{2}}\big{)} \cdot k_{M}^{k_{2}} \tag{7}\]  
For a loop Type C and D the constraint equation can be written as follows:  
\[\mathbf{H}_{(i-1)k_{2}}\big{(}q_{k_{2}}\big{)}\cdot k_{M}^{k_{2}}=\mathbf{H}_ {(i-1)k_{1}}\big{(}q_{k_{1}}\big{)}\cdot\mathbf{H}_{k_{1}i}(q_{i})\cdot r_{M}^ {i} \tag{8}\]  
Figure 4: The closed kinematic chain of the local linkages.