## 2 Kinematics of a Generalized Mechanism for The Hybrid Robots Class III  
Let us consider a manipulator which is designed with n serial links \(\left\{l_{i}\right\}_{i=1+n}\) and m local closed-loop mechanisms \(\left\{\Theta_{k}\right\}_{k=1+m}\), (\(m<n\leq 6\)) as shown in Figure 2. The serial chain of n links is the main mechanism of the robot arm, and the closed-loop mechanisms are designed as sub-mechanisms appended to the main one. The degree of freedom number (DOFs) of the designed robot equals the DOFs of the main mechanism, \(n\). The presence of the local closed-loop mechanisms does not change the DOFs of the robot. Each local closed mechanism \(\Theta_{k}\) is designed with two successive links \(l_{i-1}\) and \(l_{i}\) of the main arm and other two appended links \(l_{k_{1}}\) and \(l_{k_{2}}\). See Figures 2 and 3. In other words, each local closed-loop mechanism is designed as a four bar linkage which is composed of the links \(l_{i-1}\), \(l_{i^{\prime}}\)\(l_{k_{1}}\) and \(l_{k_{2}}\). In a local closed chain linkage, the link \(l_{i-1}\) is a local base link, and the remained links \(l_{i^{\prime}}\)\(l_{k_{1}}\) and \(l_{k_{2}}\) move relative to the base link \(l_{i-1}\). As shown in Figure 3, there exists four types of the local closed-loops, Type A, B, C and D, which can be designed by combining a couple of successive links \(l_{i-1}\) and \(l_{i}\) of the main arm and other two links \(l_{k_{1}}\) and \(l_{k_{2}}\) added to the robot architecture. In a local closed-loop Type A (Figure 3a), the two links \(l_{k_{1}}\) and \(l_{k_{2}}\) are added to the main arm, and the output link is a link \(l_{i}\). Meanwhile in Type B (Figure 3b), the two links \(l_{k_{1}}\) and \(l_{k_{2}}\) are inserted in between the links \(l_{i-1}\) and \(l_{i^{\prime}}\) but the output link is the link \(l_{k_{2}}\). When a loop Type B is added to the main kinematic chain, the link \(l_{k_{2}}\) is inserted in between \(l_{i}\) and \(l_{i+1}\). Figure 2: A generalized configuration of the hybrid robots Class III. When the two links \(l_{k_{1}}\) and \(l_{k_{2}}\) are not connected together, there exists two more combinations of the links \(l_{i-1},l_{i^{\prime}},l_{k_{1}}\) and \(l_{k_{2}}\) as shown in Figure 3c,d. In a loop Type C, the output link is \(l_{i}\), meanwhile in Type D, the output link is \(l_{k_{2}}\). For all four types of the local linkages, the base link is \(l_{i-1}\), and the formation of all the closed-loop mechanisms does not change the DOFs of the robot. Let us consider the main kinematic chain of the hybrid robot. In Figure 2, \(O_{0}\equiv(O_{0}x_{0}y_{0}z_{0})\) is the reference frame, and \(O_{1}\equiv(O_{1}x_{1}y_{1}z_{1})\) to \(O_{n}\equiv(O_{n}x_{n}y_{n}z_{n})\) are the local coordinate systems which are attached to all the links of the main arm. Let us denote \(\mathbf{q}=\left[ \begin{array}{cccc}q_{1}&q_{2}&q_{3}&\ldots&q_{n}\end{array}  \right]^{\mathrm{T}}\) as the vector of joint variables on the main arm, correspondingly. As usual, the geometric and kinematic parameters of a link \(l_{i}\) are denoted as \(\theta_{i^{\prime}}\), \(d_{i^{\prime}}\)\(a_{i}\) and \(\alpha_{i}\). By using the Denavit-Hartenberg notation, the transformation matrix for the end-effector (EEF) can be calculated as follows:  
\[\mathbf{H}_{0E}=\mathbf{H}_{01}(q_{1})\mathbf{H}_{12}(q_{2})\mathbf{H}_{23}(q _{3})\ldots\mathbf{H}_{(n-1)n}(q_{n}) \tag{1}\]  
where \(\mathbf{H}_{(i-1)i}\big{(}q_{i}\big{)}\) is the transformation matrix for a link \(l_{i}\). In other words, \(\mathbf{H}_{0E}\) can be expressed as follows:  
\[\mathbf{H}_{0E}=\left[ \begin{array}{cc}\mathbf{A}(\mathbf{q})&\mathbf{r}( \mathbf{q})\\ 0&1\end{array} \right] \tag{2}\]  
where \(\mathbf{A}(\mathbf{q})\) is the rotation matrix, and \(\mathbf{r}(\mathbf{q})\) is the translation vector for the EEF. Equation (2) describes the kinematic relationship of the robot. If we denote \(\mathbf{p}_{E}=\left[ \begin{array}{cccc}x_{E}&y_{E}&z_{E}&\alpha&\beta& \gamma\end{array}