**5. Forward kinematics of the open-chain module**

According to the Subsection 3.3 a 2+2 parallel modules disposed in a serial structure could be set up. We are going to start from a complete serial model of the robot. The Figure 1 shows this set up of the robot. An equivalent model is shown in the Figure 9.

To solve forward kinematics of the open-chain, Denavit-Hartenberg algorithm (Denavit & Hartenberg, 1955; Uicker et al., 1964; Hartenberg & Denavit, 1964) has been used. In order to solve transformation matrices, *The Robotics Toolbox for Matlab* (Corke, 1996) has been used. Firstly, and following D-H convention, we could establish the coordinate axes as follows in the Figure 10. Secondly, the transformation matrix will be solved:

$$\begin{aligned} \;^0T\_1 = \begin{bmatrix} -\sin\phi\_1 & 0 & \cos\phi\_1 & 0\\ \cos\phi\_1 & 0 & \sin\phi\_1 & 0\\ 0 & 1 & 0 & H\_1\\ 0 & 0 & 0 & 1 \end{bmatrix} \end{aligned} \tag{44}$$

$$\begin{aligned} \;^1T\_2 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & L\_2 \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{aligned} \tag{45}$$

$$\begin{aligned} \;^2T\_3 = \begin{bmatrix} \cos\phi\_3 & 0 & \sin\phi\_3 & 0\\ \sin\phi\_3 & 0 & -\cos\phi\_3 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \end{aligned} \tag{46}$$

Design and Postures of a Serial Robot Composed by Closed-Loop Kinematics Chains 139

A preliminary study of the workspace that could reach a set up of 2+2 parallel modules arranged in a serial mode, with actuated rotational joints, have been performed. The goal

was to check if it was able to climb a three-dimensional cross-linked structure.

Fig. 10. 3D model of the serial robot with D-H convention axis

the vector will be every final point of the previously described.

has been shown in the Figures 6 and 7 with the simulated model.

For one of the possible workspaces, some of the mathematical combinations of the robot actuators have been obtained, and a vector with all of them has been generated. This vector has been used to obtain different final points of the end-effector. Therefore, every element of

We have obtained a 210 elements vector (Figure 11) as a result of the ten joints of the real model. In this vector, the interferences between links were not taken into consideration. On the other hand, a cross-linked structure and a robot model have been simulated through SolidWorks. The goal was to check if the robot was able to reach enough workspace points and, at the same time, to perform a plane change in the cross-linked structure, and all of this

**6. Robot workspace** 

$${}^{5}T\_{4} = \begin{bmatrix} \cos\phi\_{4} & -\sin\phi\_{4} & 0 & \cos\phi\_{4} \ast H\_{4} \\ \sin\phi\_{4} & \cos\phi\_{4} & 0 & \sin\phi\_{4} \ast H\_{4} \\ \sin\phi\_{4} & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{47}$$

$${}^{4}T\_{5} = \begin{bmatrix} \cos\phi\_{5} & 0 & -\sin\phi\_{5} & 0 \\ \sin\phi\_{5} & 0 & \cos\phi\_{5} & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{48}$$

$${}^{5}T\_{6} = \begin{bmatrix} \cos\phi\_{6} & 0 & -\sin\phi\_{6} & 0 \\ \sin\phi\_{6} & 0 & \cos\phi\_{6} & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{49}$$

$${}^{6}T\_{7} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & L\_{6} \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{50}$$

$${}^{6}T\_{8} = \begin{bmatrix} \cos\phi\_{6} & 0 & -\sin\phi\_{6} & 0 \\ \sin\phi\_{6} & 0 & \cos\phi\_{6} & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{51}$$

Below, the Table 4 indicates the D-H Parameters of the 8 DoF of the model:


00 01

Table 4. D-H Parameters table of the serial model
