**3.1 Parallel module**

If we focus in the study of a simple closed-chain module, the Figure 2(a) shows such structure, that is composed by two prismatic actuated joints, with two central sliders to limit the lateral movement, and in turn a free rotational joint in the upper side to provide a rotational movement of the end-effector of the module.

As a consequence of the use of a discrete workspace, the actuated joints will be binary, and these modules will have a d1 translation and a 45 degrees rotation (Figure 2(b)), when their configuration will be P1 ON, P2 OFF or vice versa.

The Figure 2(c) refers to both actuators P1 and P2 set to ON.

Design and Postures of a Serial Robot Composed by Closed-Loop Kinematics Chains 129

*Actuator P1 Actuator P2 Actuator P3 Actuator P4 Translation Rotation*  0 0 0 0 0 0º 0 0 0 1 d1 -45º 0 0 1 0 d1 45º 0 0 1 1 d2 0º 0 1 0 0 d1 -45º 0 1 0 1 d1+ d1 -90º 0 1 1 0 d1+ d1 0º 0 1 1 1 d1+ d2 -45º 1 0 0 0 d1 45º 1 0 0 1 d1+ d1 0º 1 0 1 0 d1+ d1 90º 1 0 1 1 d1+ d2 45º 1 1 0 0 d2 0º 1 1 0 1 d2+ d1 -45º 1 1 1 0 d2+ d1 45º 1 1 1 1 d2+ d2 0º

Table 2. Positions from two parallel modules in a serial mode

Fig. 3. Two parallel modules to draw up a serial structure

be observed in the Table 2.

joints. In this pose, it can be observed that the central sliders indicate the distance d2 between

However, it could be needed to reach some points in the workspace that are inclined at 90 degrees according to the base. In order to achieve this, and according to the Table 2, it could be observed in the Figure 5(b), that it can be obtained an inclination of 90 degrees, if the two

In a similar way, if a 45 degrees inclination according to the base link was needed to reach, we could perform an actuation to ON of one of the linear actuators (figure 5(a)). This could

the base and the end-effector, obviously, they will be at its highest point of stretching.

mirror drives of each parallel module are switched alternately to ON and OFF.

a) Isometric view of the parallel module

b) π/4 rotation: Actuator 2 ON

c) Actuator 1 and 2 ON

Fig. 2. Some views of the closed-chain module

Obviously, the different positions that this module is able to reach are given in Table 1:


Table 1. Positions from one parallel module
