After designing the configuration of the transmission systems, it has been possible to perform a kinematic analysis. This analysis allows us to know the angular position of the servomotors in order to place the control sector and therefore the node of the represented trajectory in a specific point.

As for the first version of the module we performed the kinematic analysis by using the geometrical method. Furthermore, in order to easily implement the control software, we have formulated also an algebraic analysis, which allows managing all the degrees of freedom by means of matrixes.

Geometrical Method

As regards the geometrical formulation, it is similar to the one formulated for the first version of the module. The kinematic scheme of the module results simpler than the previous one due to the direct actuation of the joints (Fig. 6.3).

Considering an absolute frame of reference (X-Y-Z), the position of the control point can be defined by means of a coordinate set E(x_{E}, y_{E}, z_{E}, т_{Е}, в_{Е}), which is known and represent the starting point of the analysis.

As we have done for the previous analysis, in order to simplify the process, it is useful to obtain the coordinates of the point P. In this way, it will be possible to perform the analysis on a single plane and then superimpose the effects.

Therefore, the coordinates of point P are:

It is now possible to analyse the planar mechanism represented by the vectors V_{3} and V_{4} and, therefore, to obtain the value of the angles ^_{1} and

2 with the same procedure illustrated in the Sect. 5.4. As regards the translation in the Z direction, it is possible to obtain its value directly from the developed control software that will be described in Sect. 6.5.