# A Conceptual Model for Joint Graphic Representation of
## 5 MMCS Design Examples
### Virtual Servo Axes Applied to a Telescopic Servo Axis  
Although servo axes of this type do not have a physical actuator directly associated, they can receive motion commands and respond to them with an ideal behavior. It means that the motion controller kinematically simulates its position and speed for each moment. For this purpose, the position or speed setpoint that the motion controller calculates is used as position or speed feedback. Virtual servo axes are usually used as masters of real servo axes for multiple purposes, such as to equalize delays in applications with multiple slaves or to handle elements of a process that are not servo axes but where it is interesting to treat them as such or even to generate the theoretical trajectory of the TCP of a tool and then calculate the next coordinates according to the ACS of each servo axis, after applying the corresponding inverse transformation according to the kinematic configuration of the machine. Figure 16: Parallel servo axes machine (gantry configuration). (a) 3D view; (b) MMCS top view. The latter is the case studied below for a very simple transformation, but it serves as an example for the application of virtual servo axes. Figure 18a shows the image of a telescopic servo axis, formed by a servo axis whose mobile carriage has a second servo axis mounted on it, which, in turn, displaces the tool. In this case, these are servo axes driven by a belt. Its simplified mechanical representation by applying the process described in Section 4 is shown in Figure 18b, with both carriages at home position. They are named X1 and X2. For this example, the effective stroke of the base servo axis is 1/3 of the total and the one of the other servo axis is 2/3, the total effective stroke being the same as the sum of the effective strokes of both servo axes, i.e., \((\mathrm{L_{GX1}-L_{CX1}})+(\mathrm{L_{CX2}-L_{CX2}})\). The specifications of the movements can now be added to the previous representation. The MMCS resulting is shown in Figure 19. In this case, each movement is represented in an individual MMCS. A first idea may be to think about programming a motion command to position the TCP by using two individual and simultaneous motion commands to each of the individual servo axes, and taking into account that the origin of the ACS of the mobile servo axis depends on the position of the fixed servo axis, as it moves it with its carriage. However, positioning the TCP with respect to a virtual servo axis simplifies the motion commands of the TCP and abstracts the program implementation of mechanical details. It means that the virtual servo axis would be equivalent to a single real physical servo axis, whose effective stroke was the combined one. For doing so, it would be enough to define a  
Figure 17: Two linear coordinated movements of the gantry machine. (**a**) Combination of mechanical simplification and path representation; (**b**) program in PLCopen. Figure 18: Telescopic servo axis. (**a**) 3D view; (**b**) MMCS view. itural servo axis as a master of the two real servo axes by means of an electronic gear. The ratio would be 1:3 for the base servo axis and 2:3 for the mobile one, i.e., the proportion of each servo axis with respect to the total effective stroke. It is important to note that, in this case, the mechanical representation of real elements is combined with the representation of a logical resource such as the virtual servo axis. Its origin or zero is matched to the origin of the base servo axis for simplicity. The proportion that corresponds to each of them with respect to the movement of the virtual servo axis is indicated along with the arrow of movement of each servo axis. The virtual servo axis is named with the prefix V and it is drawn with discontinuous lines to differentiate it from the real ones, as presented in Section 4.4. In the case of a group of virtual servo axes describing the theoretical trajectory of a TCP, a triangle with white filling could be used. Its application in such a case is explained below.