# A MULTI-PERSPECTIVE SIMULATOR FOR VISUALIZING AND ANALYZING THE KINEMATICS AND SINGULARITIES OF 2UPS/U PARALLEL MECHANISMS

This paper presents an educative graphical tool for simulating and studying parallel mechanisms with architecture 2UPS/U (U denotes "universal joint", whereas P and S denote "prismatic" and "spherical" joints, respectively). This 2-degrees-of-freedom parallel mechanism consists of two bodies connected through a passive universal joint such that it is possible to control the relative orientation between these bodies by means of two linear actuators connected in parallel between them. The simplicity and richness of this mechanism make it very interesting from an educative point of view, since it allows us to easily illustrate and teach different concepts related to the forward kinematics and singularities of parallel mechanisms.

After introducing the 2UPS/U mechanism, this paper presents and describes a graphical simulation tool developed in Java programming language for studying diverse kinematic aspects of this parallel mechanism. This tool is part of the web-based virtual laboratory of parallel robots PaRoLa (http://arvc.umh.es/parola). The presented tool is an intuitive graphical user interface which allows the user to simulate and study the forward kinematic problem of the 2UPS/U parallel mechanism, as well as to visualize its singularities and analyze their relationship with the forward kinematic problem (which is the problem consisting in determining the relative orientation between the two bodies of the 2UPS/U mechanism for given lengths of the two linear actuators). The tool also allows the user to modify the design of this mechanism (i.e., its dimensions), to analyze how the design affects the kinematics and singularities.

The presented tool offers several graphical windows which allow the user to analyze the forward kinematic problem of this mechanism from multiple perspectives: the problem as the intersection of the planar curves that represent the kinematic constraints of this mechanism, the problem as the computation of the zeroes of the characteristic polynomial of the mechanism, and the problem as a polynomial system in the complex domain. The objective of showing multiple and complementary perspectives of the same problem is to facilitate the comprehension of this problem by the students. Using the presented tool, the students can analyze the same problem from different perspectives, understanding the relationships between different perspectives and identifying special situations in which some of these perspectives degenerate and fail to properly describe the studied problem.

After describing the tool and its capabilities, some examples of usage of this tool are presented in order to illustrate its usefulness.