Application of Barycentric Coordinates and the Jacobian Matrix to the Analysis of a Closed Structure Robot

A new approach is presented to study the kinematic properties of stationary robots with a closed structure. It combines the application of conventional methods from kinematics with geometric parameters represented in a barycentric coordinate system. This allows examining the influence of the proportions of the robot’s links on its basic mechanical characteristics. Each point from the newly introduced barycentric space corresponds to a set of robots with the same link proportions. The proposed approach is used to study three aspects: the link proportions for which the robot can exist; the shape of the robot’s workspace; and the possible singular configurations. This is valuable when evaluating the qualities of existing robots and could be applied to the design of new mechanical systems. An example of a 5-link robot with a closed structure is considered. The conditions for the existence of the mechanism and the conditions under which certain types of singular configurations can occur are defined. The example reveals the great potential of combining barycentric coordinates and Jacobian properties. The barycentric coordinates of 10 robots with a 5-link closed structure known from the literature are determined, and their properties are analyzed. The results are presented graphically. An extension of the application area of the approach is discussed.