Quadratic Characteristics of Workspace Curves of Generalized Planar Cable-Driven Parallel Mechanisms

Abstract

This paper focus on the geometrical application and discussion of robotics in the field of workspace. Workspace is referred to as the space which robotic arms can achieve with their own physical limitation. In this paper, a detailed geometrical and algebraic method is presented to analyze workspace curves for generalized 4-cable planar 3-degree-of-freedom cable-driven parallel mechanisms based on the antipodal theorem. In addition, this antipodal theorem is described by Voglewede as planar cable-driven parallel mechanisms are in force-closure condition if and only if the line p1p2 is completely surrounded by two sectors formed by two pairs of lines along each cable. The algebraic expression of workspace curves is deduced, which demonstrates that these curves are conics curves or straight lines (degenerated cases). Furthermore, quadratic features of these conics curves (i.e. ellipse, parabola and hyperbola) are discussed with various simulation cases. Based on the results of case study, the type of quadratic curves of this cable-driven mechanism is dependent on the determinant of conic matrix of curve expression. This paper also shows that the static workspace curves of planar cable-driven parallel mechanism relies on the structural parameters like the installation positions of cables.