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This article proposes a deduction method of algebraic expression of static wrench-closure workspace boundary for planar 3-degree-of-freedom cable-driven parallel robots based on the antipodal theorem. Different from interval method and discretization approach, a more precise algebraic expression of workspace boundary can be obtained without undetermined region or costly computation burden to make the design process and optimization easy and feasible. In order to illustrate and study the conic characteristics of workspace boundary, a case study is conducted on a four-cable-driven 3-degree-of-freedom parallel robot. Simulation results show that the type of conic of boundary depends on the determinant of the specific deduced matrix and corresponds to which cable is collinear with the intersection line. In addition, the results also demonstrate that the boundary of workspace consists of the same type of two conics (i.e. ellipse, parabola, and hyperbola), and the region surrounded by these two quadratic curves is the smallest when the type of the border equation is parabola

Algebraic expression and characteristics of static wrench-closure workspace boundary for planar cable-driven parallel robots