On the Stability of the Quadruple Solutions of the Forward Kinematic Problem in Analytic Parallel Robots

Abstract

Many parallel robots can change between different assembly modes (solutions of the forward kinematic problem) without crossing singularities, either by enclosing cusps or alpha-curves of the planar sections of their singularity loci. Both the cusps and the alpha-curves are stable singularities, which do not disappear under small perturbations of the geometry of the robot. Recently, it has been shown that some analytic parallel robots can also perform these nonsingular changes of assembly mode by encircling isolated points of their singularity loci at which the forward kinematic problem admits solutions with multiplicity four. In this paper, we study the stability of these quadruple solutions when the design of the robot deviates from the analytic geometry, and we show that such quadruple solutions are not stable since the isolated singular points at which they occur degenerate into closed deltoid curves. However, we also demonstrate that, although the quadruple solutions are unstable, the behavior of the robot when moving around them is practically unaffected by the perturbations from the analytic geometry. This means that the robot preserves its ability to perform nonsingular transitions by enclosing the quadruple solutions, even when its geometry is not exactly analytic due to small manufacturing tolerances.