Direct Kinematic Singularities and Stability Analysis of Sagging Cable-Driven Parallel Robots

Sagging cable-driven parallel robots (CDPRs) are often modeled by using the Irvine's model. We will show that their configurations may be unstable, and moreover, that assessing the stability of the robot with the Irvine's model cannot be done by checking the spectrum of a stiffness matrix associated with the platform motions. In this article, we show that the static configurations of the sagging CDPRs are local extrema of the functional describing the robot potential energy. For assessing the stability, it is then necessary to check two conditions: The Legendre-Clebsch and the Jacobi conditions, both well known in optimal control theory. We will also 1) prove that there is a link between some singularities of the CDPRs and the limits of stability and 2) show that singularities of the platform wrench system are not singularities of the geometric model of the sagging CDPRs, contrary to what happens in rigid-link parallel robotics. The stability prediction results are validated in simulation by cross-validating them by using a lumped model, for which the stability can be assessed by analyzing the spectrum of a reduced Hessian matrix of the potential energy.