Dynamics of cable-driven parallel manipulators with variable length vibrating cables

A parametric nonlinear model of cable-driven parallel manipulators endowed with a three-dimensional end-effector is formulated and discussed in this paper. The proposed model considers the distributed stiffness, inertia, and damping of time-varying length cables and allows to study and characterize the dynamic response of manipulators equipped with a generic number n of cables. The equations of motion of each of the n cables are first derived via a total lagrangian formulation together with the compatibility equations prescribing the connectivity between the cables and the end-effector mass, while the dynamics of the end-effector are described by enforcing the balance of its linear and angular momentum. A discretization procedure, based on admissible trial functions, is used to reduce the nonlinear partial differential equations of motion of the cables to a set of ordinary differential equations. The resulting equations are coupled with those describing the motion of the end-effector and the approximate solution is calculated via numerical time integration. Direct and inverse dynamic problems are then formulated and solved for selected case-study manipulators; finally, the role on the dynamic response of the system of the main mechanical parameters and of the degree of over-actuation is discussed.