Natural oscillations of underactuated cable-driven parallel robots

Underactuated Cable-Driven Parallel Robots (CDPR) employ a number of cables smaller than the degrees of freedom (DoFs) of the end-effector (EE) that they control. As a consequence, the EE is underconstrained and preserves some freedoms even when all actuators are locked, which may lead to undesirable oscillations. This paper proposes a methodology for the computation of the EE natural oscillation frequencies, whose knowledge has proven to be convenient for control purposes. This procedure, based on the linearization of the system internal dynamics about equilibrium con_gurations, can be applied to a generic robot suspended by any number of cables comprised between 2 and 5. The kinematics, dynamics, stability and stiffness of the robot free motion are investigated in detail. The validity of the proposed method is demonstrated by experiments on 6-DoF prototypes actuated by 2, 3, and 4 cables. Additionally, in order to highlight the interest in a robotic context, this modelling strategy is applied to the trajectory planning of a 6-DoF 4-cable CDPR by means of a frequency-based method (multi-mode input shaping), and the latter is experimentally compared with traditional non-frequency-based motion planners.