Using differential-algebraic equations and natural coordinates for modelling and simulating cable-driven parallel robots

This paper proposes a comprehensive approach to the dynamic modelling of Cable-Driven Parallel Robots (CDPRs) by means of Differential-Algebraic Equations (DAEs). CDPRs are usually modelled through a minimal set of Ordinary Differential Equations (ODEs), often by making some simplification or just focusing on the unconstrained platform/end-effector dynamics. The alternative use of redundant DAEs provides several benefits since several non-ideal properties and peculiar operations of CDPRs can be easily and accurately modelled. To provide a comprehensive modelling frame, the typical components of a CDPR with rigid cables are here discussed and modelled by exploiting the concept of DAEs, which use redundant coordinates and embed kinematic constraints in the algebraic part of the equations. Through such advantageous features, it is possible to model swivelling guiding pulleys with non-negligible dimensions and mass. The use of rheonomous constraints is proposed as well, to represent in a simple way the effect of the movable exit-points, that are widely adopted in reconfigurable CDPRs. Finally, the use of Natural Coordinates is proposed for representing spatial end-effectors and modelling some challenging operations such as its overturning or the picking of heavy objects. Numerical simulations and the comparison with the results provided by a benchmark software are shown to demonstrate the accuracy and the computational efficiency of the proposed approach.