A DAE Formulation for the Dynamic Analysis and Control Design of Cranes Executing Prescribed Motions of Payloads

Abstract

The dynamic behavior and control of cranes executing prescribed motions of payloads are strongly affected by the underactuated nature of the robotic systems, in which the number of control inputs/outputs is smaller than the number of degrees-of-freedom. The outputs are specified in time load coordinates, which, expressed in terms of the system states, lead to servo-constraints on the system. The problem can then viewed from the perspective of constrained motion. It is noticed however that servo-constraints differ from passive constraints in several aspects. Mainly, they are enforced by means of control forces which may have any directions with respect to the servo-constraint manifold, and in the extreme (some of them) may be tangent. A specific methodology must be developed to solve the’ singular’ inverse dynamics problem. In this contribution, a theoretical background for the modeling of the partly specified/actuated motion is given. The initial governing equations, arising as index five differential-algebraic equations, are transformed to a more tractable index three form by projecting the dynamic equations into the orthogonal and tangent subspaces with respect to the servo-constraint manifold in the crane velocity space. A simple numerical code for solving the resultant differential-algebraic equations, based on backward Euler method, is then proposed. The feedforward control law obtained this way is enhanced by a closed-loop control strategy with feedback of the actual errors in load position to provide stable tracking of the required reference load trajectory in presence of perturbations. A rotary crane executing a load prescribed motion serves as an illustration. Some results of numerical experiments/simulations are reported.