Optimal control of an overhead crane within an energy‐momentum conserving method

The present work deals with inverse multibody dynamics problems, more precisely with optimal control problems governed by differential-algebraic equations. In our case, the control effort, which is necessary for moving a body from one position to another, will be minimized. Beside the commonly used generalized coordinates formulation of the equations of motion, the application of redundant coordinates will permit the derivation of conserving integrators. In addition to existing structure preserving integrators [2], the equations of motion, which serve as constraints within the optimal control scheme, may also be discretized with an energy-momentum conserving integrator. Schemes with this property typically exhibit superior numerical stability properties. Two different possibilities exist, when applying an energy-momentum method. The system may be reduced by applying the discrete nullspace method, which yields the elimination of the algebraic constraints [3]. This leads to a formulation with a higher degree of nonlinearity, which complicates the calculation of the gradients, that are necessary for the applied optimal control method. Alternatively, the index 3-DAEs may be used directly. Previously, this has been done in [1] within an energy preserving direct transcription method. In that case, the simple structure of the discrete equations of motion facilitates the calculation of the gradients, which may be used for calculating the discrete necessary conditions of optimality. These equations can be solved by Newton's method. We test the different formulations with the help of an underactuated overhead crane as numerical example. This example is taken from [4], where it was used in the context of trajectory tracking. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)