Finite element versus Galerkin approximation for modeling and control of a class of flexible manipulators

Overhead cranes with flexible variable-length cables represent an important class of flexible manipulators. This paper considers a generalized mathematical model of such manipulators where a set of coupled highly non-linear hybrid integral and differential equations describe its motion. The considered model, where the time dependent spatial domain is considered as dependent variable, resulted in an accurate system representation. The continuous time dependent flexible cable manipulator system is approximated by using two methods, namely; a modified finite element method and a modified Galerkin method. To investigate the performance of each method, a proportional-derivative controller is used to drive the manipulator system into a desired location such that the payload swing is damped out. It is shown that the performance of the modified Galerkin method is generally better than that of the modified finite element method. However the modified Galerkin method exhibits more sluggish response of the swing angle at the beginning of the lowering.