Dynamic simulation of planar flexible link manipulators using adaptive modal integration

In this paper a modal integration method is proposed for analyzingthe dynamic behavior of multi-link planar flexible manipulators. Anon-linear finite element method is employed to derive theequations of motion in terms of a mixed set of generalizedcoordinates of the manipulator with rigid links and deformationparameters that characterize flexible deformations of the links. Using a perturbation method the vibrational motion of themanipulator is modeled as a first-order perturbation of thenon-linear nominal rigid link motion. For that purpose theflexible dynamic manipulator model is split into two parts. Arigidified model describes the nominal rigid link motion. Alinear system linearized about the nominal trajectory describesthe vibrational motion. In order to reduce the dimension of thelinearized system, a modal reduction technique is proposed. Thenmodal integration can be applied using only a small number of lowfrequency modes. The mode-acceleration concept is used to accountfor the pseudo static contribution of the high frequency modes.Applied to the motion of a manipulator mechanism the method isreferred to as ‘adaptive modal integration’ since thetime-varying nature of the mode shape functions is taken intoaccount. A flexible two-link manipulator is analyzed to illustrate theperformance of the solution method. Comparisons between solutionsutilizing non-linear and perturbation analyzes with and withoutmodal integration show a good agreement. In a simulation with onlya few modes the accuracy is kept, whereas the computation time isdrastically reduced