Recursive Lagrangian Dynamics of Flexible Manipulator Arms via Transformation Matrices

Abstract

Improving the performance of most engineering systems requires the ability to model the system’s behavior with improved accuracy. The evolution of the mechanical arm from teleoperator and crane to present day industrial and space robots and large space manipulators is no exception. Initial simple kinematic and dynamic models are no longer adequate to improve performance in the most critical applications. 130th the mechanical system and control system require improved models for design simulation. Proposed new control algorithms require dynamic models for control calculation. Manning and programming activities as well as man-in-the-loop simulation also require accurate models of the arms. Accuracy is usually acquired at some cost. 'The application of mechanical arms to economically sensitive endeavors in industry and space also gives incentive to improve the efficiency of the formulation and simulation of dynamic models. Control algorithms and man-in-the-loop simulation require "real time" calculation of dynamic behavior. Formulation of the dynamics in an easy to understand conceptual approach is also important if maximum use of the results is to be obtained. The nonlinear equations of motion for flexible manipulator arms consisting of rotary joints connecting two flexible links are developed. Kinematics of both the rotary joint motion and the link deformation are described by 4x4 transformation matrices. The link deflection is assumed small so that the link transformation can be composed of summations of assumed link shapes. The resulting equations are presented as scalar and 4x4 matrix operations ready for programming. The efficiency of this formulation is compared to rigid link cases reported in the literature.