A STRUCTURAL FLEXIBILITY TRANSFORMATION MATRIX FOR MODELLING OPEN-KINEMATIC CHAINS WITH REVOLUTE AND PRISMATIC JOINTS

Abstract

A general approach to systematically derive the equations of motion of flexible open-kinematic chains is presented in this paper. The methodology exploits the serial characteristics of the kinematic chain by complementing the 4×4 Denavit–Hartenberg transformation matrix with a 4×4 structural flexibility matrix. The latter is defined based on a floating coordinate system which rendered the formulation applicable to both prismatic and revolute joints. The versatility of the approach is demonstrated through its implementation to formulate forward kinematic problems of manipulators with revolute and prismatic joints. Moreover, the proposed flexibility matrix is used in the development of a dynamic model for a compliant spherical robotic manipulator. This task has a dual purpose. First, it demonstrates how the flexibility matrix can be implemented in a systematic approach for deriving the equations of motion of an open-kinematic chain that account for the axial geometric shortening, the torsional vibration, and the in-plane and out-of-plane transverse deformations of the compliant member. Second, the inclusion of the torsional vibration in the equation of motion serves to broaden the scope of previous research work done on modelling open-kinematic chains. The formulation can now address dynamic problems that are not limited to the positioning but are also concerned with the orientation of rigid body payloads as they are being manipulated by robotic manipulators. The digital simulation results exhibit the interaction between the torsional vibration and the rigid body motion of the arm. Furthermore, they demonstrate a strong coupling effect between the torsional vibration and the transverse deformations of the arm whenever the payload is not grasped at its mass center by the gripper.