Complexity Analysis for Calculating the Jacobian Matrix of 6DOF Reconfigurable Machines

Abstract

The Jacobian matrix for machinery systems is challenging due to the kinematic structure, the machine behaviour, the machine configurations, and the singularity conditions. In the area of singularity, small velocities in the operational space may result in large velocities in the joint space. Therefore the control processes of machines may sometimes run into difficulties, since the inverse mapping from a Cartesian space to a joint space may cause problems. For solving the singularity of the kinematic structure, the development of the Jacobian matrix is the first priority. All of the different methods used for the Jacobian calculation are complex and require mathematical tools for symbolic calculation. It is important to select the most appropriate method with the minimum complexity for quick and accurate Jacobian matrix calculation. Thus, the authors were motivated to choose different procedures for developing the Jacobian matrix. In this study, the most general reconfigurable machinery kinematic structure, a remodelled n-DOF Global Kinematic Model, CNC-R GKM, is selected for the Jacobian matrix computation analysis. Each joint represents a combination of either rotational and/or translational joints type with any joint positive direction, with total number of 36 possible configurations in one joint. Calculation of the Jacobian matrix for this highly complex reconfigurable kinematic model, gives a comprehensive and unique results. The selected methods for Jacobian calculation are recursive Newton-Euler method and Vector cross multiplication method. The symbolic mathematical tool MAPLE 16 has been used. The comparison between these two calculation procedures has been completed with respect to the calculation complexity, and the results have been validated using the new developed CNC-Robot Global Kinematic Model (CNC-R GKM). The kinematic model and Jacobian matrix calculation used in this study are applicable in reconfigurable robotic and machines in industry, space and health care.