Jacobian Matrix Normalization - A Comparison of Different Approaches in the Context of Multi-Objective Optimization of 6-DOF Haptic Devices

Abstract

This paper focuses on Jacobian matrix normalization and the performance effects of using different criterion and techniques. Normalization of the Jacobian matrix becomes an issue when using kinematic performance indices and the matrix contains elements with non-homogenous physical units, i.e. representing both translational and rotational motions. Normalization is necessary in multi objective optimization if kinematic performance indices are used based on the full Jacobian matrix. Different methods have been proposed in literature for defining a scaling factor used to normalize the Jacobian. Based on a comparison of a few of these methods, we conclude that it is better to have the scaling factor as a design variable in the multi objective optimization. However, as an alternative, a new scaling factor is proposed based on the relationship between linear actuator motion range in joint space and rotational end effector motion in task space, a proposal underpinned by simulation, analysis and comparison of optimization results using existing normalization techniques. For optimization, performance indices for workspace, kinematic sensitivity, device isotropy and inertia are considered. To deal with the multi-objective optimization problem, genetic algorithms are employed together with a normalized multi-objective optimization function. The performances of different device configurations (depending on the normalization method and the global isotropy index used) are presented in this article.