Abstract
Abstract The foundation for the main instrument designed to study robot kinematic structures is to analyze a Jacobian matrix of partial transmission ratios (Whitney. 1972; Ovakimov and Anshin, 1972). The [6x6] Jacobian matrix relates joint velocities q ˙ = [ q 1 , … , q n ] T of the robot to translational (linear) and angular velocities V = x ˙ , y ˙ , z ˙ , w x , w y , w z T of its end-effector. Conventional application of the J-matrix involves derivation and analysis of rather cumbersome matrix element relationships so complicated that they very often seem to be inappropriate for the problems to be solved. This paper introduces a modified, so called “indicator” Jacobian matrix. Its qualitative analysis enables one to gain an insight into the following subjects: 1) on the number of degrees-of-freedom (DOF) of the manipulator end-effector: 2) on the form of the most compact solutions for a velocity inverse problem: 3) on the existence of the explicit solution for a mechanism position inverse problem and on the algorithm to obtain these solutions.
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