A Novel Approach to Deriving the Unit-Homogeneous Jacobian Matrices of Mechanisms

The condition number of a mechanism's Jacobian matrix can be considered as an indicator of its dexterity. However, the unit-inhomogeneity problem existing in the Jacobian matrix of some mechanisms with both translational and rotational degrees of freedom discourages the designers from using this index. Addressing this issue, this paper furthers the earlier work and presents a new general method for deriving the dimensionally homogeneous Jacobian matrix of mechanisms. The deriving process includes two steps: deriving the conventional Jacobian matrix, and then mapping the general velocity of the end effecter onto the linear velocities of three points on the top plate plane. A 6-DOF hybrid mechanism is used to demonstrate this deriving process, and its dexterity is initially analyzed.