Formulating Jacobian matrices for the dexterity analysis of parallel manipulators

The condition number of the Jacobian matrix has been commonly used in determining the dexterous regions of a manipulator workspace. This has been successful when applied to manipulators having either solely spherical or solely Cartesian degrees of freedom. However, for manipulators having a mix of both rotational and translational degrees of freedom, i.e., complex degree of freedom manipulators, the condition number of the Jacobian matrix may not be used due to dimensional inconsistencies with its elements. This paper furthers earlier work introduced in obtaining a Jacobian matrix which may be used to determine the dexterity of parallel mechanisms regardless of the number and type of degrees of freedom of the mechanism. The result of the method introduced in this paper is a dimensionally homogeneous Jacobian matrix mapping m actuator velocities to n independent end effector velocities. In the typical case where m = n, the Jacobian matrix is also square. As opposed to earlier works, the singular values of the Jacobian matrix obtained here have an evident physical significance. Furthermore, the ratio of the maximum and minimum singular values, i.e., the condition number may be used to measure the dexterity of the manipulator at a given pose. To illustrate the concepts introduced in this paper, the 3-PRS manipulator is analyzed.