Resumo: 
Through either method of obtaining a constrained dimensionally homogeneous Jacobian matrix (proposed by (Gosselin, 1992) or by (Pond & Carretero, 2006)) for planar mechanisms, a choice exists on which of the potential six Cartesian velocity components on the end effector be used to define the task space velocity variables. The choice has an influence on the resulting Jacobian matrix and therefore its condition number and singular values. Without constraining the Jacobian matrix, the condition number was demonstrated to be essentially meaningless, as in (Kim & Ryu, 2003). In terms of measuring dexterity, the constrained dimensionally homogeneous Jacobian matrices (J'P) are superior to the screw based Jacobian matrix (J) in that they are dimensionally consistent. Furthermore, the six matrices (J'P) are superior to the 3 ? 6 dimensionally homogeneous matrix (J') in that they are constrained, and therefore provide true dexterous information. The condition number and singular values of each of the six matrices (J'P) are different for any given pose. Therefore, dexterity measures involving only one of the six (J'P) matrices are potentially bias. Four potential strategies for dexterity measurement have been proposed based on the condition number and/or singular values of the Jacobian matrices obtained in all six cases. Each measure has a distinct physical meaning, as discussed. In sum, the Jacobian matrix formulation presented in this chapter allows, for the first time, to quantitatively compare different mechanism architectures with complex degrees of freedom in terms of dexterity. Moreover, as illustrated in this chapter, the formulation is not limited to parallel manipulators as it can also be used to quantitatively compare the dexterity of different architectures as long as the end effector is represented by an equivalent set of points. Quantitative dexterity comparisons will allow robot designers to better select proper mechanisms for specific tasks.
