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Recently a number of measures of mechanism dexterity have been proposed in the robotics literature. These are defined as functions of the configuration, and are primarily promoted as criteria which should be optimized in the planning of robot motions. In this paper, a coordinate-free version of the manipulatability index is presented. This permits the index to be evaluated in a meaningful way for a (kinematically redundant) mechanism operating under imposed functional constraints on the joint variables. Such an intrinsic quantity is useful for deciding which of several possible constraints yields the best performance of a given mechanism, and conversely, it may be employed as a constraint oriented figure-of-merit in the design of kinematically redundant manipulators. It also provides a new computational tool for finding the singularitites inherent in most approaches to redundancy resolution by means of functional constraints.