This paper presents a coordinate-invariant differential geometric analysis of manipulability for closed kinematic chains containing active and passive joints. The formulation treats both redundant and nonredundant mechanisms, as well as over-actuated and exactly actuated ones, in a uniform manner. Dynamic characteristics of the mechanism and manipulated object can also be naturally included by an appropriate choice of Riemannian metric. We illustrate the methodology with several closed chain examples, and provide a practical algorithm for manipulability analysis of general chains.

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