Robotic mechanisms in general can be of either serial-chain, parallel-chain, or hybrid (a combination of both parallel and serial chains) geometry. While it can be asserted that kinematic theories and techniques are well established for fully serial-chain manipulators, the same assertion cannot be made when it is considered in the general context. In this paper, we present a general procedure for systematic formulation and characterization of the instantaneous kinematics for a robotic mechanism with a general parallel-chain geometry. A kinestatic approach based on screw system theory is adopted in this treatment, which makes intensive use of dualistic properties and relations between the velocity and statics of rigid bodies. The equations resulted from this approach is organized into a compact matrix form (Jacobian matrix) including the constraint equations. For the non-redundant parallel manipulator, a 6 by 6 matrix can describe the relationship between joint rates and end-effector velocity. And the statically singular condition can be easily checked from the final form. An example has also been provided to demonstrate the methodology as well as its theoretical feasibility.

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