The generation of two kinds of Schatz six-revolute linkages is first delineated from the view point of geometry and then, the general analytical kinematic closed-form solutions of both types are developed by matrix algebra and its differentiation for confirming the constrained motion and further application. The full cycle range of motion about the input link for these linkages is also verified by using the matrix differential closure loop equation. Furthermore, based on the six-by-six screw coordinate transformation matrix, we establish the algebraic formulas of screw coordinates of six-revolute joint axes and the single screw axis reciprocal to the five-order screw system defined by the joint axes of linkages. Then we make their algebraic and geometric analysis with the help of 3D computer graphics. Two kinds of algebraic surfaces for both linkages’ reciprocal screws are presented for understanding the real features and for proving the fact that during their cycles of movement there are no transversal to all six-revolue joint axes permanently.

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