Based on derived matrix loop equations and their algebraic manipulations, we obtain the necessary mobility criteria, and prove that there are only three types of movable spatial 6R mechanisms with three adjacent parallel axes. The first type is a general form of Sarrus 6R mechanism. The second type can be regarded as a combination of spherical and planar 4R mechanisms. The third type is a hybrid linkage, first mentioned by Waldron. All these three types are proper linkages confirmed by iteration and optimization methods.

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