A four-bar linkage having a quadratic input-output equation can admit a constant value of the output variable for any value of the input variable. Such a linkage is termed constant-branch four-bar linkage in this paper. The general conditions governing the existence of this special class of linkages are derived through an in-depth analysis of the input-output equation. Geometric, mobility, and force-transmission properties are discussed with regard to most common types of planar, spherical, and spatial four-bar linkages. As an application example, a clutch mechanism, based on a constant-branch spatial RSSR four-bar linkage, is proposed.

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