Mobility of Single-Loop Kinematic Mechanisms Under Differential Displacement

Linear analysis of motion screws is a means for determining the mobility of a mechanism composed of the parallel combination of serial kinematic chains. Such a mechanism may have one or more degrees of freedom that vanish after a differential displacement from a reference posture. The Lie product, also called the Lie bracket, is known to give the derivative of a motion screw with respect to the displacement along an upstream screw in a serial chain. Serial chains having motion screws that are closed under the Lie product are known to retain their mobility after differential displacement. For a single-loop mechanism, which is composed of a pair of chains that are not closed under the Lie product, mobility is retained when the Lie closures of those chains are within the span of the union of motion screws of the two chains, a new result determined by applying the Baker–Campbell–Hausdorff expansion to the motion screws of the serial chain. When the Lie products have at most one dimension outside the union span, a second-order expression of mobility reduces to a quadratic form, allowing the numerical characterization of constraint singularities under that condition.