The geometry of infinitesimal mobility of closed-loop linkages

Classical mobility criteria such as the Chebyshev-Grübler-Kutzbach formula capture only first-order behaviour and may overlook geometric constraints of higher order. In this work we examine a large family of spatial closed-loop mechanisms that, despite satisfying the classical mobility count, are found to be rigid. Motivated by unexpectedly elegant geometric structure underlying this behaviour, we introduce the term hypo-paradoxical linkages to describe these mechanisms. Our analysis combines screw theory with geometric considerations to identify when such higher-order restrictions arise and how they suppress motion. Representative examples are examined, and the effects of small design perturbations on the attainable workspace are quantified. This geometric viewpoint also offers a fresh and intuitive interpretation of mobility of Bennet's mechanism. Overall, the results illustrate the limitations of traditional mobility criteria and point to a richer geometric foundation governing rigidity and motion in spatial linkages.