Design and spatial analysis of a generalized polygon-based crossing pivot for regulating performance

Compliant rotational joints have been widely applied in precision mechanisms, where over-constrained sheets are frequently employed to regulate stiffness. However, the effects of the position space of these over-constrained sheets on the stiffness variation remain insufficiently characterized. Furthermore, merely evaluating stiffness along a single direction fails to provide an objective assessment of their kinetostatic performance. To address these limitations, this paper introduces a generalized joint featuring three or five elastic sheets in a polygon arrangement, and two such joints are then combined to form a new generalized polygon-based crossing pivot. The nonlinear spatial analytical models are developed for the generalized triangular and quadrilateral joints based on the Beam Constraint Model and Free Body Diagram. These models are subsequently extended to characterize the polygon-based crossing pivot. A key advantage of the analytical model for the crossing pivot is its explicit incorporation of the position space between the two constituent triangular/quadrilateral joints. Specifically, each generalized triangular/quadrilateral joint can be placed at any position relative to the rotational center of the crossing pivot and oriented at any crossing angle between 0° and 180°. Utilizing the proposed analytical models, the geometric effects (i.e., position space) of the over-constrained sheets on the joint stiffness are systematically investigated using normalized stiffness ratios to ensure an objective comparison across different designs. The normalized twisting stiffness ratios of the triangular and quadrilateral joints are investigated under three types of geometric configurations. Subsequently, the normalized twisting stiffness ratio of the polygon-based crossing pivot comprising two representative quadrilateral joints, also called quadrilateral crossing pivot (QCP), is analyzed with respect to varying crossing positions and crossing angles. The accuracy of the above analysis is validated through nonlinear finite element analysis (FEA), yielding excellent agreement, with errors strictly bounded between 1.6% and 8.5%. Optimization results under three types of loading conditions demonstrate that the QCP can reduce center shift by 75% (or 62%, depending on the loading condition) compared to a conventional crossing pivot when out-of-plane loads exist. Finally, experimental testing on the quadrilateral joint and QCP confirms the accuracy of the analytical and FEA models. Notably, the developed prototype of the QCP enables four basic units to be assembled into 47 distinct configurations, highlighting its exceptional reconfigurability for precision applications.