Higher-Order Kinematic Analysis of a Six-Bar Mechanism with a Prismatic Joint: Centrodes and Bresse Circles

Planar linkage mechanisms remain a cornerstone of motion generation and trajectory control, yet the geometric tools that desRcribe their instantaneous behavior, namely centrodes and Bresse’s circles, have been developed almost exclusively for mechanisms with entirely revolute joints, where a sliding pair fundamentally alters the velocity and acceleration fields and disrupts the symmetries on which classical curvature theory relies. This paper presents a comprehensive higher-order kinematic analysis of a planar six-link, single-degree-of-freedom mechanism in which a slider-crank stage and a rocker stage are coupled through a shared prismatic joint that acts simultaneously as output and input. Using vector algebra and a matrix-based loop-closure formulation, the position, velocity, and acceleration analyses are derived in closed form, yielding angular velocity ratios, the instantaneous centers of rotation and acceleration of both coupler links, and their inflection and stationarity circles. The analysis reveals a distinctive geometric constraint on the slider-side coupler’s instantaneous center, a decoupling of the curvature loci of the two couplers, and degenerate configurations, linked to coupler instantaneous-stop and rocker dead-point conditions, that arise at joint-invariant crank angles. Implemented as a computational algorithm and demonstrated on a carton flap-closing mechanism and cross-validated against independent multibody simulation, the framework confirms favorable transmission and dead-point clearance behavior, extending curvature-theory tools to linkages containing sliding pairs.