Configuration representation in conformal geometric algebra for reconfigurable mechanisms

Motion bifurcation is a key characteristic of reconfigurable mechanisms, reflecting their ability to switch between motion branches. Conformal geometric algebra (CGA), by incorporating both geometric structure and input angle parameters, provides a unified framework for representing bifurcations and simplifying the modeling process. Based on CGA theory, this paper proposes a novel method for the motion bifurcation of reconfigurable mechanisms. First, feature points of motion axes of Grassmann line geometry are extracted to construct motion and constraint conditions, forming a motion-constraint model within the CGA framework. Second, using CGA invariant operations, a configuration transformation method is established, revealing the relationship between configuration changes and axis motion. Furthermore, the spherical four-bar mechanism is analyzed, where an input angle parameter model and bifurcation judgment criterion are proposed, enabling a CGA-based representation of its reconfigurable characteristics. A geometric parameter model is also developed for the line-plane symmetric Bricard mechanism. Under unconstrained conditions, two types of Bricard 6R motion branches are identified. Plane 4R, spherical 4R, and Bennett branches are found to share collinear joint axes under geometric constraints. The method verifies the feasibility of CGA in representing motion bifurcation.