Algebraic Type Synthesis of Reconfigurable Mechanisms via Gröbner Cover

Abstract

Conventional type synthesis of reconfigurable mechanisms mainly relies on case-specific geometric methods, which are mostly confined to single-loop mechanisms and structurally regular topologies. To overcome this limitation, this paper proposes an algebraic framework for the type synthesis of reconfigurable mechanisms, extending synthesis capabilities to multi-loop mechanisms. By combining redundant constraint detection, parameterized higher-order kinematic constraints, and Gröbner Cover theory, the framework formulates the synthesis process as a systematic algebraic decomposition of the parameter space. Applied to a planar four-bar linkage, the framework yields a reconfigurable mechanism that preserves its original planar and coaxial motion modes while introducing new spherical and 7R motion modes. Subsequently, the framework is applied to a 3-RPR parallel mechanism, with joints inserted to generate 3-UPU topologies. The resulting Gröbner Cover decomposition yields 12 distinct reconfigurable topologies, of which 3 are symmetric and 9 are asymmetric. Of these, 5 topologies contain an inserted joint numerically observed to remain passive across all sampled motion branches, suggesting its removability for structural simplification. This framework establishes a mathematical foundation for the systematic algebraic synthesis of multi-loop reconfigurable mechanisms.