Construction, Reduction, and Decomposition of Planar Mechanisms via an Extended Pebble Game Framework

This paper presents an advanced computational framework for the analysis and decomposition of planar mechanisms using an extended pebble game algorithm. The framework efficiently handles mechanisms comprising revolute, prismatic, and rolling joints by transforming geometric constraints into a constraint graph. A two-phase process is introduced: first, a reduction phase, where redundant constraints are eliminated, and second, a decomposition phase, where the system is broken into minimal rigid substructures using Assur graph decomposition. Additionally, an edge-cluster dictionary is introduced to extract the skeletal structure of the mechanism, optimizing the constraint-solving sequence. This method extends traditional pebble game approaches by incorporating conceptual vertices and edges to represent distance, linear, and angle-angle constraints, allowing it to analyze complex planar geared linkages. The framework maintains a quadratic time complexity, ensuring suitability for real-time kinematic simulations and CAD applications. The methodology is validated through case studies demonstrating its effectiveness in decomposing and solving planar linkage systems with multiple actuators. It supports the arbitrary combination of planar circular gears and linkages, as well as varying numbers of actuators, enabling comprehensive mobility analysis and kinematic simulation of these mechanisms.