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

This article introduces a computational framework for analyzing geometric constraint systems in planar mechanisms, built upon an extended pebble game framework tailored to the point-based model. The proposed approach synthesizes key ideas from classic geometric constraint-solving methods: degrees-of-freedom are tracked using pebbles, and directed edges capture algebraic dependencies between variables. Each geometric constraint - such as distance, line, angle, or angular relationships - is encoded as one or more edges in the constraint graph, consistent with the point-based modeling formalism. The analysis is two-phased: a reduction phase that eliminates redundant constraints and a decomposition phase that partitions the system into minimal, rigid Assur graphs. To enhance solving efficiency, the method adopts a skeleton-first, body-next strategy, supported by constraint-type-specific rules. As a result, the framework achieves the following: (1) robust handling of arbitrary mixtures of planar geometric constraints, including those involving rolling joints and circular gears; (2) efficient performance with an overall time complexity of O(|V|²) for reduction and decomposition, enabling real-time simulation suitable for a web-based computer-aided design application. By introducing conceptual edges and rule-based deduction from the point-based model, this method offers a unified and scalable tool for mobility analysis, constraint reduction, and structure-preserving decomposition of complex planar linkages.