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

Abstract 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.