Jacobian linear subspace-based determination of kinematic paths for multi-degree-of-freedom multi-loop mechanisms

Spatial closed-loop and multi-loop mechanisms are common in kinematic analysis and mechanism design. However, tracing their multi-degree-of-freedom kinematic paths under intricate geometric constraints remains challenging. Traditional branch-and-prune methods suffer from exponential computational complexity and box-clustering effects, whereas conventional Singular Value Decomposition (SVD)-based approaches struggle to accommodate multi-loop coupling and variable topologies. To overcome these limitations, this paper presents a systematic path-tracing algorithm based on the fundamental linear subspaces of a globally coupled Jacobian matrix. Integrating SVD with breadth-first search, the method exploits the right null space for motion prediction, the left null space for formulating higher-order constraint equations at singular configurations, and the row and column spaces for geometric error correction. Within this framework, higher-order kinematic analyses are reformulated as systems of quadratic equations, enabling the automatic identification of singular sets, path bifurcations, and mode transitions without explicit analytical derivations. The proposed methodology exhibits near-linear computational growth with increasing solution accuracy while preserving geometric fidelity even with relatively large step sizes. Its effectiveness and robustness are demonstrated through applications to several representative mechanisms, including classical spatial, parallel, and DNA-inspired origami mechanisms.