A Dual-Quaternion Framework for Bennett-Limit Diagnostics in Rigid Kresling Origami FOLD–TWIST–FOLD Robots

This paper presents a dual-quaternion (DQ) framework for the rigid-kinematic modeling and validation of rigid hexagonal Kresling origami robots executing a prescribed FOLD–TWIST–FOLD motion. Triangular panels are modeled as rigid bodies, and crease lines are modeled as fixed revolute axes. Exact spatial 4R closure is decomposed into a primal orientation closure and a dual transported-translation closure. Under explicit non-degeneracy, paired-normal transport, and branch assumptions, the projection of the dual closure recovers a Bennett-type axis ratio. For Kresling, this result is applied only in the intersecting-axis limit d = 0, serving as a local axis-geometry diagnostic rather than a sufficient global closure criterion. The baseline three-cell module (Ns = 6, R = 48 mm, H0 = 60 mm, initial twist 30°) executes a (−2 mm, +5°, −2 mm) actuation command over 37 states, preserving rigid-edge/panel and DQ consistency within the prescribed tolerances. Comparative benchmarks show that DQ and homogeneous-transform mappings achieve practically identical accuracy, although homogeneous transforms run faster in the tested MATLAB 2025b workload; Direct LM is faster for computing the endpoint solution, whereas DQ-parametric homotopy provides state-by-state path certification. A generalized implementation evaluates 27 configurations (21 accepted, six rejected), with all six controlled actuation-order permutations accepted. The framework serves as a pre-prototyping rigid-kinematic qualification tool that separates local axis compatibility from global rigid-origami feasibility.