Harmonic and polar screw invariants in single-case knee gait dynamics

Introduction: Knee stability during gait depends on coordinated interaction between segmental motion and environmental support. This study introduces a screw-theoretic framework for interpreting knee joint dynamics through harmonic and polar relations linking instantaneous screw axes (ISAs) and ground reaction forces (GRFs). Materials and methods: A proof-of-concept reanalysis was conducted using a previously published Grand Challenge instrumented-knee dataset from one adult male participant with a reconstructed knee. ISAs of the thigh, shank, and knee were reconstructed during stance, and screw pitch and reciprocal organization were examined in relation to measured ground reaction and knee contact forces. Results: In this single case, the knee ISA formed a structured bundle within the screw system generated by the thigh and shank. The knee pitch varied from approximately − 0.7 to 21.4 mm/rad and approached near-zero values toward terminal stance. The shank screw remained tightly constrained near zero pitch, whereas the thigh screw showed broader pitch modulation. Previously reported validation of the same reciprocally connected knee model showed root-mean-square errors (RMSEs) of 148.1 N and 147.3 N for lateral and medial contact forces, respectively, with a predicted peak medial force of 1092 N. Conclusions: The findings support the feasibility of describing knee function through screw-theoretic relations between motion and support. Harmonic conjugacy and polar reciprocity are proposed as geometric principles for evaluating natural and artificial knee systems, anterior cruciate ligament (ACL) reconstruction, rehabilitation, and slip–fall risk. Because the analysis is based on a single dataset, multi-subject and perturbation-based validation is required.