Contact-rich physical human--robot interaction (pHRI) imposes time-varying demands associated with physical interaction, motor regulation, and physiological response, motivating continuous assessment of interaction workload. This paper presents an online multimodal assessment framework that integrates interaction wrench, planar tool-center-point (TCP) kinematics, and skin conductance level (SCL) into four interpretable workload-related factors. Their relative contributions are adjusted using path curvature to reflect changes in motion demand and task progression to account for gradual physiological variation over time. The framework was evaluated with 24 participants across 18 controlled combinations of temperature, acoustic noise, and illuminance under two admittance-control modes. Strict leave-one-subject-out (LOSO) evaluation used standardized pupil diameter ($\mathrm{PD}_z$) as an independent physiological reference and included comparisons with static variants and representative state-of-the-art learning-based baselines. The proposed framework achieves a cohort-mean $30\,\mathrm{s}$ block-wise Spearman correlation of $ρ_{30}=0.308$ with the physiological reference, with positive subject-level correspondence in 23 of 24 participants. Its overall performance is comparable to the state-of-the-art learning-based baseline. At the same time, our framework keeps the assessment process transparent through explicit workload-related factors and defined weighting rules, while outperforming the corresponding fixed-weight formulation. The framework also maintains consistent performance across the two tested admittance-control modes. These results support a transparent and interpretable approach to continuous interaction workload assessment in contact-rich pHRI.

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