Absement: Quantitative Assessment of Metabolic Cost during Quasi-Isometric Muscle Loading

Abstract

Small deviations during nominally isometric loading can be separated into a sustained mean offset and fluctuations about that offset. We develop a local quasi-static model that connects these video-accessible kinematic quantities to metabolic energy. Muscle activation is eliminated through joint-moment equilibrium, and the resulting metabolic power is reduced to a smooth function of a locally invertible muscle-length coordinate. For the deviation \(x(t)=\ell(t)-\ell_0\), the reduced energy satisfies \[ \mathcal{E}_{\mathrm{met}}[\ell] = P_0T + C_1\Delta\mathcal{A}_{\ell} + C_2\mathcal{M}_{2,\ell} + R_3, \qquad |R_3| \leq KT\|x\|_{L^\infty(0,T)}^3, \] where \(\Delta\mathcal{A}_{\ell}=\int_0^T x(t)\,dt\) is signed deviation absement and \(\mathcal{M}_{2,\ell}=\int_0^T x(t)^2\,dt\) is the second raw integral moment. Equivalently, if \(\mu_\ell\) and \(\sigma_\ell^2\) are the mean and variance of the observed length deviation, then \(\Delta\mathcal{A}_{\ell}=T\mu_\ell\) and \(\mathcal{M}_{2,\ell}=T(\mu_\ell^2+\sigma_\ell^2)\). A video-based protocol can therefore estimate the required predictors without differentiating the recorded trajectory. Within the autonomous quasi-static model, periodic variation does not require a separate cycle-specific predictor: its mean contributes through absement and its dispersion through the second moment. This moment-based protocol, rather than the Taylor expansion alone, provides the experimentally testable result: residual dependence on frequency after control for the first two moments would identify the limit of the quasi-static reduction.