Mechanical power for trail and mountain running — Introduction of a parametric model

Here, we present a model for calculating the mechanical power delivered by the metabolism while running in alpine settings. The aim is to quantify power for various conditions in trail and mountain running, using data from modern consumer sports watches and complementary acceleration sensors. After extending an existing analytical model by including collision losses, speed-dependent elastic energy storage, a force-rate contribution, and an additional power component accounting for upper-body movement, we generalize it to running uphill and downhill. We introduce additional power components to account for the increased metabolic work required to run on uneven technical sections. On flat terrain and when assuming an elasticity coefficient as motivated from the literature, our approach predicts a comparable output between cycling and running for athletes similarly trained in both disciplines. As for running uphill and downhill, we obtain a very good agreement with the measured metabolic cost. This corresponds to the first time an algebraic model explains this inclination dependency. We evaluate our model using data from an exemplary trail run recorded with a standard consumer setup. This work provides a transparent, physics-based model grounded in scientific principles, offering an interpretable, reproducible, and adaptable framework for estimating mechanical power across diverse trail running conditions.