Metrics, in plain words
How Inertial computes grade-adjusted pace
We use the Minetti 2002 polynomial for the energy cost of running at a gradient, evaluated over 30 m distance windows, and divide moving time by the equivalent flat distance. Gradients are clamped to plus or minus 45%, which is the range the polynomial was fitted over.
| What | Number | The qualifier |
|---|---|---|
| Cost model | Minetti et al. 2002 | published science only, no proprietary curve |
| Flat reference cost | 3.6 J/kg/m | the polynomial at gradient 0 |
| Grade window | 30 m of distance | suppresses altitude noise before differencing |
| Valid gradient range | -45% to +45% | clamped to the fitted domain |
The formula
Minetti and colleagues measured the energy cost of running across slopes from -45% to +45% and fitted a fifth-order polynomial. We use it unchanged:
C(i) = 155.4i^5 - 30.4i^4 - 43.3i^3 + 46.3i^2 + 19.5i + 3.6i is the gradient as rise over run. C is joules per kilogram per metre. At gradient zero it gives 3.6, which is our flat reference.
Each segment of your run contributes run x C(i) / 3.6 to an equivalent flat distance. Grade-adjusted pace is your moving time divided by that total.
| Gradient | C(i) | Multiplier vs flat |
|---|---|---|
| -10% | 2.6 | 0.72 |
| -5% | 2.9 | 0.81 |
| 0% | 3.6 | 1.00 |
| +5% | 4.7 | 1.31 |
| +10% | 6.0 | 1.66 |
Why a 30-metre window
Altitude from a barometer or from satellites is noisy. Differencing two noisy altitudes over a short distance produces a wildly wrong gradient.
So we do not compute a gradient until at least 30 m of distance has passed since the last altitude sample we used. Any tail shorter than that at the end of a run is counted as flat, which is a small and unbiased approximation.
What this gets wrong
Downhills. The polynomial says a moderate descent is cheaper than flat, and it is. But past a point, braking costs more than gravity gives, and the model does not capture the eccentric muscle damage that makes a fast descent hurt tomorrow.
Trails. Uneven ground costs energy that has nothing to do with gradient. Adjusted pace on technical terrain will flatter you.
Bad altitude data. If a device reports altitude poorly, the gradients are wrong and so is the output. We return no value at all rather than a made-up one when there is no altitude stream.
What we will not do
We do not calibrate grade adjustment against your heart rate. That approach exists and it is more personal, but it is also inside a patent boundary we have chosen to stay outside. Published science only.
Sources
- Minetti AE et al. Energy cost of walking and running at extreme uphill and downhill slopes. J Appl Physiol 93(3):1039-1046, 2002 primary
- Vernillo G et al. Biomechanics and physiology of uphill and downhill running. Sports Med 47(4):615-629, 2017 primary
- Giovanelli N et al. Energetics of vertical kilometer foot races. Eur J Appl Physiol 116(6):1179-1187, 2016 primary
Changelog
- — First published.