Fatigue and effort estimation

"The muscle told me the same story twice: it is trying harder, and it is getting slower. I only had to notice that both sentences pointed at the word tired."

A Perceptive AI Agent

Why a tiring muscle changes its electrical signature

Surface EMG is the summed electrical activity of many motor units firing under the electrode. Two physiological changes during sustained effort reshape that sum in opposite frequency directions, and both are exploitable. First, as metabolic byproducts accumulate and muscle-fiber membranes acidify, the conduction velocity of the action potentials propagating along the fibers slows down. A slower-traveling waveform is a stretched-in-time waveform, and stretching in time compresses in frequency: the whole power spectrum slides toward lower frequencies. Second, to keep producing force as individual fibers weaken, the nervous system recruits additional motor units and increases firing rates, so the summed amplitude climbs. The signature of fatigue during a constant-force hold is therefore unmistakable and paradoxical: amplitude up, frequency down, force flat or dropping.

This is why a single number is never enough. Amplitude alone cannot distinguish "working harder" from "fatiguing," because both raise it. Spectrum alone cannot measure effort level, because a light and a heavy contraction can share a median frequency early on. You need the joint motion of the two, and the classic way to visualize it is the JASA plane (joint analysis of spectrum and amplitude): plot amplitude change against spectral change, and the quadrant you land in separates increased force (both up), decreased force (both down), fatigue (amplitude up, spectrum down), and recovery (amplitude down, spectrum up). One scalar hides the diagnosis; the two-dimensional motion reveals it.

Measuring the spectral slide

The workhorse fatigue indices summarize where the power sits. The mean frequency and median frequency of the power spectral density \( P(f) \) are

During a fatiguing isometric contraction both fall roughly linearly, and the slope of that decline (in Hz per second, or normalized as percent per minute) is the single most used fatigue rate. Median frequency is preferred over mean frequency in practice because it is more robust to additive noise and to the power-line and movement artifacts that plague field recordings. For higher sensitivity, the Dimitrov spectral fatigue index weights the spectrum toward low frequencies by taking the ratio of a negative-order spectral moment to a high-order one, amplifying exactly the low-frequency accumulation that early fatigue produces. The code below fits a fatigue rate to a simulated sustained hold.

import numpy as np
rng = np.random.default_rng(0)

fs, dur = 1000, 30.0                      # 1 kHz sEMG, 30 s sustained hold
t = np.arange(int(fs * dur)) / fs
# Fatigue model: dominant frequency slides 90 Hz -> 60 Hz, amplitude rises 1.0 -> 1.6
f_inst = 90 - 30 * (t / dur)
amp    = 1.0 + 0.6 * (t / dur)
emg = amp * np.sin(2 * np.pi * np.cumsum(f_inst) / fs)
emg += 0.15 * rng.standard_normal(emg.size)          # measurement noise

def median_freq(x, fs):
    f = np.fft.rfftfreq(x.size, 1 / fs)
    p = np.abs(np.fft.rfft(x * np.hanning(x.size))) ** 2
    c = np.cumsum(p)
    return f[np.searchsorted(c, c[-1] / 2)]

win = fs                                              # 1 s windows, no overlap
mdf = np.array([median_freq(emg[i:i + win], fs)
                for i in range(0, emg.size - win, win)])
rms = np.array([np.sqrt(np.mean(emg[i:i + win] ** 2))
                for i in range(0, emg.size - win, win)])
tw = np.arange(mdf.size)                              # window index in seconds
slope = np.polyfit(tw, mdf, 1)[0]                     # Hz per second
print(f"MDF fatigue slope: {slope:+.2f} Hz/s   (negative = fatiguing)")
print(f"RMS trend:         {np.polyfit(tw, rms, 1)[0]:+.3f} /s (positive = amplitude rising)")

Run it and the median-frequency slope comes out clearly negative while the RMS trend is positive: the JASA fatigue quadrant, recovered from raw samples. In a real study you would regress MDF against time per subject and report the slope, because the absolute frequency depends on electrode placement, subcutaneous fat, and muscle, none of which are comparable across people. The rate of change within a recording travels better than any absolute value, a theme that returns in Chapter 12 and again for cross-subject transfer in Section 32.5.

From EMG to effort and force

Effort estimation maps EMG amplitude to a percentage of maximal capacity or to an absolute force. The starting point is normalization: raw millivolts are meaningless across sessions and people, so you express amplitude as a fraction of a maximum voluntary contraction (%MVC) recorded for that muscle and placement. Even normalized, the amplitude-to-force relation is not a straight line: it is roughly linear for small muscles and distinctly nonlinear for large ones, it depends on muscle length and contraction speed, and it saturates near maximal effort. This is why serious effort models learn a regression rather than assume proportionality, using the amplitude, spectral, and envelope features of Chapter 8 as inputs and a measured force or perceived-exertion rating as the target.

The uncertainty of that map matters as much as its point estimate. A prosthesis that reports "40 percent effort, plus or minus 5" behaves very differently from one that reports "40, plus or minus 30," and the difference should gate how aggressively a downstream controller acts. The calibrated-uncertainty machinery of Chapter 18 applies directly: an effort estimate without a trustworthy interval is a number no safety-critical loop should act on.

Dynamic contractions and cumulative fatigue

The clean spectral-compression story assumes a stationary signal: a fixed-force isometric hold where the spectrum is well defined and slides slowly. Real movement violates that assumption constantly. During a dynamic contraction the muscle length, the electrode-to-fiber geometry, and the force all change within a second, so a single windowed PSD mixes several physiological states and the median-frequency estimate becomes unreliable. The remedy is the time-frequency toolkit of Chapter 7: an instantaneous median frequency tracked with short-time or wavelet analysis, or a model that conditions the fatigue read on the current joint angle and load so that like is compared with like. Comparing the spectrum at the top of a lift against the spectrum at the bottom is comparing two different muscles.

Over a long shift or a rehabilitation session, fatigue is a slow-drift detection problem, not a snapshot. A robust monitor does not threshold an instantaneous median frequency; it estimates the trend and raises a flag when the downward slope steepens or fails to recover during rest, which is exactly the change-detection framing of Chapter 12. Framing fatigue as change detection on a slope, rather than classification on a window, is what makes it survive the noise of real recordings.

What's Next

In Section 32.5, we confront the reason absolute EMG features never travel: electrode placement, skin, and anatomy differ across sessions and people, so a model trained on you rarely works on me. We build the calibration, normalization, and domain-adaptation strategies that let fatigue and gesture models survive a new day and a new body.