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The Auger Engineering Radio Array (AERA), part of the Pierre AugerObservatory, is currently the largest array of radio antenna stations deployedfor the detection of cosmic rays, spanning an area of $17$ km$^2$ with 153radio stations. It detects the radio emission of extensive air showers producedby cosmic rays in the $30-80$ MHz band. Here, we report the AERA measurementsof the depth of the shower maximum ($X_\text{max}$), a probe for masscomposition, at cosmic-ray energies between $10^{17.5}$ to $10^{18.8}$ eV,which show agreement with earlier measurements with the fluorescence techniqueat the Pierre Auger Observatory. We show advancements in the method for radio$X_\text{max}$ reconstruction by comparison to dedicated sets of CORSIKA/CoREASair-shower simulations, including steps of reconstruction-bias identificationand correction, which is of particular importance for irregular or sparse radioarrays. Using the largest set of radio air-shower measurements to date, we showthe radio $X_\text{max}$ resolution as a function of energy, reaching aresolution better than $15$ g cm$^{-2}$ at the highest energies, demonstratingthat radio $X_\text{max}$ measurements are competitive with the establishedhigh-precision fluorescence technique. In addition, we developed a procedurefor performing an extensive data-driven study of systematic uncertainties,including the effects of acceptance bias, reconstruction bias, and theinvestigation of possible residual biases. These results have beencross-checked with air showers measured independently with both the radio andfluorescence techniques, a setup unique to the Pierre Auger Observatory.

Radio Measurements of the Depth of Air-Shower Maximum at the Pierre Auger Observatory