Logistic regression is usually fitted via maximum likelihood, but it can be used here. We need to adjust the ML technique in order to get robust estimates. The robust package implements several algorithms that can be chosen using the method= option, such as: Mallows, cubif, and misclass. Essentially, Mallows downweights the observations that have covariates with a high leverage and misclass assumes that there are certain observations that will be missclassified. 

Detailed algorithms can be found on the following websites:  https://www.jstor.org/stable/2345763?seq=1#page_scan_tab_contents and https://pdfs.semanticscholar.org/0687/32a9fb924a9360254e6ed9384707c8ee92bf.pdf.