A simple and possible approach for doing robust principal component analysis could be to replace the covariance matrix with its robust counterpart. We could then extract the eigenvalues/eigenvectors out of it and get the principal components, but this was found not to work too well.

The robust principal component analysis algorithm used for this is quite complex. A good reference on the computational details used for robust covariance matrix estimation can be found on https://arxiv.org/pdf/1506.00691.pdf.