This article studies a robust expectation maximization (EM) algorithm to solve the problem of hypersphere fitting. This algorithm relies on the introduction of random latent vectors having independent von Mises-Fisher distributions defined on the hypersphere and random latent vectors indicating the presence of potential outliers. This model leads to an inference problem that can be solved with a simple EM algorithm. The performance of the resulting robust hypersphere fitting algorithm is evaluated for circle and sphere fitting with promising results in terms of both estimation performance and computation time.