Robustness analysis is an emerging field in the domain of uncertainty quantification. It consists of analysing the response of a computer model with uncertain inputs to the perturbation of one or several of its input distributions. Thus, a practical robustness analysis methodology should rely on a coherent definition of a distribution perturbation. This paper addresses this issue by exposing a rigorous way of perturbing densities. The proposed methodology is based the Fisher distance on manifolds of probability distributions. A numerical method to calculate perturbed densities in practice is presented. This method comes from Lagrangian mechanics and consists of solving an ordinary differential equations system. This perturbation definition is then used to compute quantile-oriented robustness indices. The resulting Perturbed-Law based sensitivity Indices (PLI) are illustrated on several numerical models. This methodology is also applied to an industrial study (simulation of a loss of coolant accident in a nuclear reactor), where several tens of the model physical parameters are uncertain with limited knowledge concerning their distributions.