This article proposes a new skewed outlier-robust localization algorithm that is based on time-difference of arrival (TDOA) measurements at an airport. A new outlier-robust filtering framework is derived based on the skew Gaussian-gamma mixture (SGGM) distribution, where the state, a mixing parameter, a shape parameter, a scale matrix, and the degrees of freedom (DOFs) are inferred simultaneously using variational Bayesian (VB) approach. An interacting multiple-model (IMM) filter with different kinematic system models is implemented to handle the multimodal dynamics of the vehicle, yielding the IMM-SGGM algorithm. In particular, a new measurement likelihood based on the SGGM distribution is derived utilizing VB inference for the combination procedure in the proposed IMM-SGGM algorithm. Car-mounted experiments using TDOA measurements at an airport were conducted to verify the effectiveness of the proposed algorithm. The performance of the proposed IMM-SGGM algorithm is evaluated through comparisons with the state-of-the-art approaches. The experimental results demonstrate that the proposed IMM-SGGM algorithm has better localization accuracy and robustness to skewed outlier measurements than the state-of-the-art approaches.