  Ego-localization is a crucial task for autonomous vehicles. On the one hand,
it needs to be very accurate, and on the other hand, very robust to provide
reliable pose (position and orientation) information, even in challenging
environments. Finding the best ego-position is usually tied to optimizing an
objective function based on the sensor measurements. The most common approach
is to maximize the likelihood, which leads under the assumption of normally
distributed random variables to the well-known least squares minimization,
often used in conjunction with recursive estimation, e. g. using a Kalman
filter. However, least squares minimization is inherently sensitive to
outliers, and consequently, more robust loss functions, such as L1 norm or
Huber loss have been proposed. Arguably the most robust loss function is the
outlier count, also known as maximum consensus optimization, where the outcome
is independent of the outlier magnitude. In this paper, we investigate in
detail the performance of maximum consensus localization based on LiDAR data.
We elaborate on its shortcomings and propose a novel objective function based
on Helmert's point error. In an experiment using 3001 measurement epochs, we
show that the maximum consensus localization based on the introduced objective
function provides superior results with respect to robustness.
