This paper investigates the observability properties of a mobile robot that moves on a planar surface by satisfying the unicycle dynamics and that is equipped with exteroceptive sensors (visual or range sensors). In accordance with the unicycle dynamics, the motion is powered by two independent controls, which are the linear and the angular speed, respectively. We assume that both these speeds are known. We consider the case when the robot motion is affected by a disturbance (or unknown input) that produces an additional (unknown and time dependent) robot speed along a fixed direction. The goal of the paper is to obtain the observability properties of the state that characterizes the robot configuration. The novelty of this observability analysis is that it takes into account the presence of an unknown and time dependent disturbance. Previous works that analyzed similar localization problems, either did not consider the presence of disturbances, or assumed disturbances constant in time. In order to deal with an unknown and time dependent disturbance , the paper adopts a new analytic tool [18]. This analytic tool is the solution of a fundamental open problem in control theory (the Unknown Input Observability problem in the general nonlinear case). We show that the application of this analytic tool is very simple and can be implemented automatically. Additionally, we simulate the aforementioned system and we show that a simple estimator based on an Extended Kalman Filter provides results that fully agree with what we could expect from the observability analysis.