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Using internal and external sensors to provide position estimates in a two-dimensional space is necessary to solve the localization and navigation problems for a robot or an autonomous vehicle (AV). Usually, a unique source of position information is not enough, so researchers try to fuse data from different sensors using several methods, as, for example, Kalman filtering. Those methods need an estimation of the uncertainty in the position estimates obtained from the sensory system. This uncertainty is expressed by a covariance matrix, which is usually obtained from experimental data, assuming, by the nature of this matrix, general and unconstrained motion. We propose in this paper a closed-form expression for the uncertainty in the odometry position estimate of a mobile vehicle, using a covariance matrix whose form is derived from the cinematic model. We then particularize for a nonholonomic Ackerman driving-type AV. Its cinematic model relates the two measures being obtained for internal sensors: the velocity, translated into the instantaneous displacement; and the instantaneous steering angle. The proposed method is validated experimentally, and compared against Kalman filtering.