## 1 Introduction  
Today's most outstanding technology for road vehicle positioning is global navigation satellite systems (GNSS). Among all GNSS, the US GPS (global positioning system) is the most popular. GNSS systems provide absolute positioning based on the information gathered at the GNSS receiver from the GNSS satellite constellation. Because of this, GNSS is subject to relevant problems such as lack of satellite visibility, multipath problems due to the reflection of the satellite signals on surfaces around the receiver's antenna, jamming, or spoofing [1]. For this reason, it is recommended that the positioning system features some other sensors that, together with the GNSS, provide an overall better performance. The most common configuration is the GNSS, plus the odometry of the vehicle and inertial sensors for attitude determination (a gyro). The values of travelled distance and the heading provided by the odometry and the gyro respectively, work together to provide a dead reckoning (DR) position that supports the GNSS in case of no coverage, and add redundant values to improve the reliability of the whole system [2]. The extra cost of this sensor set is relatively low, as vehicles have already embedded odometry and gyroscopes.  
The main concept of DR is that the estimation of the vehicle position is based on the previous one, as spatial increments are calculated from the sensor values. Consequently, the necessary integration process leads to drifts when no absolute updates are obtained. Odometry errors may be due to mechanical or electrical (quantization) issues; tire wear, which changes the wheel diameter over time; and slippages and slides. Among these, the latter is the one that accounts for larger errors in the provision of instant positioning. However, GNSS/DR fusion algorithms do not usually account for slide errors. Therefore, when it happens, the solution is not only more inaccurate, but also inconsistent,since the errors considered by the algorithm are underestimated. This may lead to the malfunction of applications that are based on the confidence of the position solution. For instance, in lane-level applications, the confidence on the system on being on a specific lane at a given instant is crucial: the knowledge of the goodness of the position is as important as the position itself to decide whether a certain action must be taken. This is the concept of position integrity [3].  
This article presents a multiple-model particle filter (MMPF) based solution that features two different models running in parallel, one that accounts for slides errors, and another one that does not, choosing at any time which model represents better the vehicle behavior. The simple concept behind this approach lays on the fact that particles will follow different behaviors depending on their assumed model and probability density function, and only those that are a better match with the vehicle motion will remain strong (with relevant weights), serving as the germ for new offspring particles.  
The rest of the paper is organized as follows. Section 2 analyzes the most relevant related works. Later, the model for the slide error is introduced in Section 3. Sections 4 and 5 present the formulations of both the MMPF implementation and the consistency checks, respectively. Section 6 shows and discusses the results. Finally, Section 7 concludes the paper.