# Improving Smartphone GNSS Positioning Accuracy Using Inequality Constraints
## 3 Kalman Filter with Inequality Constraints
### Solution of Inequality Constraint
calculation, we applied observations of the three smartphones simultaneously. In this section, we detail our calculation of the cumulative distribution function (CDF) to clearly depict and compare the positioning accuracy of the three smartphones. Since the horizontal error in smartphone positioning requires more concentration, we only considered horizontal positioning performance in the following analysis. Additionally, the horizontal CDF error is displayed in Figure 14, in which the left, middle, and right panels represent the CDF errors of HP40, HP30, and XMI8, respectively. The orange lines in the figure represent the original results, the blue lines represent the results using the distance constraint, and the red dashed lines represent the references of 50% and 95%

Figure 13: Trajectory calculated using different approaches. The blue lines represent results without constraint, the red lines represent results with direction constraint, and the green lines represent references for (**a**) HP40, (**b**) HP30, and (**c**) XMI8. CDF. Intuitively, the horizontal positioning accuracy of all the smartphones is improved using the constraint. This means that it is beneficial for all devices in the system using distance information. The enhancement of HP30 is especially significant. This is because the original accuracy of HP30 was not satisfactory, which could have been influenced by the considerable multipath error. After constraint, the performance of all smartphones is similar, as the distance is fixed. Considering the 95% CDF, the improvement percentages of the three smartphones are 34.27%, 75.58%, and 23.66%. Figure 15 displays the results of the dynamic datasets. Compared with the aforementioned experiment on the static datasets, the improvement is not obvious. This is because the dynamic data collection environment is ideal most of time. Therefore, three smartphones can achieve a satisfactory positioning in this case, and the enhancement may not always work over the data collection period. Numerically, the improvement percentages are 15.90%, 5.55%, and 0.17%, respectively. ## 6 Conclusions

Taking advantage of multi-sensor-embedded and cutting-age communication techniques, most smartphones can obtain multi-source information, and some information can be expressed in the form of inequalities. Using extra inequalities could improve the positioning accuracy and stability of smartphones. In this paper, we introduce an inequality constraint estimator and three applications using inequality constraints on smartphone GNSS positioning, including a vertical velocity constraint, direction constraint, and distance constraint. Real-world collected datasets from a Huawei P40, a Huawei P30, and a Xiaomi M18 were utilized for the method's validation. It is proven that velocity and position estimation performance can be improved in the up component using vertical velocity constraints. In particular, the position series was smoother after applying this method, as it was indirectly influenced. However, horizontal enhancement was not obvious in this application. To verify the direction constraint, we selected a period in which data were collected for analysis along a straight road. Compared with the original results, the trajectories using constraints are much closer to the reference trajectory. This means that the horizontal positioning accuracy is improved. Numerically, the horizontal RMSE result improvement percentages were 16.77%, 14.57%, and 31.09% for HP40, HP30, and XMI8, respectively. Additionally, we considered the inter-smartphone

Figure 14: Horizontal CDF error of three smartphones using static datasets. The orange lines represent results without constraint and the blue lines represent results using the distance constraint. Figure 15: Horizontal CDF error for three smartphones using dynamic datasets. The orange lines represent results without constraint and the blue lines represent results using the distance constraint. distance inequality constraint.