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### Session C6: Harsh Urban and Indoor GNSS

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**Factor Graph Optimization-Based RTK-GNSS for Urban Positioning**

*Taro Suzuki, Chiba Institute of Technology*

**Date/Time:** Friday, Sep. 20, 2:12 p.m.

**I. Introduction**

Positioning methods that use the double-differenced (DD) GNSS carrier phase measurements between reference stations and between satellites can estimate absolute positions with centimeter accuracy by estimating the integer ambiguity of the DD carrier phase. These integer ambiguity estimation-based positioning methods, called real-time kinematic (RTK) or post-processing kinematic (PPK) GNSS, are widely used in surveying, vehicle and robot automation, and drone mapping applications.

In RTK GNSS, the process is usually to estimate the ambiguity of the DD carrier phase in float, and then convert the float ambiguity to an integer. If this integer ambiguity cannot be resolved, or if an incorrect integer ambiguity is estimated, the positioning accuracy will be significantly degraded. However, in urban environments, multipath effects often make it impossible to correctly estimate integer ambiguity due to noise in the carrier phase measurements. Various ambiguity resolution algorithms, such as integer least squares, have been proposed to convert float ambiguities to integers, but the success rate of integer ambiguity resolution is highly dependent on the accuracy and variance of the previously estimated float ambiguities.

The Kalman filter (KF) has been widely used as a method for obtaining float solutions using the DD carrier phase for RTK-GNSS. On the other hand, state estimation methods based on optimization techniques such as factor graph optimization (FGO), which has been developed in the robotics field, have recently attracted attention in the GNSS field. GNSS positioning methods using FGO have superior position estimation accuracy in both real-time and post-processing compared to methods using KFs.

In addition, attention has been focused on methods that use carrier phase observation as a factor, which is more accurate than the method that uses a factor of pseudorange measurements, which is often used in FGO. One such study is to use FGO for integer ambiguity estimation of the DD carrier phase. For example, in a previous study [1-3], the DD carrier phase ambiguity is added to the factor graph as an estimated state, and a float solution is estimated by optimization using the DD carrier phase factors. However, in these methods, the integer ambiguity resolution is not integrated into the graph optimization calculation, and FGO only estimates the float ambiguities and the float solution.

Therefore, we propose a new method for estimating the ambiguity of the DD carrier phase using FGO to estimate the position with high accuracy in urban environments. The proposed method is characterized by the inclusion of an integer ambiguity resolution of the DD carrier phase within the optimization calculation, and the integer ambiguity constraint changes the graph structure, resulting in a higher determination rate of the integer ambiguity than conventional methods.**II. Proposed Method**

The key of the proposed method is to estimate the float ambiguity and variance of the DD carrier phase by using a factor graph, convert the ambiguity to an integer in the iterative optimization calculation, and add a new integer constraint to the graph when the integer ambiguity is obtained, thus dynamically changing the graph structure. When integer ambiguity is obtained, the ambiguity and 3D position can be strongly constrained to the correct integer ambiguity and correct position. By using relative constraints between positions (e.g., by Doppler or time-differenced carrier phase (TDCP) constraints) and relative constraints between ambiguities, it is possible to improve the final ambiguity resolution success rate and position estimation accuracy by propagating the added integer constraints to other state variables.

The proposed method is implemented in the following steps.

(1) Construct a factor graph from GNSS observations. Here, (i) single-differenced (SD) carrier phase factor, (ii) DD carrier phase factor, (iii) DD pseudorange factor, (iv) SD TDCP factor, and (v) SD Doppler factor are used to construct the graph. The SD carrier phase ambiguity (estimated in floats) of each signal for each satellite, 3D position, and SD receiver clock bias are estimated as the estimation state. The SD receiver clock bias is added to the estimated state because of the SD TDCP and SD Doppler factors, which do not differenced the observations between satellites.

(2) Perform the first optimization computation step during the optimization iterations. For each state, the ambiguity estimates and covariance matrices are computed for all satellites. Then, in each state, the integer ambiguity is resolved using the LAMBDA method, which is an integer least squares ambiguity solution method. A ratio test is used to determine whether the integer ambiguity estimation is successful or not.

(3) If the ambiguity is successfully converted to an integer, an integer constraint (integer ambiguity factor) is added to the ambiguity successfully converted to an integer, and a fixed position factor is added to its 3D position to change the graph structure.

(4) (2) The optimization calculation and integer ambiguity resolution, (3) the addition of integer constraints to the graph, are repeated until convergence.

By incorporating integer ambiguity resolution into the optimization computation, the above approach allows the benefits of successful integer ambiguity resolution to be fully utilized for other state estimation, thus improving the success rate of integer ambiguity resolution in an urban environment.**III. Experiments and Anticipated Results**

The proposed method is evaluated in static and mobile tests in an urban environment. Three methods will be compared: (1) RTK-GNSS using conventional KF, (2) a method that estimates integer ambiguity after estimating the float solution by FGO, and (3) a proposed method that includes the ambiguity resolution inside the FGO. For the static test, a GNSS antenna was set up in a multipath environment surrounded by buildings, and data acquisition was conducted for one hour. For the mobile test, a GNSS antenna receiver was mounted on a vehicle, and the data set was acquired while driving in an urban environment. Here, Applanix's POS/LV, a vehicle-mounted GNSS/INS combined position and attitude estimation system, was used as the position reference to evaluate the accuracy of the proposed method.

The RTK-GNSS using conventional KF results in a low ambiguity fix success rate due to multipath in urban environments. The expected result of the proposed method is that the use of FGO improves the accuracy of the float solution compared to KF, and furthermore, the inclusion of the integer ambiguity resolution method within the optimization improves the overall ambiguity fix success rate from relative constraints between states. In addition, we plan to show that the accuracy of position estimation greatly exceeds that of conventional RTK-GNSS.**IV. Conclusion**

In this paper, we propose an optimization-based RTK-GNSS method. The proposed method estimates the carrier phase ambiguity by FGO, performs ambiguity resolution inside the optimization calculation, adds integer constraints to the graph, and optimizes the graph while changing the graph structure. The proposed RTK-GNSS using FGO will be quantitatively evaluated and shown to perform better than the conventional RTK-GNSS using KF in the final paper.**REFERENCES**

[1] Gao, Han, et al. "Robust GNSS real-time kinematic with ambiguity resolution in factor graph optimization." Proceedings of the 2022 International Technical Meeting of The Institute of Navigation. 2022.

[2] Wang, Xuanbin, et al. "Factor graph optimization-based multi-GNSS real-time kinematic system for robust and precise positioning in urban canyons." GPS Solutions 27.4 (2023): 200.

[3] Geng, Jianghui, Chiyu Long, and Guangcai Li. "A Robust Android GNSS RTK Positioning Scheme Using Factor Graph Optimization." IEEE Sensors Journal (2023).

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