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Session B6: Emerging PNT Consumer Applications

Super GNSS Odometry: Long-Horizon Relative Constraints with Time Differenced Carrier Phase Error Correction
Daiki Niimi, An Fujino, Meijo University; Taro Suzuki, Future Robotics Technology Center, Chiba Institute of Technology; Junichi Meguro, Meijo University
Location: Prince David Room
Date/Time: Thursday, Apr. 16, 2:55 p.m.

In recent years, methods utilizing time-differenced carrier phases (TDCP), which is the time difference of global navigation satellite system (GNSS) carrier phase measurements, have been researched for high-precision relative positioning of a standalone receiver. TDCP enables precise estimation because two-epoch time differencing largely cancels changes in the integer ambiguities and in carrier-phase error sources, including satellite orbit and clock errors, ionospheric delay, and tropospheric delay. This principle relies on the assumption that, over very short time intervals, the temporal variations of these errors are negligibly small and the integer ambiguity remains constant. Consequently, a jump in the carrier phase measurement, known as a cycle slip, prevents integer ambiguities from remaining constant and leads to significant degradation in relative position accuracy. Furthermore, drift error accumulated through the integration of relative displacements also presents a significant challenge.

To address these challenges, a conventional method [1] proposed a factor graph optimization (FGO) based method which tightly integrates TDCP measurements. This approach explicitly estimates the cycle slip amount for each satellite as a state variable, enabling the exact detection of cycle slips. Furthermore, it successfully suppressed time-integration drift by imposing constraints on the total displacement between two distant epochs using TDCP intervals of up to 60 seconds. Experiments using an open dataset [2] demonstrated its effectiveness, achieving an accuracy of 45 cm or better over a 25-minute flight. However, this method limited the long-duration TDCP constraints to a maximum of 60 seconds. This limitation was imposed because the accumulation of various uncorrected errors degrades the precision of TDCP as the time difference expands.

Therefore, this study aims to achieve high-precision TDCP over intervals exceeding 60 seconds by comprehensively correcting these errors. This enhancement will enable densely adding accurate, long-duration TDCP constraints between states throughout the FGO graph, thereby effectively suppressing drift and achieving high-accuracy relative positioning. To accomplish this objective, this research is structured in two main phases. First, we conduct a detailed budget analysis of the satellite orbit and clock errors, ionospheric delay, and tropospheric delay on long-duration TDCP and quantitatively evaluate the contribution of existing correction methods. Second, based on the findings of this analysis, we construct a new FGO-based relative positioning method that incorporates error corrections.

For the analysis phase, we utilize static data collected in an open-sky environment to compute TDCP for intervals up to 1000 seconds. During this process, the satellite with the highest elevation at each epoch is set as the reference satellite to eliminate the influence of the receiver clock error. With this setup, if all error sources are perfectly corrected, the TDCP residual should ideally approach zero. Consequently, comparing the variance of the TDCP residuals across various combinations of corrections will reveal which strategies are the most effective. The specific evaluation methods for each error are as follows:

1) Satellite orbit and clock errors: We apply the precise orbits and clocks from the International GNSS Service (IGS) to evaluate the error budget and correction contribution.

2) Ionospheric delay: We first visualize the error budget using the geometry-free (GF) linear combination, and then compare the corrective effects of the Klobuchar model against the ionosphere-free (IF) linear combination.

3) Tropospheric delay: We compare two methods, one using the empirical Saastamoinen model, and the other using the zenith total delay (ZTD) and gradients estimated using precise point positioning (PPP). In both cases, the zenith delay is projected onto each line-of-sight (LOS) direction using a mapping function to determine the correction amount.

The results of our analysis confirm that the growth of TDCP residuals with increasing time difference is dominated by satellite orbit error over short intervals. In contrast, over longer intervals, the primary errors degrading TDCP, in descending order of magnitude, are tropospheric delay, ionospheric delay, satellite orbit error, and satellite clock error. Furthermore, the analysis confirms that corrections based on empirical models for tropospheric and ionospheric delays are insufficient for minimizing the TDCP residuals. We conclude that achieving high-precision, long-duration TDCP requires a configuration that introduces precise orbits and clocks, corrects for tropospheric delay using estimated ZTD and gradients, and computes the TDCP from the IF linear combination of carrier phases.

Based on these conclusions, we construct a new FGO-based relative positioning model incorporating these error corrections. As input data for the FGO, we first utilize the precise ephemeris provided by IGS to correct for satellite orbit and clock errors. Next, to eliminate the influence of ionospheric delay, we compute the IF linear combination from dual-frequency carrier phases and use this resultant value for the TDCP calculation.

In the FGO state variables, the conventional method [1] defined the receiver's 3D position and clock error, along with the cumulative cycle slip amount for each satellite. In the proposed method, we augment the state variables by adding the time-varying components of ZTD and gradients. This allows us to explicitly estimate the time variations of tropospheric delay in the LOS direction for each satellite within the FGO framework. Furthermore, the estimation of cumulative cycle slips is continuously integrated to ensure robustness against cycle slips in long-duration TDCP. The state constraints utilize TDCP, which is corrected for satellite orbit/clock errors and ionospheric delay, and further incorporates the time variations of tropospheric delay into the model. Here, the most significant feature of the proposed graph structure is that, in addition to using TDCP for conventional constraints between consecutive epochs, TDCP constraints between non-sequential epochs separated by long intervals exceeding 60 seconds are also densely added throughout the entire graph. These long-duration, non-sequential constraints directly connect distant nodes in the graph, functioning as loop closures and powerfully suppressing the drift error associated with time integration.

For the evaluation, we will assess the trajectory estimation accuracy of the proposed method using the open dataset [2]. It is anticipated that, compared to the conventional method [1], our proposed approach will significantly reduce drift error and substantially improve the trajectory estimation accuracy.

[1] T. Suzuki, "GNSS Odometry: Precise Trajectory Estimation Based on Carrier Phase Cycle Slip Estimation," in IEEE Robotics and Automation Letters, vol. 7, no. 3, pp. 7319-7326, July 2022.

[2] S. Cao, X. Lu and S. Shen, "GVINS: Tightly Coupled GNSS–Visual–Inertial Fusion for Smooth and Consistent State Estimation," in IEEE Transactions on Robotics, vol. 38, no. 4, pp. 2004-2021, Aug. 2022.

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