# An Algorithm to Assist the Robust Filter for Tightly Coupled RTK/INS Navigation System
## 1 Introduction
This is because of the above-mentioned function of INS: TC RTK/INS integration can provide more accurate, reliable, and continuous navigation solutions in urban areas [40; 41; 42]. As mentioned above, the GNSS/INS integration is usually based on EKF. The process and observation noises are both assumed to be zero-mean multivariate Gaussian noises. The parameters of the covariance matrices of the noise have a large influence on the performance of EKF. However, urban areas are often GNSS-hostile areas, which leads to gross errors, including noise and multipath effects in GNSS measurements. The GNSS faults, also known as GNSS outliers, can cause degeneration or even divergence of the GNSS/INS integration [43]. As a result, a reliable algorithm to deal with the GNSS outliers is important, especially for high-precision GNSS techniques, including RTK [39]. Many researchers have studied algorithms to down weight and exclude the outlying measurements to weaken their effects. Test-based outlier detection and adaptively robust estimation are two main streams to handle outlying measurements [44]. Test-based outlier detection generally utilizes the filter's predicted residuals or innovations to form test statistics [45]. The innovation \begin{table}
\begin{tabular}{l c c c} \hline \hline \begin{tabular}{c} **Characteristics** \\ \end{tabular} & **LC** & **TC** & **DC** \\ \hline \begin{tabular}{c} EKF measurements \\ \end{tabular} & \begin{tabular}{c} GNSS outputs \\ (position and velocity) \\ \end{tabular} & \begin{tabular}{c} Raw observations \\ (pseudo range, phase, Doppler) \\ \end{tabular} & \begin{tabular}{c} Loop parameters \\ (code error, phase error) \\ \end{tabular} \\ \hline \begin{tabular}{c} Minimum number of \\ available satellites \\ \end{tabular} & Four & One & One \\ \hline \begin{tabular}{c} Aiding receiver \\ \end{tabular} & No & No & Yes \\ \hline \begin{tabular}{c} Acquisition and tracking \\ performance improvement \\ \end{tabular} & No & No & Yes \\ \hline \begin{tabular}{c} Disturbance resistance \\ \end{tabular} & No & No & Yes \\ \hline \
\begin{tabular}{c} Cost and complexity \\ \end{tabular} & Low & High & Highest \\ \hline
\begin{tabular}{c} Possible to use INS to \\ exclude outlying \\ measurements \\ \end{tabular} & No & Yes & Yes \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The characteristics of three integration strategies. -based Detection, Identification, and Adaptation (DIA) technique is representative and has been widely used in GNSS/INS integration [46; 47]. The adaptation procedure in DIA means eliminating the state's bias caused by the identified outlier. DIA and the GNSS receiver autonomous integrity monitoring (RAIM) have a strong connection. Many RAIM algorithms are developed from DIA, such as eRAIM [48]. The adaptively robust estimation methods aim to make the state estimations insensitive to outliers [44]. These methods generally adjust the noises' covariance matrices to meet the actual situation. The Kalman filter based on the robust estimation can be called RKF. The Sage-Husa filter uses the time-varying noise statistical estimator to adjust the process noise and the observation noise, which is widely used in GNSS/INS integration [49; 50]. Many researchers introduce the M-estimation based on the Bayesian estimation and the principle of equivalent weight to make their filters more robust [51]. These filters utilize the covariance inflation factor to inflate the noise covariance matrix parameters corresponding to the possible measurement outliers [52]. The classic equivalent weight functions include Huber, Tukey, and IGG-III, which are both widely used in GNSS/INS integration [53; 54]. Some adaptively robust filters simultaneously use an adaptive factor to inflate the covariance matrix of the predicted states, which can weaken the effects of the state modeling error [55]. Notably, a problem called excessive robustness can happen in these robust filters. Only when the gross errors contaminate the measurements should the covariance of the corresponding measurement be adjusted.