## 1 Introduction
SLAM has a strong ability to establish the model of the surrounding environment by applying visual sensors, which is a complementary strategy for GNSS disturbance scenes [17]. Nevertheless, SLAM has limitations for weak characteristic environments, such as piping and tubes, resulting in error accumulation and scene degradation [18]. The collaborative navigation method is a feasible solution. Through the collaboration constraints measured by multi-robots, the state can be updated based on pose constraint factors and achieve high positioning accuracy [19]. Regarding favorable environment estimation, [PERSON] et al. constructed a prior map by LiDAR to assist in estimating an inertial/GNSS tight-coupled navigation system when GNSS is in good condition. The scene optimizer and sliding window also improve the system's performance [20]. Therefore, the factor graph algorithm can satisfy the multi-source asynchronous fusion function and improve the comprehensive system's performance in various conditions. Nevertheless, it is still inevitable that the system suffers from sensor failure, which significantly reduces the navigation accuracy of the global system. With the increasing number and variety of sensors in vehicle navigation systems, the possibility of sensor faults is also increasing, resulting in poor system estimation and positioning accuracy. Fault detection, isolation, and recovery (FDIR) methods, which generally obtain system state and identify faults to isolate or restore sensors by determining whether the index of sensor measurements exceeds the target threshold, become significant to the system reliability and safety [21; 22; 23]. The residual chi-square statistical method is typically applied in navigation fault detection system, and the detection function is constructed to calculate the index through Kalman filter (KF) method, which detects the positioning faults of sensors based on the prediction information and determines whether a fault exists [24]. [PERSON] et al. constructed the IMU/GNSS tightly coupled system based on EKF and detected the system faults with the assistance of LiDAR real-time estimation [25]. In urban areas, GNSS signals may suffer from the multipath effect or non-line-of-sight (NLOS) signal interference due to reflections, and the whole system tendsto provide poor positioning results. The pseudo-range errors are well calculated and account for signal strength, satellite elevation angle, and coordinate information. These algorithms have been developed based on the KF algorithm under the assumption that GNSS measurements come with Gaussian uncorrelated white noise. But the existence of colored noise in GNSS measurements, typical for positioning with low-cost GNSS receivers, will degrade the performance of KF-based fault detection and exclusion (FDE) algorithms. To obtain more accurate estimation results, [PERSON] et al. proposed an improved considering colored noise model to improve the FDE performance [26]. However, satellites still suffer from multipath signals, and the error model cannot be constructed. Multiple model adaptive estimation (MMAE) provides a feasible solution to avoid directly estimating unknown faults. Considering that the dynamic model of the localization is highly nonlinear, [PERSON] et al. provided the navigation strategy of fusing multi-sensor measurements with an interactive multiple model (IMM) algorithm based on unscented Kalman filters (UKF) and EKF [27]. [PERSON] et al. proposed a hybrid robust Gaussian belief propagation method as a cooperative positioning system for transportation navigation, which applies an IMM structure to fuse FDE with [PERSON]'s M-estimation method and combines the advantages of both without drawbacks [28]. Given the decline in estimation and diagnostic accuracy caused by the fixed transformation probability models of IMM, [PERSON] et al. combined an adaptive interacting multiple model filter and federated Kalman algorithm to achieve an outlier detection function [29].