## 1 Introduction  
The current operating Global Navigation Satellite System (GNSS) includes the Global Positioning System (GPS), Globalnaya Navigazionnaya Sputnikovaya Sletrema (GLONASS), BeiDou (BDS), and Galileo. The precise position, velocity, and time (PVT) generated from real-time kinematics (RTK) are crucial for engineering surveys, fleet monitoring, intelligent transportation systems, geographical information systems, guidance and control, etc. [1, 2]. By 2024, more than 110 satellites with different frequencies are expected to be accessible for multi-GNSS. Compared with a single constellation, multiple constellations and frequencies could improve the accuracy, continuity, availability, and integrity significantly [3], while enhancing the robustness against noise and outliers [4]. Multi-GNSS applications usually apply loosely and tightly coupled models for data curation [5, 6]. Both of them could achieve similar performance [7] in precise point positioning [8] and multi-sensor fusion such as integrating GNSS with the Inertial NavigationSystem, Simultaneous Location and Mapping, Lidar, and 5G [9; 10; 11; 12; 13; 14; 15]. Although the tightly coupled model shares one common pivot satellite for each constellation, it is challenging to improve the model strength due to the presence of double difference inter-system bias (DISB) originating from receivers [16]. In this study, the loosely coupled model is used due to its usability [17]. Even if dual-frequency (DF) data is available [18; 19; 20], the ambiguity resolution is still impacted in urban environments due to signal blockage and interference [18; 19; 20; 21]. The traditional extended Kalman filter is sensitive to outliers and noise [22], as it is based on the second-order statistics (e.g., variance, correlation, etc.) that exist in the Gaussian noise assumption and minimum mean square error (MSE) criterion [23]. Thus, the following limitations still need to be addressed: (1) The unmodeled non-Gaussian noise and heavy-tail noise originated not only from the outliers and gross errors but also the missed data due to signal blockage and deformation [24]. (2) The lack of complete prior knowledge of system dynamics and observations may eventually cause divergence [25]. Extensive attempts have been made to address these problems. One solution is the particle filter (PF), which is capable of estimating the posterior probability density function (PDF) by massive particles [26]. Multi-model filters such as the Gaussian sum filter (GSF) are another solution, which parallelly implement and interactively combine several filters to estimate the system states. A computationally economical filter was studied by integrating the robust M estimation into the GSF framework [27]. Heavy-tailed distribution-based filters and the \(H_{\infty}\) filter have also been studied in [28]. However, attention still needs to be paid, as most of them are deficient in universality and efficiency [29]. Recently, correntropy has received growing attention in signal processing, posture estimation, and machine learning [30; 31; 32; 33]. It is a measurable metric of local similarity and is established on Gaussian kernel functions. Specifically, higher dimensional data matching and error detection can be realized, as Gaussian kernel functions enrich data features by transforming the observation into the Hibert space with higher dimensions. Thus, the maximum correntropy between the input and output is robust against various types of noise and even arbitrarily large outliers, and can be achieved according to the maximum correntropy criterion (MCC). Although the kernel bandwidth (KBW) is the key parameter for implementing the MCC, existing studies mostly treat it as an exogenous parameter from empirical experiments, rather than an endogenous variable of the system [33; 34; 35; 36]. In this paper, a nonlinear adaptive Kalman filter (KF) based on MCC with adaptive KBW is proposed. The Gaussian hypothesis and MSE criterion are further relaxed, which aims to improve the adaptability and robustness [37].