# Maximum Correntropy Unscented Kalman Filter for Spacecraft Relative State Estimation
## 1 Introduction  
The relative state estimation problem is rather important in space communication networks, including inter-satellite link (ISL) establishment, communication support, multiple spacecraft formation flying, and spacecraft rendezvous and docking [1; 2; 3; 4]. Many relative motion scenes make use of the radar or/and ladar to obtain the relative measurement information among spacecrafts, which is necessary to determine the relative motions. As for the state estimation problem, the Kalman filter (KF) is often used for a linear dynamic system, which is, in essence, a recursive minimum \(l_{2}\)-norm linear filter [5; 6; 7]. However, from [8], we know that the relative state estimation in a circular reference orbit involves a linear state Clohessy-Wiltshire (CW) equation and a nonlinear measurement equation. Especially, when the reference orbit is elliptic, the dynamic state equation is also nonlinear [9]. To handle the nonlinear problems, the widely used methods are the extended Kalman filter (EKF) [10; 11; 12] and unscented Kalman filter (UKF) [11; 12; 13]. The EKF is a popular nonlinear extension of KF, which makes use of the first order Taylor series expansions to approximate the nonlinear system and applies the KF to this approximation. Nevertheless, the approximation is crude and it may lead to filter divergence if the function exhibits highly non-linear characteristics. Furthermore, the derivation of the Jacobian matrices is cumbersome and often results in the implementation difficulties. In UKF, the probability distribution of the state is approximated by a set of deterministically selected sigma points and propagated thoughthe non-linear process and measurement equations. UKF does not need to calculate the cumbersome Jacobian matrices and provides derivative-free higher-order approximations to achieve more accurate performances than the EKF. Unfortunately, the EKF and UKF may fail to perform well when the measurements are disturbed by some heavy-tailed non-Gaussian noises, which arises naturally in many real applications of engineering. The main reason for this is that both methods are based on the minimum \(l_{2}\)-norm technique and thus exhibit sensitivity to heavy-tailed noises [14]. To address this problem, some robust methods have been proposed. The [PERSON]'s generalized maximum likelihood methodology is very common [15; 16], which is a combined minimum \(l_{1}\) and \(l_{2}\)-norm technique. Besides [PERSON]'s robust statistics, the optimization criteria in information theoretic learning (ITL) [17; 18] provide an alternative effective way, using information theoretic quantities estimated directly from the data as the optimization costs instead of using the usual second-order statistical measures (such as variance and covariance). The ITL costs can capture higher-order statistics of data, and can be used as a robust adaptation cost to achieve excellent performance in a number of applications [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. Particularly, an optimization criterion based on correntropy, called the maximum correntropy criterion (MCC) [17; 18], has recently been successfully applied in robust adaptive filtering in the presence of heavy-tailed non-Gaussian noises [17; 28; 29; 30; 31; 32]. In this paper, a non-linear Kalman type filter based on MCC has been derived, namely the maximum correntropy unscented Kalman filter (MCUKF). In MCUKF, the UT is applied to obtain a predicted state estimation and covariance matrix, and a non-linear regression model under MCC is used to reformulate the measurement information. Then, the UT is adopted to the measurement equation to obtain the filter state and covariance matrix. Similar to the original UKF, the new filter also has a recursive structure and is suitable for online implementation. It is worth noting that the proposed MCUKF is different from the algorithm in [33]. The organization of the paper is as follows. In Section 2, we provide a short review of the MCC and the traditional UKF. In Section 3, the MCUKF algorithm is derived.