# A Multiple-Step, Randomly Delayed, Robust Cubature Kalman Filter for Spacecraft-Relative Navigation
## 1 Introduction
The Kalman filter (KF) is the most commonly used estimation technique for the assumption of linear systems and Gaussian noise [12]. However, many problems do not satisfy the linearity assumption, and non-linear systems are more common in practical engineering. Vision-based spacecraft-relative navigation uses the optical camera to obtain measurements, which are modeled with non-linear equations. Among various non-linear filters, the extended Kalman filter (EKF), as a broadly available method, linearizes non-linear systems using the first-order Taylor series expansions. Unfortunately, this can result in significant truncation errors, and the procedure of deriving the Jacobi matrix is tedious [13]. A family of sigma-point filters and particle filters can avoid the loss of high-order terms, and therefore, they exhibit better performances than the EKF. The sigma-point filter approximates the probability density distributions (PDFs) of states through a group of defined sigma points and propagates the mean and covariance of random variables through non-linear processes. [PERSON] has proposed the unscented Kalman filter (UKF) based on an intuitive statistical information transformation [14]. [PERSON] adopted the spherical-radial cubature rule to calculate the means and covariances of state variables after non-linear propagation and proposed the cubature Kalman filter (CKF) [15]. [PERSON] proposed the high-degree cubature Kalman filter (HCKF) based on the Genz's code [16] and the generalized Gauss-Laguerre quadrature (GGLQ) rule [17]. [PERSON] proposed the particle filter (PF) using the sequential Monte Carlo method, which is a powerful state estimation method for handling non-Gaussian noise with arbitrary non-linear models and arbitrary noise distributions [18]. However, the weights of sigma points could be negative in the unscented transformation (UT), so UKF does not behave stably in high-dimensional systems. As compared to UKF and CKF, HCKF has a very high computational burden. Particle filters suffer from particle degradation and a heavy computational burden. Considering both filtering stability and computational demands, CKF is the more suitable choice for solving non-linear filtering problems in this work. In practice, various unknown factors, such as environmental factors and equipment failures, can disturb navigation systems, and when data are transmitted along unreliable communication channels, the phenomenon of delayed measurements cannot be ignored [19]. The estimation accuracy is dramatically reduced when filters use delayed measurements. Handling the challenges associated with randomly delayed measurements and ensuring the accuracy of state estimation are crucial for spacecraft-relative navigation. [PERSON] derived a Gaussian approximation (GA) filter using Gaussian approximation of the posterior predictive PDFs of states and delayed measurements [20]. [PERSON] adopted Bernoulli random variables (BRV) to re-represent the likelihood PDF in exponential multiplicative form and approximated the state vector with variational Bayesian methods in order to propose an improved one-step, randomly delayed KF [21]. [PERSON] proposed a modified adaptive EKF (MAEKF) algorithm, which adopts adaptive estimation to alleviate the effects of modeling uncertainty and error [22]. [PERSON] used BRV to build measurement information and proposed a randomly delayed EKF. However, these filters are all restricted to one-step or two-step delay [23; 24]. [PERSON] proposed a multiple-step randomly delayed CKF (MRD-CKF) that computes the likelihood function through marginalizing delay variables to mitigate the random delay and loss [25]. In the aforementioned research, however, the impact of outliers on the navigation accuracy is not considered, and this also degrades the filter significantly. Therefore, this paper extends the above-mentioned researches. Outliers (contaminated Gaussian distribution) degrade the effectiveness of Gaussian filters. Relevant robust filters have been developed to address this problem.