# A Multiple-Step, Randomly Delayed, Robust Cubature Kalman Filter for Spacecraft-Relative Navigation
## 1 Introduction  
Formation flights, on-orbit servicing, asteroid exploration, rendezvous and docking, active debris removal, and other space-based missions commonly require high-precision position and attitude estimations [1]. [PERSON] et al. has proposed a satellite navigation system that combines inter-satellite links and a batch-sequential filter, and it reconstructs high-quality orbits for the constellation nodes and provides high-level autonomous navigation for Mars exploration [2]. [PERSON] et al. has designed the space exploration rover with a positioning and mapping system which is equipped with light detection and ranging instruments, and it could achieve safe navigation in rough terrain and avoid obstacles [3]. [PERSON] et al. has processed stereo images captured during the rover's travel with 3D visual odometry, where 2D image keypoints are identified to estimate the relative position and attitude between every step. This method has the advantage of mitigating trajectory inconsistencies from dead-reckoning techniques [4]. [PERSON] et al. has developed a vision-based spacecraft-relative navigation system using a position sensing diode (PSD) [5]. The vision-based navigation (VISNAV) system has the virtues of light weight, small size, and low power consumption [6]. It is usually adopted to determine the relative positions and attitudes of spacecraft that are within several hundred meters from each other. Optical sensors and beacons are essential components of the VISNAV system, and they enable-optimal vision or adaptive vision. An optical sensor consists of a wide-angle lens and a position-sensing diode. A PSD possesses the merits of measuring the intensity and position of a light-point simultaneously. In vision-based spacecraft-relative navigation systems, light-emitting diodes (LEDs) are mounted as optical beacons on the chief spacecraft, and the deputy spacecraft is equipped with PSD-based optical sensors. The energy emitted from the optical beacons is then focused on the PSD through a wide-angle lens, so the optical signal is transformed into an electrical signal through electronic processing to acquire the image information of the target. In the more sophisticated VISNAV systems, the PSD can be programmed to only recognize a specified light source [7]. In VISNAV systems, the relative motion equations are built in the local-vertical-local-horizontal (LVLH) coordinate frame to obtain the motion state of targets. Generally, the beacon position is defined and known in the chief spacecraft body frame, and the position vector from the deputy spacecraft with respect to chief spacecraft is defined within the LVLH coordinate frame. Typically, the research in this field has assumed that the chief body is consistent within the LVLH coordinate system as well. However, this assumption is not rigorous and is just a special case. To eliminate the above assumptions, an efficient method is to estimate the attitudes of the two spacecrafts with respect to the LVLH coordinate frame, and the relative attitudes could then be obtained as well. The spacecraft's orientation relative to the reference coordinate frame can be determined through the estimation of its attitude [8]. Euler angles, rotation vectors, direction cosine matrices, and quaternions are usually used to describe spacecraft attitude. Specifically, quaternions are useful for describing spacecraft attitude kinematics due to their non-singularity and the bi-linear kinematics equation [9]. To ensure that quaternions meet the normalization constraint, we adopted quaternions for propagation and updating and introduced the generalized Rodrigues parameter (GRP) [10] to denote the local attitude error. Jitter, vibration, and multiple disruptions can degrade the measurements' quality [11]. Vision-based spacecraft-relative navigation systems are subjected to randomly delayed measurements, and measurement noise is always disturbed by outliers. These factors motivated us to develop a filter that could simultaneously handle randomly delayed measurements and outliers.