Angles-only Relative Navigation Research Articles

A nonlinear filter based on GSK for relative navigation using relative orbital elements

Due to the sensitivity of the relative orbital elements model to the measurement noise, the non-stationary heavy-tailed noise(NSHT) induced by the time-varying environment during the relative navigation usually leads to filter divergence. To address this problem, a new nonlinear filter based on Gaussian-Student's-Multivariate K(GSK) mixture distribution is proposed in this paper. A Dirichlet stochastic mixture vector fusing Gaussian, Student's t, and Multivariate K distributions is introduced, thus proposing a GSK mixture distribution modeling measurement likelihood; then the Kullback-Leibler Divergence (KLD) of the true posteriori probability density function(PDF) and the approximate posteriori PDF are minimized by a variational Bayesian(VB) technique to solve for the state and parameter approximate a posteriori estimations, and finally a new nonlinear filter based on the GSK mixture distribution is derived for angles-only relative navigation in time-varying environments. Simulation outcomes indicate that the filter can realize state estimation in non-stationary states effectively with 45.16% higher estimation accuracy than the existing advanced filters.

Fast angles-only relative navigation using polynomial dynamics

A fast and efficient method for initial relative orbit determination from bearing-angle measurements is introduced. The range ambiguity problem for angles-only relative navigation is addressed by modeling nonlinear effects with a novel second-order mapping from relative orbit elements (ROE) to relative position coordinates. This model is used to form a system of polynomial constraint equations linking the line-of-sight measurements to the ROE. An efficient method for solving this system is developed around the insight that the ROE scale with the ratio of the inter-spacecraft separation to the orbit radius and are therefore small for most applications of interest. The method uses a truncated expansion of the quadratic formula to recursively eliminate unknowns, reduce the dimension of the system, and ultimately acquire an approximate solution. Strategies for improving robustness, efficiency, and accuracy are developed and the method is applied to general second-order systems as well as to a broad range of intial relative orbit determination scenarios. Modifications to the constraint equations and a solution algorithm are introduced to address the challenge of bias in the bearing-angle measurements.

Rendezvous and proximity operations design of an active debris removal service to a large constellation fleet

This work presents the design of rendezvous and close proximity operations for an active debris removal service for maintenance and disposal of a large constellation fleet. The design drivers of the service are safety and robustness of the operations, capability to remove a satellite under every condition, and high level of autonomy to reduce the service-related operations’ cost. Accordingly, the most challenging case of a rendezvous with a uncooperative non-collaborative target satellite is considered and a strategy to reduce its rotational motion is also presented. As for the rendezvous, the focus is on the relative trajectory design. First a guidance solution capable to maximise the observability property of the angles-only relative navigation problem is developed. Second, a set of convenient relative trajectories to enable target inspection for various illumination conditions is selected and an approach to ensure safe transitions among them is settled. Both these designed strategies can be readily implemented on autonomous spaceborne systems. Simulations are run for two constellation fleets, differing in mass of the single s/c and orbit’s altitude and inclination.

Cooperative Angles-Only Relative Navigation Algorithm for Multi-Spacecraft Formation in Close-Range

As to solve the collaborative relative navigation problem for near-circular orbiting small satellites in close-range under GNSS denied environment, a novel consensus constrained relative navigation algorithm based on the lever arm effect of the sensor offset from the spacecraft center of mass is proposed. Firstly, the orbital propagation model for the relative motion of multi-spacecraft is established based on Hill-Clohessy-Wiltshire dynamics and the line-of-sight measurement under sensor offset condition is modeled in Local Vertical Local Horizontal frame. Secondly, the consensus constraint model for the relative orbit state is constructed by introducing the geometry constraint between the spacecraft, based on which the consensus unscented Kalman filter is designed. Thirdly, the observability analysis is done and the necessary conditions of the sensor offset to make the state observable are obtained. Lastly, digital simulations are conducted to verify the proposed algorithm, where the comparison to the unconstrained case is also done. The results show that the estimated error of the relative position converges very quickly, the location error is smaller than 10 m under the condition of 10−3 rad level camera and 5 m offset.

Absolute Triangulation Algorithms for Space Exploration

Images are an important source of information for spacecraft navigation and for three-dimensional (3-D) reconstruction of observed space objects. Both of these applications take the form of a triangulation problem when the camera has a known attitude and the measurements extracted from the image are line-of-sight (LOS) directions. This work provides a comprehensive review of the history and theoretical foundations of triangulation. A variety of classical triangulation algorithms are reviewed, including a number of suboptimal linear methods (many LOS measurements) and the optimal method of Hartley and Sturm (only two LOS measurements). It is shown that the optimal many-measurement case may be solved without iteration as a linear system using the new linear optimal sine triangulation (LOST) method. Both LOST and the polynomial method of Hartley and Sturm provide the same result in the case of only two measurements. The various triangulation algorithms are assessed with a few numerical examples, including planetary terrain relative navigation, angles-only optical navigation at Uranus, 3-D reconstruction of Notre-Dame de Paris, and angles-only relative navigation.

Precise line-of-sight modelling for angles-only relative navigation

This work presents a precise analytical model to reconstruct the line-of-sight vector to a target satellite over time, as required by angles-only relative navigation systems for application to rendezvous missions. The model includes the effects of the geopotential, featuring: the analytical propagation in the mean relative orbital elements (up to second-order expansion), the analytical two-way osculating/mean orbital elements’ conversion (second-order in J2 and up to a given degree and order of the geopotential), and a second-order mapping from the perturbed osculating elements’ set to the local orbital frame. Performances are assessed against the line-of-sight reconstructed out of the precise GPS-based positioning products of the PRISMA mission. The line-of-sight modelled over a far-range one day long scenario can be fitted against the true one presenting residuals of the order of ten arc-seconds, which is below the typical sensor noise at far-range.

Satellite Relative Motion Modeling and Estimation via Nodal Elements

In this paper, a new parameterization of the relative motion between two satellites orbiting a central body is presented. The parameterization is based on the nodal elements: a set of angles describing the orbit geometry with respect to the relative line of nodes. These are combined with classical orbital elements to yield a nonsingular relative motion description. The exact nonlinear, perturbed dynamic model resulting from the new parameterization is established. The proposed parameter set captures the fundamental Keplerian invariants, while retaining a simple relationship with local orbital coordinates. An angles-only relative navigation filter and a collision avoidance scheme are devised by exploiting these features. The navigation solution is validated on a case study of an asteroid flyby mission. It is shown that a collision can be detected early on in the estimation process, which allows one to issue a timely evasive maneuver.

Observability criterion of angles-only navigation for spacecraft proximity operations

This research studies the angles-only relative navigation problem for spacecraft proximity operations when the camera offset from the vehicle center-of-mass allows for range observability. Emphasis is placed on developing an analytic observability criterion for the camera offset within the context of the Clohessy-Wiltshire (CW) dynamics and evaluating the performance of the navigation algorithm in nonlinear dynamics environment. Further, the newly observability criterion is mathematically derived by changing the analyzing problem of the nonlinear observability into the solving problem of linear equations, and the criterion can be a powerful guideline to be followed to install the camera and execute attitude maneuvers. Moreover, navigation filtering algorithm based on an Unscented Kalman Filter and systematic nonlinear Monte Carlo simulation framework are established to verify the analytic observability criterion and evaluate the performance of the proposed algorithm through typical proximity missions. The sensitivity of the navigation accuracy to spacecraft trajectories and camera offset is presented and discussed. The conclusion is that the camera offset is required to have out-of-plane component and cannot be parallel to the line-of-sight vector in order to supply range observability.

Analytical Approach for Orbital Evasion with Space Geometry Considered

This paper researches an optimal problem of orbital evasion with considering space geometry by using an analytical approach. Firstly, an angles-only relative navigation model is built and the definition of completely nonobservable maneuver is proposed. After algebraic analysis of relative space geometry, it is proved that the completely nonobservable maneuver is nonexistent. Based on this, the angle measurements of orbit without evasion are set as reference measurements and an analytical solution is derived to find the minimum difference between measurements and the reference measurements in a constant measuring time. Then, an object function using vector multiplication is designed and an optimization model is established so as to prove the optimality of analytical solution. At last, several numerical simulations are performed with different maneuver directions, which verify the effectiveness of the analytical method of this paper for orbital evasion problem. This method offers a new viewpoint for orbital evasion problem.

Angles-only relative navigation and closed-loop guidance for spacecraft proximity operations

This research investigates angles-only relative navigation and closed-loop guidance algorithm for spacecraft mid-range orbital proximity operations when the orbital maneuver allows for range observability. Emphasis and contribution are on developing angles-only relative navigation and guidance coupling algorithm in the context of Clohessy–Wiltshire and Tschauner–Hempel dynamics. Observability analysis of the relative state is done and the general mathematical expression of the observable condition is obtained. Coupling relationship between the angles-only relative navigation and the multi-pulse sliding guidance is discussed and its analytic expression is derived. A novel closed-loop guidance scheme is designed based on the coupling relationship and unscented kalman filter. Two-body Monte Carlo simulations are conducted to evaluate the validity and test the performance of the closed-loop system. The sensitivities of the navigation and guidance accuracy to the line-of-sight angles accuracy, initial separation and initial state uncertainties, number of pulses, and dynamics are presented and discussed.

Trajectory Design for Improving Observability of Angles-Only Relative Navigation between Two Satellites

This study focuses on designing a relative rendezvous trajectory for angles-only navigation between two satellites. The main purpose of this paper is to design a rendezvous trajectory which provides better observability for the whole mission period. The second purpose is to improve fuel consumption efficiency. For the trade-off between the observability maneuver and fuel consumption, it is important to minimize the fuel consumption while ensuring sufficient observability since a small-size satellite has a tight mass budget. To design such a trajectory, this study proposes implementing the observability constraint at each optimization node. Then the cost function is aimed at minimizing fuel consumption. A numerical optimization technique is implemented to solve the optimization problem. Simulation for the estimator is performed to verify the designed trajectory. To get physical insight into the relationship between relative trajectory and observability, an observability analysis is also presented using a geometrical relationship between two satellites and the Fisher information matrix as a measure of observability.

Optimal multi-objective trajectory design based on close-looped control for autonomous rendezvous

This paper considers the problem of optimal multi-objective trajectory design for autonomous rendezvous. Total velocity cost and relative state robustness of close-looped control are selected as the objective functions. Based on relative dynamics equations, the state equations and measurement equations for angles-only relative navigation between spacecrafts are set forth. According to the method of linear covariance analysis, the close-looped control covariance of the true relative state from the reference relative state is analyzed, and the objective functions of relative state robustness are formulated. Considering the total velocity cost and the relative state robustness, the multi-objective optimization algorithm of NSGA-II is employed to solve this multi-impulsive rendezvous problem. Lastly, the validity of the objective functions and the covariance results are demonstrated through 100 times Monte Carlo simulation.

Research on the strategy of angles-only relative navigation for autonomous rendezvous

This paper considers the problem of angles-only relative navigation for autonomous rendezvous. Methods for determining degree of observability (DOO) and latent range information of orbital maneuver are proposed for analyzing and enhancing the precision of relative position and velocity estimation. The equations of angles-only relative navigation are set forth on the condition that optical camera is the only viable sensor for relative measurement, and expressions for the DOO of relative navigation are obtained by using the Newton iterative method. The latent range information of orbital maneuver is analyzed, which is employed to enhance the DOO of angles-only relative navigation. Simulation result shows that DOO is effective to describe the observability level of relative position and velocity, and the latent range information is useful in enhancing the DOO of the angles-only relative navigation.