Relative Orbital Elements 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.

Formation keeping control for deorbiting an uncooperative satellite by laser ablation

This paper proposes the formation keeping control law for deorbiting debris by a laser ablation. Laser ablation is vital technology for contactless active debris removal, where a chaser satellite with a laser system irradiates laser pulses to a target object to generate the ablation force for deorbiting. The deorbiting force decelerates the target, and the chaser must maintain its relative position and continue irradiating. In other words, both the chaser and the target are supposed to be deorbited simultaneously, where both have accelerations. Although conventional formation flying missions assume that only a chaser maneuvers, the formation flying in this paper considers that both a chaser and a target have accelerations. Thus, this paper derives the relative equations of motion between the chaser and the target in powered flight and their analytical solution using relative orbital elements. A control law based on the analytical solution is proposed, which determines the timings and directions of the laser ablation and the electrical thrust so that the formation periodically returns to a desired formation. Numerical simulations first examine the control law in two cases with different maneuver timings. Then, a Monte Carlo simulation is performed to verify the effectiveness of the control law for a variety of desired formations.

Distributed satellite system autonomous orbital control with recursive filtering

In this article, we propose a recursive orbital elements filter for autonomous control of Distributed Satellite Systems (DSS) that significantly reduces the variance of relative orbital elements between the observed and the desired satellite orbits. Leveraging satellite kinematics and control inputs data, combined with a model of relative dynamics, the filter provides smooth and continuous orbital control, while minimizing propellant consumption. In conjunction with Precise Point Positioning (PPP) navigation, the proposed filter enables onboard continuous low-thrust control compatible with high-performance electric propulsion. We also propose a restricted transverse/normal control law that simplifies the thruster's configurations and/or attitude manoeuvres required for propulsion pointing. The applicability and validity of our proposed techniques are verified by numerical simulations with two case studies: a constellation for Differential Interferometric Synthetic Aperture Radar (DInSAR) for global infrastructure monitoring; and a maritime domain awareness mission based on along-track interferometric synthetic aperture radar which requires single-pass interferometry for responsive ship traffic surveillance, and the coverage of a very large maritime zone with high revisit rates.

Simplified Maneuvering Strategies for Rendezvous in Near-Circular Earth Orbits

The development of autonomous guidance control and navigation systems for spacecraft would greatly benefit applications such as debris removals or on-orbit servicing, where human intervention is not practical. Within this context, inspired by Autonomous Vision Approach Navigation and Target Identification (AVANTI) demonstration, this work presents new guidance algorithms for rendezvous and proximity operations missions. Analytical laws are adopted and preferred over numerical methods, and mean relative orbital elements are chosen as state variables. Application times, magnitudes and directions of impulsive controls are sought to minimize propellant consumption for the planar reconfiguration of the relative motion between a passive target spacecraft and an active chaser one. In addition, simple and effective algorithms to evaluate the benefit of combining in-plane and out-of-plane maneuvers are introduced to deal with 3D problems. The proposed new strategies focus on maneuvers with a dominant change in the relative mean longitude (rarely addressed in the literature), but they can also deal with transfers where other relative orbital elements exhibit the most significant variations. A comprehensive parametric analysis compares the proposed new strategies with those employed in AVANTI and with the global optimum, numerically found for each test case. Results are similar to the AVANTI solutions when variations of the relative eccentricity vector dominate. Instead, in scenarios requiring predominant changes in the relative mean longitude, the required ΔV exhibits a 49.88% reduction (on average) when compared to the original methods. In all the test cases, the proposed solutions are within 3.5% of the global optimum in terms of ΔV. The practical accuracy of the presented guidance algorithms is also tested with numerical integration of equations of motion with J2 perturbation.

Astrodynamics-Informed Kinodynamic Sampling-Based Motion Planning for Relative Spacecraft Motion

This paper presents a sampling-based spacecraft relative motion planning algorithm to reconfigure a spacecraft from a given initial state to a desired final state, subject to kinodynamic (simultaneous kinematic and dynamics) constraints. We leverage theoretical advancement in the field of sampling-based motion planning in robotics and extend their application to achieve motion planning in astrodynamics. We present an astrodynamics-informed kinodynamic motion planning (AIKMP) algorithm that takes advantage of a spacecraft’s natural motion to compute a safe, fuel-efficient motion plan for spacecraft reconfiguration in a cluttered environment. This algorithm is developed in the relative orbital element space of the deputy spacecraft around a chief spacecraft that provides very useful geometric insight to relative spacecraft motion, as well as helps in generating an overall smooth final transfer trajectory without demanding additional postprocessing of the transfer solution. Through several example scenarios, we demonstrate that the AIKMP algorithm is regularly able to compute safe and fuel-efficient solutions to relative transfer problems. We also show that this algorithm iteratively improves on its computed solutions and thus holds the capability of finding near-optimal transfer solutions given a sufficient number of algorithm iterations—without requiring an initial guess of the solution.

Relative and absolute on-board optimal formation acquisition and keeping for scientific activities in high-drag low-orbit environment

In this paper, a novel Model Predictive Control (MPC) technique for multi-satellite formation flying geometry acquisition and maintenance in high-drag environment is presented. The proposed MPC relies on a linearized and convexified quasi-nonsingular Relative Orbital Elements (ROE) model based on state transition matrices propagation, allowing to include the effect of perturbations in the prediction to optimize fuel efficiency and tracking accuracy. The formation is controlled with respect to a non-decaying orbiting point to perform absolute and relative station keeping simultaneously. For this purpose, a new dedicated plant matrix to include drag effects on ROE in the propagation is derived and validated with respect to numerical results. In all simulations, the satellites are assumed to be equipped with a single low-thrust propulsion unit, therefore, specific constraints are included in the controller to obtain a feasible solution in a real operational scenario. Moreover, a collision avoidance constraint is added in case of close proximity operations exploiting a linear mapping between the set of ROE and cartesian coordinates expressed in the Local-Vertical-Local-Horizontal (LVLH) reference frame. The controller response is simulated in several realistic mission contexts with a high-fidelity orbital propagator and the results are validated for fuel efficiency by comparing them to similar approaches available in literature and to optimal solutions obtained respectively with a direct single shooting algorithm and with a closed-form impulsive formulation.

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.

Second-Order State Propagation for Continuous-Thrust Relative Motion in $J_{2}$-Perturbed Elliptic Orbits

The second-order analytical state propagation under an arbitrary continuous thrust is proposed for relative motion in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$J_{2}$</tex-math></inline-formula> -perturbed elliptic chief orbits. The mean relative orbital elements are used to describe the relative motion. The second-order relative motion equations are obtained by performing the Taylor expansion of the nonlinear relative motion equations about the chief orbit. The original thrust acceleration is approximated by a truncated Fourier series in terms of eccentric anomaly of the chief orbit. Based on the first-order state propagation and the thrust acceleration Fourier-series expansion, the second-order propagation of the mean relative orbital elements is obtained. Numerical simulations show that the state propagation errors of the second-order solution are reduced by 1 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sim$</tex-math></inline-formula> 2 orders of magnitude when compared with the first-order solution.

A Geometrical, Reachable Set Approach for Constrained Pursuit–Evasion Games with Multiple Pursuers and Evaders

This paper presents a solution strategy for deterministic time-optimal pursuit–evasion games with linear state constraints, convex control constraints, and linear dynamics that is consistent with linearized relative orbital motion models such as the Clohessy–Wiltshire equations and relative orbital elements. The strategy first generates polytopic inner approximations of the players’ reachable sets by solving a sequence of convex programs. A bisection method then computes the optimal termination time, which is the least time at which a set containment condition is satisfied. The pursuit–evasion games considered are games with (1) a single pursuer and single evader, (2) multiple pursuers and a single evader, and (3) a single pursuer and multiple evaders. Compared to variational methods, this reachable set strategy leads to a tractable formulation even when there are state and control constraints. The efficacy of the strategy is demonstrated in three numerical simulations for a constellation of satellites in close proximity in low earth orbit.

Continuous Low-Thrust Maneuver Planning for Space Gravitational Wave Formation Reconfiguration Based on Improved Particle Swarm Optimization Algorithm

This study proposes a three-spacecraft formation reconfiguration strategy of minimum fuel for space gravitational wave detection missions in the high Earth orbit (105 km). For solving the limitations of measurement and communication in long baseline formations, a control strategy of a virtual formation is applied. The virtual reference spacecraft provides a desired relative state between the satellites, which is then used to control the motion of the physical spacecraft to maintain the desired formation. A linear dynamics model based on relative orbit elements' parameterization is used to describe the relative motion in the virtual formation, which facilitates the inclusion of J2, SRP, and lunisolar third-body gravity effects and provides a direct insight into the relative motion geometry. Considering the actual flight scenarios of gravitational wave formations, a formation reconfiguration strategy based on continuous low thrust is investigated to achieve the desired state at a given time while minimizing interference to the satellite platform. The reconfiguration problem is considered a constrained nonlinear programming problem, and an improved particle swarm algorithm is developed to solve this problem. Finally, the simulation results demonstrate the performance of the proposed method in improving the maneuver sequence distribution and optimizing maneuver consumption.

Mission Design and Orbit-Attitude Control Algorithms Development of Multistatic SAR Satellites for Very-High-Resolution Stripmap Imaging

This study designs a multistatic synthetic aperture radar (SAR) formation-flying system for very-high-resolution stripmap imaging (VHRSI) using manufacturable SAR microsatellites. Multistatic SAR formation specifications for VHRSI are derived based on the SAR image theory. For the simultaneous multi-satellite operation, the advantages of the autonomous orbit and attitude control are prominent in terms of the workload of the ground station or the efficient performance of missions. Therefore, the autonomous relative-orbit-control algorithm using relative orbital elements is developed to maintain the designed multistatic SAR formation. Additionally, an autonomous attitude-control algorithm for multistatic SAR imaging is designed by applying the optimal right-ascension of the descending node (RADN) sector concept. Finally, the resolution improvement of VHRSI is verified through multistatic SAR imaging simulations. The multistatic SAR formation is designed with three satellites separated by 7.5 km each in the along-track direction. Autonomous relative orbit control maintains the relative position error within 45 m (3σ). Additionally, the autonomous attitude control simulation verifies that the satellites perform attitude maneuvers suitable for the operation mode, and the pointing error is maintained within 0.0035° (3σ). The spatial resolution of the multistatic SAR system for VHRSI is 0.95 × 0.96 m, which satisfies the very-high-spatial-resolution requirement.

Self-Organizing Control of Mega Constellations for Continuous Earth Observation

This work presents a novel self-organizing control method for mega constellations to meet the continuous Earth observation requirements. In order to decrease the TT&amp;C pressure caused by numerous satellites, constellation satellites are not controlled according to the designed configurations but are controlled with respect to intersatellite constraints. By analyzing the street-of-coverage (SOC) of coplanar constellation satellites, the continuous coverage constraint of the mega constellation is transformed into constraints of the right ascension of ascending node (RAAN) and relative motion bound between every two adjacent coplanar satellites. The proposed continuous coverage constraint can be satisfied by most ongoing or planned mega constellations. Artificial potential functions (APFs) are used to realize self-organizing control. The scale-independent relative orbital elements (SIROEs) are innovatively presented as the self-organizing control variables. Using the Gaussian equations and Lyapunov’s theory, the stability of the APF control in quadratic form is proven, from which it can be concluded that the APF control variables of the controlled satellite should have the same time derivative as the target satellite states under two-body Keplerian motion condition, and SIROEs are ideal choices. The proposed controllers and self-organizing rules are verified in the sub-constellation of the GW-2 mega constellation by simulation. The results demonstrate the goodness in control effect and ground coverage performance.

Manoeuvre planning algorithm for satellite formations using mean relative orbital elements

Formation flying missions have increased their importance thanks to the better performances that can be achieved with a distributed system of satellites. One of the most demanding challenges linked to formation flying missions is the formulation of a robust control plan. Due to the recent developments in electrical propulsion, controllers that apply a continuous low-thrust to each spacecraft in the formation have been proposed, achieving precise control. This idea is used in this manuscript, where an open-loop manoeuvre planning algorithm is developed, to provide the control accelerations to achieve a formation reconfiguration in a given temporal frame. The proposed algorithm is based on an innovative control technique, which relies on the mean relative orbital elements as state variables of a first-order dynamic system. It includes the effect of the gravitational field of the Earth. Finally, the algorithm is tested with two different test cases of formation flying reconfiguration, presenting and analysing their results. The test case scenarios considered are based on possible future mission concepts in Low Earth Orbit, aiming at improving the control performances for the mission design.

Modeling and control of scale-independent relative orbital elements for near-circular orbits

This paper innovatively proposes six scale-independent relative orbital elements, that can be used for dynamics analysis and control for relative motion with near-circular orbits and arbitrary inter-satellite separation. Scale-independent relative orbital elements are deduced from the first-order decomposed relative motion model. These elements reveal the twisted elliptical characteristics of large-range relative motion and standard elliptical characteristics of close-range relative motion. The error of the proposed relative orbital elements keeps zero for circular orbits and the error is irrelevant to inter-satellite range. Based on the scale-independent relative orbital elements, the configuration design equation is established. Given the configuration requirements, the initial orbital elements of the deputy can be calculated. The autonomous configuration control problem is studied by means of artificial potential functions. Scale-independent relative orbital elements are used as control variables, and the large-range reconfiguration and collision avoidance control are realized with low control consumption.

Design of optimal low-thrust manoeuvres for remote sensing multi-satellite formation flying in low Earth orbit

This paper presents a strategy for optimal manoeuvre design of multi-satellite formation flying in low Earth orbit environment, with the aim of providing a tool for mission operation design. The proposed methodology for formation flying manoeuvres foresees a continuous low-thrust control profile, to enable the operational phases. The design is performed starting from the dynamic representation described in the relative orbital elements, including the main orbital perturbations effects. It also exploits an interface with the classical radial-transversal-normal description to include the maximum delta-v limitation and the safety condition requirements. The methodology is applied to a remote sensing mission study, Formation Flying L-band Aperture Synthesis, for land and ocean application, such as a potential high-resolution Soil Moisture and Ocean Salinity (SMOS) follow-on mission, as part of a European Space Agency mission concept study. Moreover, the results are applicable to a wide range of low Earth orbit missions, exploiting a distributed system, and in particular to Formation Flying L-band Aperture Synthesis (FFLAS) as a follow-on concept to SMOS.

Safe, Delta--Efficient Spacecraft Swarm Reconfiguration Using Lyapunov Stability and Artificial Potentials

Spacecraft swarms can achieve mission objectives otherwise impossible by monolithic satellites. To this end, swarms need to autonomously reconfigure their relative motion while complying with spacecraft constraints. Convex optimization provides safety and efficiency for arbitrary reconfigurations at high computational cost. Comparatively, closed-form solutions are computationally efficient but do not guarantee optimality and compliance to constraints in general cases. This work proposes a novel control architecture with Lyapunov functions and artificial potentials based on relative orbit elements that promise the numerical efficiency of closed-form solutions and the general applicability of convex optimization solvers. The novel Lyapunov functions are designed using analytical lower bounds calculated using the current control tracking errors. When the functions are used with a feedback control scheme modeled after the optimal, closed-form solutions for binary system reconfiguration, the resulting controller is both computationally and efficient. To address safety and state constraints in general, reference governors based on artificial potential functions enable computationally efficient collision avoidance using only neighbor state information. The controllers are validated in example mission simulations representative of Starling-1 and the Van Allen Swarm. In these missions, the controllers demonstrate safe, -efficient, and computationally tractable swarm reconfiguration, enabling future swarming missions.

On Autonomous Reconfiguration of SAR Satellite Formation Flight With Continuous Control

Satellites clusters for Earth observation purposes entail precise orbit monitoring and control to satisfy application needs efficiently. Fast technological progress in GNSS receivers, intersatellite communications/vision links, and propulsion technologies have inspired and fostered autonomous orbit determination and control techniques, allowing more efficient ground segment operations. Moreover, new distributed satellite missions have been enabled by autonomous formation flying, with unprecedented Earth observation potential. These missions may adopt varying observational modes associated with different relative orbital configurations and also transit among these modes as required by mission control. In this article, we consider a hypothetic distributed synthetic aperture radar (SAR) mission with a main SAR illuminator, as CONAE's SAOCOM 1 satellites, and one or more companions able to adopt different bistatic SAR geometries, enabling interferometry, tomography, or specular acquisition. To approach this problem, we formulate a basic task for any actively controlled companion satellite (deputy satellite) consisting of following a reference maneuver relative to the chief satellite orbit. This task is performed by means of a feedback control law based on newly introduced relative orbital elements (ROEs) and suitable to continuous actuators (e.g., electric propulsion) with bounded thrust. As a main result, we prove the asymptotic stability of the maneuver orbit tracking task for low Earth, nonequatorial, and near-circular chief orbits. This required a reformulation of the relative orbit tracking problem and the associated ROEs. A control allocation along the orbit is also proposed to reduce the propellant consumption and the transient time between SAR modes. The theoretical results are validated with simulated controlled transitions between different acquisition modes.

Generalized Angles-Only Navigation Architecture for Autonomous Distributed Space Systems

This paper addresses the development and verification of an estimation architecture for autonomous relative navigation in multi-satellite systems using angles-only measurements. Unlike traditional angles-only navigation approaches, which are generally hindered by poor handling of observability constraints, this prototype solution is designed to be flexible and applicable to a multitude of relevant missions. The estimator converges accurately and robustly in varying orbital environments without requiring maneuvers by exploiting system nonlinearities using an unscented Kalman filter that is streamlined by formulating the measurement and dynamics models using relative orbital elements. While the majority of prior work has focused on only solving for the relative motion of a single target, the approach here enables estimation of several additional parameters. Key design trades are evaluated for estimating additive sensor biases, target ballistic properties in environments with strong nonconservative perturbations, adaptive process noise statistics, and the absolute orbit of the observing satellite. The architecture is generalized for relative navigation with multiple targets, and for decentralized estimation using multiple coordinated observers. These filter strategies are used to inform the development of a novel on-orbit demonstration of angles-only navigation known as the Starling Formation-flying Optical eXperiment (StarFOX) in partnership with NASA. Key hardware-in-the-loop algorithm verification results for StarFOX-specific scenarios are provided.

Optimal spacecraft swarm reconfiguration through chief orbit refinement

This work presents a novel approach to minimizing delta-v for spacecraft swarm reconfiguration through refinement of the real or virtual chief spacecraft orbit, which is a closed orbit of arbitrary eccentricity. Given a desired relative motion configuration, the optimal target end-state of the swarm is solved for to provide minimum delta-v expenditure. The relative orbital elements (ROE) representation of relative motion is leveraged to produce two quadratically constrained linear programs (QCLPs). One program minimizes the sum of delta-v usage across the swarm, and the second minimizes the maximum delta-v usage among the swarm or co-orbiting spacecraft. The two objectives are designed to balance delta-v usage across the swarm to maximize mission lifetime. Furthermore, both optimization programs can be solved efficiently with interior point methods. While the minimization is applicable to any ROE set, the quasi-nonsingular ROE are exploited in this work to produce analytical solutions for arbitrary swarm geometry reconfigurations in non-equatorial orbits. If analytical solutions are not available, guidance is provided to minimize the effort required by convex optimization solvers in finding solutions. The delta-v savings enabled by the refinement of the swarm’s reference orbit are demonstrated using a state-of-the-art solver that produces the minimum reachable delta-v for a given reconfiguration. Therefore, leveraging a small refinement in the reference orbit for swarm reconfiguration through a QCLP can provide significant delta-v savings, extending mission lifetime.

Linear transfer guidance based on Lyapunov method

Low-thrust propulsion technology is widely used in space missions. However, traditional methods are not appropriate for the general case. In this study, a linearization Lyapunov method for constant thrust control is proposed. Based on relative orbit elements (ROEs), analytical constant-thrust linear equations are introduced. Moreover, via Lyapunov method, the linear constant-thrust control strategy is obtained. Numerical simulations are conducted to demonstrate the validation of the proposed method. Given different thrust magnitudes, the accuracy meets the requirement in these cases. Thus, it proves to be a practical choice for engineering applications.