Relative Orbital Elements Research Articles

Neural-aided GNC reconfiguration algorithm for distributed space system: development and PIL test

This paper presents a neural-aided Guidance, Navigation & Control algorithm for reconfiguration of distributed space systems. The guidance algorithm is based on Artificial Potential Fields (APF) in the Relative Orbital Elements (ROE) space. Since the relative orbit determination measurements are typically referred to the Cartesian metrics (e.g. range or range rate), a linear mapping between the set of ROE and the Cartesian coordinates expressed in the Local-Vertical-Local-Horizontal (LVLH) reference frame is derived. The navigation and control algorithms rely on the relative dynamics expressed in the same ROE set of coordinates. To cope with uncertainties and nonlinearities of the system, a Radial Basis Function Neural Network (RBFNN) is employed to reconstruct the perturbed dynamics. The Artificial Neural Network (ANN) is coupled with an adaptive Extended Kalman Filter for state estimation. A feedback control is designed to track the desired state, whose stability is analyzed using Lyapunov theory. The guidance, navigation and control algorithms are tested in a high-fidelity numerical orbit propagator. Moreover, the algorithm is tested in relevant Processor-In-the-Loop (PIL) simulations using a TI C2000-Delfino MCU F28379D. The results demonstrate the effectiveness of the algorithm for relative reconfiguration maneuvers involving relative distances ~102 m with limited fuel consumption and constrained available thrust (⩽1mN). In particular along-track maneuvers, relative plane change and formation enlargement are analysed in the paper, showing the comparison between the proposed algorithm and the legacy one without the neural network. The benefit of implementing a neural network is particularly highlighted when the nonlinearities or unmodelled terms in the on-board dynamics become prominent.

Input-output linearized spacecraft formation control via differential drag using relative orbital elements

The derivation, implementation, and performance assessment of a differential drag based algorithm, designed for spacecraft formation control using a relative orbital elements formulation, is presented. In contrast to similar approaches found in literature, the underlying motion dynamics model used to derive the control law includes J 2 perturbation as well as atmospheric drag perturbation based on a nonlinear density model. The basic idea of the control algorithm is to algebraically transform the nonlinear plant into a linear substitute. The resulting control law can be interpreted as combined feedforward and feedback PID control, applied to the linear substitute model. Motion dynamics and perturbations are accounted for in a feedforward branch while a PID feedback branch undertakes the cancellation of errors caused by model uncertainties and unmodeled physics. For linearized plant models, the computation of a feedforward branch is straight forward from the state transition matrix. In the nonlinear case proposed in this paper, the inclusion of a feedforward signal is achieved using the input-output linearization technique, yielding a nonlinear and globally valid control law. Sample numerical simulations are presented to support the validity of the controller and to demonstrate its performance compared to a simple PID controller. Simulation results show enhanced tracking performance of the mean along-track separation reference signal and reduced control effort. Moreover, input–output linearization enables the control of a large set of operating points using one set of control parameters.

Linear Cotangential Transfers and Safe Orbits for Elliptic Orbit Rendezvous

This paper presents the theory for linear cotangential transfers and safe orbits for elliptic orbit rendezvous. Expressions for the transfer angle and the required ’s are derived. Singularities in the algorithm can occur if the two orbits intersect. Alternative maneuvers for such singular cases are developed. The linear cotangential transfer algorithm is compared with the nonlinear cotangential transfer and the algorithm is found to be very similar. The development of the linear cotangential transfer leads to a new set of relative orbital elements that are well suited for defining safe trajectories. The characteristics of safe trajectories are discussed and a linear safety checking algorithm is developed. Finally, the combination of the cotangential transfers and safe orbits is used to define safe rendezvous trajectories for elliptical orbit rendezvous.

Minimization of Delta-v for Satellite Swarm Maintenance Using a Virtual Chief

This article presents a novel methodology for minimizing delta-v costs associated with relative orbit-keeping of a satellite swarm. The minimization is achieved through the selection of a virtual chief spacecraft for the reference of the swarm. By casting the minimization in relative orbital elements (ROE), the proposed approach leverages state-transition matrices (STMs) and analytical representations of delta-v usage. The STMs and analytical expressions are used to represent the swarm delta-v usage as a function of the virtual chief spacecraft parameters. The minimization of the resulting swarm delta-v function is manipulated to take the form of a convex program that is applicable to various delta-v formulations, ROE sets, and dynamic environments, generalizing approaches used for binary satellite systems to larger swarms. The novel quadratically constrained, linear objective convex program can be solved efficiently with available algorithms, such as interior point methods or commercial solvers. Furthermore, a new analytical delta-v lower bound for 6 degrees-of-freedom reconfigurations of quasi-nonsingular ROEs is developed. By leveraging the novel delta-v lower bound in the optimization, analytical solutions to the optimization for various orbit environments that include the perturbing effects of J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> or drag are produced. For the problems that do not have an analytical solution, geometric insight is provided to reduce computational effort in solving the optimization. The presented solutions and insights are validated through a convex optimization solver, demonstrating the accuracy and value of the analysis. Additionally, a high-fidelity simulation demonstrates fuel savings in a realistic mission scenario.

Spacecraft swarm dynamics and control about asteroids

This paper presents a novel methodology to control spacecraft swarms about single asteroids. This approach enables the use of small, autonomous swarm spacecraft in conjunction with a mothership, reducing the need for the Deep Space Network and improving performance in future asteroid missions. The methodology is informed by a semi-analytical model for the spacecraft relative motion that includes relevant gravitational effects without assuming J2-dominance as well as solar radiation pressure. The dynamics model is exploited in an Extended Kalman Filter (EKF) to produce an osculating-to-mean relative orbital element (ROE) conversion that relies on minimum knowledge of the asteroid gravity. The resulting real-time relative mean state estimate is utilized in a new formation-keeping control algorithm. The control problem is cast in mean relative orbital elements to leverage the geometric insight of secular and long-period effects in the definition of control windows for swarm maintenance. Analytical constraints that ensure collision avoidance and enforce swarm geometry are derived and enforced in ROE space. The proposed swarm-keeping algorithms are tested and validated in high-fidelity simulations for a reference asteroid mission.

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.

New fly-around formations for an elliptical reference orbit

Fly-around technology, as applied to maintain a chasing spacecraft flying periodically around the chief spacecraft, has received widespread attention in recent years. The conventional impulsive or continuous-thrust control approaches for maintaining non-natural formations put a high demand on the engine and limit the applicability of engineering. In this study, which is based on the relative orbital elements, we propose a series of analytic natural fly-around formations for the two-body case, which can revisit specific points or be maintained within a certain range. In addition to maintaining the fly-around formation when considering the J2 perturbation, we present a two-stage constant thrust control method solved with the linearized sensitive matrix. Furthermore, based on this method, we present several maintaining strategies that guarantee a periodic revisit to a specified relative position. We conduct numerical simulations to demonstrate the efficacy of these proposed methods.

Analytical Framework for Precise Relative Motion in Low Earth Orbits

This work presents a practical and efficient analytical framework for the precise modeling of the relative motion in low Earth orbits. Developed to support the design and verification of relative guidance navigation and control algorithms devoted to spacecraft rendezvous for active debris removal applications, only the orbital perturbation due to nonspherically symmetric mass distribution is considered. The relative motion is modeled in mean relative orbital elements, revisiting the available formulations to include the first-order expansion of the effects due to any even zonal harmonics and the second-order expansion of the unperturbed and terms. Mean/osculating orbital element conversions are obtained by merging a second-order Hamiltonian approach applied to the problem with the Kaula linear perturbation method for the remaining terms of the geopotential. The paper describes the main building blocks of the framework as well as their interfaces because the key aspect to achieve precision is to set up a fully consistent environment. The results show the achievable accuracy under realistic operational conditions for possible guidance and navigation applications.

Hovering Formation Control Based on Two-Stage Constant Thrust

Hovering technology, applied to maintain a chasing spacecraft at a specified position relative to a reference orbit, has received widespread attention in recent years. Conventional impulsive or continuous-thrust control approaches require high demand for the engine and have limited applicability in practice. In this study, a new hovering formation controlled by two-stage constant thrust is proposed. The mathematical model of the two-stage constant-thrust control problem is first formulated, and a rapid method using sensitivity matrix is presented to solve the problem. Then, based on relative orbital elements, the strategy of hovering formation by two-stage constant control in the two-body case is obtained. To avoid the drift of hovering formation caused by perturbation, an improved strategy is presented to guarantee periodic revisits to the specified relative position. Numerical simulations are conducted to demonstrate the efficiency of the proposed method.

Bounded flight and collision avoidance control for satellite clusters using intersatellite flight bounds

Abstract A novel satellite cluster control method by means of a set of artificial potential functions is proposed in this paper. To avoid the huge difficulty of designing target configuration for a large-scale satellite cluster, the proposed cluster control method relies on intersatellite flight bounds so that satellites can autonomously converge to a given boundary range, rather than a set of exact orbit configuration. The analytic relative motion bounds in terms of relative orbital elements are deduced and used as variables in artificial potential functions. The accumulated control effort is minute, that shows great potential in micro satellite cluster control. On the other hand, collision avoidance as a key requirement for satellite cluster, also gets attention in this paper. For a long-term perspective, the minimum intersatellite flight bound rather than the real-time relative distance between satellites is introduced into the repulsive artificial potential function. Satellite members can adjust their control effort in real time according to nearby satellites, which reflect the characteristic of swarm intelligence. Monte Carlo simulation reveals the superiority in fuel consumption of the proposed collision avoidance control.

Establishment and control of spacecraft formations using artificial potential functions

The automated control of spacecraft formations in specified relative orbits is a challenging problem that is computationally intensive using traditional methods of numerically integrating trajectories and applying optimal control theory. In this paper, relative orbital elements and artificial potential functions are combined into a methodology that allows full control of the relative orbits of spacecraft formations. This method enables the intuitive design of geometrically complex formations, and allows collision avoidance during the formation establishment. The effectiveness of the control algorithm is illustrated with the numerical simulation of the establishment of various formation geometries.

Linear Models for Spacecraft Relative Motion Perturbed by Solar Radiation Pressure

Many scientific applications require the implementation of spacecraft formation-flying around Earth and other celestial bodies. The control of these formations calls for simple relative dynamics models able to accurately and efficiently incorporate secular and long-periodic effects of relevant perturbations. This paper presents novel linear analytical models of the effect of solar radiation pressure (SRP) on the relative motion of two spacecraft in arbitrary eccentric orbit for two state definitions based on relative orbital elements. The new models are valid both around Earth and around other celestial bodies such as near-Earth asteroids. The linear models are derived by augmenting the state with force model parameters and Taylor expanding the differential equations of relative motion, including the secular and long-periodic effects of SRP. New state transition matrices including and SRP are formalized and validated through comparison with a high-fidelity orbit propagator around Earth and around a near-Earth asteroid. In addition, the dominant trends caused by SRP on the relative orbital elements are captured in closed-form. The new models have accuracy in the range of meter to few tens of meters in all orbit scenarios considered and can be leveraged for formation guidance, control, and design.

Spacecraft formation flying reconfiguration with extended and impulsive maneuvers

This paper presents the control solutions to the spacecraft formation reconfiguration problem when impulsive or extended maneuvers are considered, and the reference orbit is circular. The proposed approach for the derivation of the control solutions is based on the inversion of the linearized equations of relative motion parameterized using the mean relative orbit elements. The use of mean relative orbit elements eases the inclusion of perturbing accelerations, such as the Earth’s oblateness effects, and offers an immediate insight into the relative motion geometry. Several maneuvering schemes of practical operational relevance are considered and the performance of the derived impulsive and piecewise continuous control solutions are investigated through the numerical propagation of the nonlinear relative dynamics. Finally, the benefits of the new extended maneuvers strategies are assessed through a comparison with the corresponding impulsive one.

Baseline analysis and control of spaceborne repeat-pass InSAR based on relative orbital elements

Baseline analysis and control of spaceborne repeat-pass InSAR based on relative orbital elements

Minimum-Fuel Control Strategy for Spacecraft Formation Reconfiguration via Finite-Time Maneuvers

This paper addresses the minimum-fuel spacecraft formation reconfiguration maneuver in perturbed near-circular orbits. In this study, only the deputy is assumed to be maneuverable and capable of providing a piecewise constant control thrust. The optimal control problem is formulated as a mixed-integer linear programming problem. To make the proposed strategy compliant with the requirements deriving from a realistic flight scenario, the collision avoidance and the maneuvering time constraints dictated by mission operations are included in the mathematical formulation. A linear dynamics model based on relative orbit elements parameterization and its associated closed-form solution are used to impose the boundary conditions, avoiding the dynamics integration within the optimization process. Several examples are shown to demonstrate the effectiveness of the proposed approach.

Self-organizing control for satellite clusters using artificial potential function in terms of relative orbital elements

Self-organizing control for satellite clusters is a challenging and promising problem which has drawn considerable attention in the recent past. The artificial potential function method has been widely used for self-organizing control due to its elegant mathematical analysis and simplicity. This paper proposes a set of self-organizing control rules for satellite clusters described by artificial potential functions, so that the reconfiguration, uniform distribution and collision avoidance operations can be achieved spontaneously. It may work regardless of the failure of satellites, attendance of new members or existence of space debris, and the corresponding communication and control system are completely distributed. Particularly, the artificial potential functions are written in terms of relative orbital elements derived from T–H equations, so that the self-organizing control can reflect relative motion dynamics, and guide the satellite along a fuel-efficient trajectory. The proposed method is especially suitable for highly distributed micro-satellite clusters and large-scale clusters to track moving targets, with few fuel cost and relatively high control accuracy, and it is applicable to clusters in either deep space or near-earth space. The stability of the control was proved by Lyapunov second method, and verified by Monte Carlo simulation. Finally, the comparison between self-organizing control and fuel-optimal control was made to demonstrate the performance properties.

Concurrent co-location maneuver planning for geostationary satellites

This paper details the development of a planning algorithm for multiple co-located geostationary satellites to perform station keeping and momentum unloading maneuvers concurrently. The objective is to minimize the overall fuel consumption while guaranteeing a safe separation distance between the satellites within a specific geostationary slot, as well as managing their stored angular momentum to maintain their nadir pointing orientation. The algorithm adopts the leader-follower architecture to define relative orbital elements of the satellites equipped with four gimbaled on-off electric thrusters, and solves a convex optimization problem with inequality constraints, including momentum unloading requirements, to determine the optimal maneuvers. The proposed algorithm is verified, in terms of fuel consumption, constraints enforcement and satellites performance, using numerical simulations that take into account dominant perturbations in the geostationary environment.

Ambiguous relative orbits in sequential relative orbit estimation with range-only measurements

This paper describes a manifold of ambiguous spacecraft relative orbits that arise in sequential relative orbit estimation. The development herein assumes linear relative dynamics, a circular reference orbit, and range-only measurements. Using a formulation based on relative orbit elements, the ambiguous orbits are categorized into two cases: mirror orbits, which conserve the size and shape but transform the orientation of the true relative orbit, and deformed orbits, which both distort the shape and change the orientation. A special case, that of central ambiguous relative orbits, which are geometrically symmetric relative to the chief's local-vertical-local-horizontal frame is also discussed. The multiplicity of mirror ambiguous orbits, deformed ambiguous orbits and central ambiguous orbits are shown to be three, four and infinity, respectively. Numerical results using an extended Kalman filter are provided to confirm the existence of these ambiguous orbits. Furthermore, the observability is studied analytically with a nonlinear observability criterion using Lie derivatives. It is also shown by numerical results that the inclusion of nonlinearities in the filter model can help resist the tendency of an extended Kalman filter to converge to the ambiguous relative orbits. Finally, the persistence of these ambiguous orbits under unmodeled chief eccentricity error and J2 perturbation is studied.

Robust and Safe -Spacecraft Swarming in Perturbed Near-Circular Orbits

This paper presents a new guidance and control methodology for spacecraft swarms based on relative orbital elements, that generalizes flight-proven techniques used for binary formations to accommodate a large number of spacecraft. The formation design problem is cast in the relative orbital elements to allow derivation of linear constraints that ensure passively bounded relative motion in perturbed near-circular orbits. This approach also enables derivation of analytical constraints that ensure collision avoidance including uncertainty at a computational cost that scales only linearly with the number of spacecraft. Additionally, two swarm formations are identified that ensure a user-specified minimum separation between all spacecraft in either the orbit plane or the plane perpendicular to the flight direction. To counteract the effects of differential drag, nonlinear state space control laws are developed that require actuation in only the (anti-)flight direction. These control laws are employed in a hybrid passive/active control architecture that only actuates the spacecraft when necessary to ensure collision avoidance, allowing periodic passive drifts of up to several days. The proposed formation designs and control architectures are validated in simulations of a reference mission scenario using a high-fidelity numerical orbit propagator.

A simple method to design non-collision relative orbits for close spacecraft formation flying

A set of linearized relative motion equations of spacecraft flying on unperturbed elliptical orbits are specialized for particular cases, where the leader orbit is circular or equatorial. Based on these extended equations, we are able to analyze the relative motion regulation between a pair of spacecraft flying on arbitrary unperturbed orbits with the same semi-major axis in close formation. Given the initial orbital elements of the leader, this paper presents a simple way to design initial relative orbital elements of close spacecraft with the same semi-major axis, thus preventing collision under non-perturbed conditions. Considering the mean influence of J2 perturbation, namely secular J2 perturbation, we derive the mean derivatives of orbital element differences, and then expand them to first order. Thus the first order expansion of orbital element differences can be added to the relative motion equations for further analysis. For a pair of spacecraft that will never collide under non-perturbed situations, we present a simple method to determine whether a collision will occur when J2 perturbation is considered. Examples are given to prove the validity of the extended relative motion equations and to illustrate how the methods presented can be used. The simple method for designing initial relative orbital elements proposed here could be helpful to the preliminary design of the relative orbital elements between spacecraft in a close formation, when collision avoidance is necessary.