Relative Orbit Research Articles

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.

Roto-Translational Control of Spacecraft in Low Earth Orbit Using Environmental Forces and Torques

In this paper, relative orbit and attitude adaptive controllers are integrated to perform roto-translational maneuvers for CubeSats equipped with a Drag Maneuvering Device (DMD). The DMD enables the host CubeSat with modulation of aerodynamic forces/torques and gravity gradient torque. Adaptive controllers for independent orbital and attitude maneuvers are revisited to account for traslational-attitude coupling while compensating for uncertainty in parameters such as atmospheric density, drag/lift coefficients, location of the Center of Mass (CoM) and inertia matrix. Uniformly ultimately bounded convergence of the attitude error and relative orbit states is guaranteed by Lyapunov-based stability analysis for the integrated roto-translational maneuver. A simulation example of an along-track formation maneuver between two CubeSats with simultaneous attitude control using only environmental forces and torques is presented to validate the controller.

Relative Position and Attitude Estimation of Non-cooperative Spacecraft Based on Stereo Camera and Dynamics

The relative position and attitude estimation of non-cooperative spacecraft is essential to high-precision guidance, navigation, and control systems. This paper mainly proposes a method to measure the relative position, velocity and attitude of non-cooperative spacecrafts using stereo camera and dynamics. First, using stereo camera observe the spacecraft, and extract the feature points of images. Then, calculate the 3D position of the feature points, and match the relationship between the feature points to obtain geometric constraint information between frames. Based on this, the relative position, velocity, attitude and angular velocity of the target spacecraft are calculated. According to relative orbit dynamics and relative attitude dynamics model, design extended Kalman filter to update the measurement results. The simulation results show that the estimation accuracy is better than 0.01m, the relative velocity accuracy is better than 0.001m/s, and the relative attitude accuracy is better than 0.01°. This method can provide technical support for the relative navigation of non-cooperative spacecraft.

Approximate analytic solution of nonlinear Riccati spacecraft formation flying dynamics in terms of orbit element differences

Generally, spacecraft relative orbit is described using Hill frame coordinates with which it is difficult to obtain precise orbit geometry. The orbit description is made simpler and spacecraft relative trajectory easier to track using relative orbit element differences with the chief spacecraft true latitude (θ) and the difference between deputy and chief spacecraft true latitude (δθ) being the only time-varying parameters. Via the approach of parameterized orbit element differences and transformation, this paper develops approximate analytic solutions of nonlinear Riccati spacecraft formation flying relative motion dynamics. The algorithm for the computations of the δθ solutions is developed and it is based on well-structured and defined step-by-step instructions. As shown by the numerical simulations, the Riccati model gave a better approximation with lesser error making it amenable for the spacecraft formation flying, rendezvous analysis, spacecraft guidance, navigation and control.

Are we observing an NSC in course of formation in the NGC 4654 galaxy?

ABSTRACT We use direct N-body simulations to explore some possible scenarios for the future evolution of two massive clusters observed towards the centre of NGC 4654, a spiral galaxy with mass similar to that of the Milky Way. Using archival HST data, we obtain the photometric masses of the two clusters, M = 3 × 105 M⊙ and M = 1.7 × 106 M⊙, their half-light radii, Reff ∼ 4 pc and Reff ∼ 6 pc, and their projected distances from the photometric centre of the galaxy (both <22 pc). The knowledge of the structure and separation of these two clusters (∼24 pc) provides a unique view for studying the dynamics of a galactic central zone hosting massive clusters. Varying some of the unknown cluster orbital parameters, we carry out several N-body simulations showing that the future evolution of these clusters will inevitably result in their merger. We find that, mainly depending on the shape of their relative orbit, they will merge into the galactic centre in less than 82 Myr. In addition to the tidal interaction, a proper consideration of the dynamical friction braking would shorten the merging times up to few Myr. We also investigate the possibility to form a massive nuclear star cluster (NSC) in the centre of the galaxy by this process. Our analysis suggests that for low-eccentricity orbits, and relatively long merger times, the final merged cluster is spherical in shape, with an effective radius of few parsecs and a mass within the effective radius of the order of $10^5\, \mathrm{M_{\odot }}$. Because the central density of such a cluster is higher than that of the host galaxy, it is likely that this merger remnant could be the likely embryo of a future NSC.

Coupled Relative Orbit and Attitude Control Augmented by the Geomagnetic Lorentz Propulsions

An electrostatically charged spacecraft is subject to the Lorentz force and torque when moving in the geomagnetic field. By active modulation of the surface charge, the induced Lorentz force and torque can be used for orbit and attitude control, respectively. Because of the electromagnetic mechanism, the orbit and attitude motion of such spacecraft are naturally coupled, and the induced Lorentz force and torque are both instantaneously underactuated. Thus, other kinds of actuation are required to render the system fully actuated and controllable. A hybrid control system is then proposed in this paper for Lorentz-augmented spacecraft relative orbit and attitude control. The relative orbit motion is controlled by the combination of the Lorentz force and the thruster-generated control force, and the relative attitude motion is governed by the combination of the Lorentz torque and the magnetic torque. By using the adaptive backstepping control method, a closed-loop control scheme is designed for the hybrid control system to deal with the dynamic coupling, underactuation, unknown external disturbances, and model approximation errors simultaneously. The parameter adaptation laws are derived via the Lyapunov theory to guarantee the stability of the closed-loop system. Optimal distribution laws of the hybrid control inputs are then analytically solved. Finally, numerical examples are simulated in a -perturbed environment to verify the feasibility of the proposed hybrid control system and the validity of the closed-loop control scheme.

Fully Decentralized Cooperative Navigation for Spacecraft Constellations

This article proposes a method to decentralize the navigation burden and improve the fault tolerance for a spacecraft constellation. The constellation body reference system is introduced, which is the perifocal frame of one satellite in the constellation. The structure of the proposed navigation method is constructed to enable each spacecraft to estimate its own orbit in this body reference system. This step is essentially the relative orbit determination (OD) based on inter-satellite range measurements. Thereafter, the approach to transfer an orbit from the constellation body reference system to an inertial reference system is developed. The essential requirements on absolute measurements to realize the coordinate transfer are presented. By dividing the absolute OD into relative OD and coordinate transfer, each navigation subsystem operated in a spacecraft can be independent with others, and the absolute measurements collected by any spacecraft can contribute to the absolute OD of the whole constellation. The proposed method applies to constellations in any geometric configuration. A Walker constellation is taken as an example for numerical simulations. The results show that the proposed method has a lower computation burden compared to an integrated navigation system. With the same type of absolute measurements, the proposed method has higher accuracy and convergence velocity than conventional decentralized algorithms. When a spacecraft occurs with fault, the orbit results of other spacecraft are not affected using the proposed method, which is beyond the ability of conventional methods.

Optimal Multi-Target Overflight Using Ground-Track Adjustment

This study proposes a near-optimal, impulsive control method for overflying ground targets. A set of impulses, which adjusts the satellite ground trajectory to enable the overflight, is derived. The proposed impulsive solution is expressed in closed-form which only consists of orbital revolution numbers and target overflight errors. It allows not only simplifying the search for a minimum delta-v solution but also facilitating its extension to the multiple-target overflight problem. It is also provided in this paper an insight connecting the overflight problem to a relative orbit transfer problem controlling a phase angle or mean anomaly. Numerical examples for single-, two-, and three-target overflight verify that the controlled trajectory with the proposed impulsive solution successfully overflies the designated targets within a prescribed time. It is also demonstrated that the total delta-v required for the proposed method is near-optimal based on the performance comparison with a numerical optimization-based method which employs a very large set of initial guesses.

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.

New families of periodic orbits in the 4-body problem emanating from a kite configuration

By using analytical and numerical tools we show the existence of families of quasiperiodic orbits (also called relative periodic orbits) emanating from a kite configuration in the planar four–body problem with three equal masses. Relative equilibria are periodic solutions where all particles are rotating uniformely around the center of mass in the inertial frame, that is the system behaves as a rigid body problem, in rotating coordinates in general these solutions are quasiperidioc. We introduce a new coordinate system which measures (in the planar four–body problem) how far is an arbitrary configuration from a kite configuration. Using these coordinates, and the Lyapunov center theorem, we get families of quasiperiodic orbits, an by using symmetry arguments, we obtain periodic ones, all of them emanating from a kite configuration.

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.

Hybrid Guardian Map-based adaptive control of spacecraft formation flying on highly elliptical orbits in the restricted three-body problem

In this paper, a single parameter Guardian Map-based simple adaptive guidance and control architecture for spacecraft formation flying is developed. When compared with traditional vehicles, the dynamics of spacecraft features larger non-linearities, tighter coupling and higher uncertainty as the eccentricity of an orbit increases. In addition, most formation flying applications involve following bounded relative trajectories that are only applicable to circular orbits and hence becomes an unnatural trajectory when used on eccentric orbits, resulting in unnecessary fuel consumption. Making use of a combination of nonlinear Simple Adaptive Control and Guardian Map theories, a novel autonomous adaptive guidance and control system is developed such that the desired closed-loop stability of the system and asymptotic convergence is guaranteed throughout the orbit while tracking fuel-efficient natural trajectories that are applicable to eccentric orbits. Simulation results show various new types and sizes of bounded natural relative orbits to illustrate the increased performance and robustness of the proposed adaptive controller compared to a conventional non-adaptive linear quadratic regulator found in literature.

A Submillimeter Level Relative Navigation Technology for Spacecraft Formation Flying in Highly Elliptical Orbit.

Spacecraft formation flying (SFF) in highly elliptical orbit (HEO) has attracted a great deal of attention in many space exploration applications, while precise guidance, navigation, and control (GNC) technology—especially precise ranging—are the basis of success for such SFF missions. In this paper, we introduce a novel K-band microwave ranging (MWR) equipment for the on-orbit verification of submillimeter-level precise ranging technology in future HEO SFF missions. The ranging technique is a synchronous dual one-way ranging (DOWR) microwave phase accumulation system, which achieved a ranging accuracy of tens of microns in the laboratory environment. The detailed design and development process of the MWR equipment are provided, ranging error sources are analyzed, and relative orbit dynamic models for HEO formation scenes are given with real perturbations considered. Moreover, an adaptive Kalman filter algorithm is introduced for SFF relative navigation design, incorporating process noise uncertainty. The performance of SFF relative navigation while using MWR is tested in a hardware-in-the-loop (HIL) simulation system within a high-precision six degrees of freedom (6-DOF) moving platform. The final range estimation errors from MWR using the adaptive filter were less than 35 m and 8.5 m/s for range rate, demonstrating the promising accuracy for future HEO formation mission applications.

Closed-Form Optimal Impulsive Control of Spacecraft Formations Using Reachable Set Theory

This paper addresses the spacecraft relative orbit reconfiguration problem of minimizing the delta- cost of impulsive control actions while achieving a desired state in fixed time. The problem is posed in relative orbit element (ROE) space, which yields insight into relative motion geometry and allows for the straightforward inclusion of perturbations in linear time-variant form. Reachable set theory is used to translate the cost-minimization problem into a geometric path-planning problem and formulate the reachable delta- minimum, a new metric to assess optimality and quantify reachability of a maneuver scheme. Next, this paper presents a methodology to compute maneuver schemes that meet this new optimality criterion and achieve a prescribed reconfiguration. Though the methodology is applicable to any linear time-variant system, this paper uses a state representation in ROE to derive new globally optimal maneuver schemes in orbits of arbitrary eccentricity. The methodology is also used to generate quantifiably suboptimal solutions when the optimal solutions are unreachable. Further, this paper determines the mathematical impact of uncertainties on achieving the desired end state and provides a geometric visualization of those effects on the reachable set. The proposed algorithms are tested in realistic reconfiguration scenarios and validated in a high-fidelity computational simulation environment.

Parametric Approach for Satellite Relative Orbit Designs and Constellations

This paper proposes a dynamics solution that creates an identical constellation pattern of multiple satellites with the parameterization of relative motion dynamics. The parameterization illustrates that the complexity of relative motion dynamics is interpreted by a Ptolemaic model of an epicycle and deferent system. All possible relative orbits can be represented by a hypo-epi curve or a deformed curve on the x-y plane and a simple z-axis harmonic oscillator motion. The designed relative orbits are used to create identical constellation patterns of multiple satellites, called parametric constellations in this study, by closed-form formulas and satellite phasing rules obtained from the geometrical relationship and periodic conditions. In conclusion, the resulting solution contributes to solving technical dynamics and tracking problems by simply designing complex relative motions in free space.

Closed form parametric solutions of nonlinear Abel-type and Riccati-type spacecraft relative motion

In comparison to the conventional approach of using Cartesian coordinates to describe spacecraft relative motion, the relative orbit description using Keplerian orbital elements provides a better visualization of the relative motion due to the benefit of having only one term (anomaly) that changes with time out of the six orbital elements leading to the reduction of the number of terms to be tracked from six, as in the case of Hill coordinates, to one. In this paper, under certain assumptions and transformations, the spacecraft relative equations of motion, in terms of orbital element differences, is approximated into the nonlinear first kind Abel-type and Riccati-type differential equations. Furthermore, we present methodologies for the formulation of the close form analytical solutions of the approximated equations. As shown by the numerical simulations, the closed form solutions and the nonlinear equations are in conformity with Riccati-type equations having higher errors than the Abel-type equations. This shows that the Abel-type equation, a third order polynomial, approximated the relative motion better than the Riccati-type equation, a second order polynomial. The resulting new analytical solutions gave better insight into the relative motion dynamics and can be used for the analysis of spacecraft formation flying, proximity and rendezvous operations.

Application of Lagrangian coherent structures to Coulomb formation on elliptic orbit

Coulomb thrusting presents attractive prospects in many astronautical missions from the standpoint of close-proximity formation flying. In this study, a dynamical model of Coulomb formation and its Hamiltonian is derived on the basis of the Tschauner–Hempel equation. As a nonautonomous system, it is more complicated than the dynamical model based on the Clohessy–Wiltshire equation. The Lagrangian coherent structure (LCS), a useful tool for describing the dynamical behavior of nonautonomous systems, is used to study the Coulomb formation dynamics, design, and reconfiguration. Simulation results show that, in the autonomous case of Coulomb formation, the LCS coincides with the invariant manifolds in the proper Poincare section. When it is extended to the nonautonomous case with a small-eccentricity reference orbit, the global morphology of the Coulomb formation dynamics remains almost unchanged. Based on the property that the LCS can act as the transport barrier in dynamical systems, it is used to construct homoclinic and heteroclinic orbits and search for invariant relative orbits and transfer trajectories for formation design and reconfiguration.

Rotation based analytic range-only initial relative orbit solution for natural periodic motion

Range-only Initial relative orbit determination (IROD) for natural-periodic relative motion (periodically coasting flight) during close-in proximity suffers from a mirror-solution (also named orbital ambiguity) problem if linear relative dynamics is used to propagate the orbit. Approaches proposed in previous work were to perform specific orbital maneuvers, measure by double range-sensor or utilize prior information so as to avoid mirror solutions. Alternatively, if the range-sensor offset from the vehicle center-of-mass (COM) is included, the range-only IROD problem may be uniquely solved. This research develops a novel solution to the range-only IROD problem for natural periodic motion in the context of Clohessy–Wiltshire dynamics, where the range-sensor offset is included to provide state observability to exclude the mirror solutions and certain attitude maneuver is introduced to construct analytic solution. Therefore, the range-only IROD is reduced to a problem of solving linear equations. The uniquely analytic solution is achieved based on these equations. Three rotation schemes for attitude maneuver are analyzed among which the y-mirror rotation scheme is proven to be effective for de-ambiguity. The proposed algorithm is verified and tested by a set of Monte Carlo simulations. The sensitivity of the solution accuracy to different orbits, range-sensor offset, ranging accuracy, rotating accuracy and measurement span is presented and discussed.

GRACE-FO precise orbit determination and gravity recovery

The gravity recovery and climate experiment follow-on (GRACE-FO) satellites, launched in May of 2018, are equipped with geodetic quality GPS receivers for precise orbit determination (POD) and gravity recovery. The primary objective of the GRACE-FO mission is to map the time-variable and mean gravity field of the Earth. To achieve this goal, both GRACE-FO satellites are additionally equipped with a K-band ranging (KBR) system, accelerometers and star trackers. Data processing strategies, data weighting approaches and impacts of observation types and rates are investigated in order to determine the most efficient approach for processing GRACE-FO multi-type data for precise orbit determination and gravity recovery. Two GPS observation types, un-differenced (UD) and double-differenced (DD) observations in general can be used for GPS-based POD and gravity recovery. The GRACE-FO KBR observations are mainly used for gravity recovery, but they can be also used for POD to improve the relative orbit accuracy. The main purpose of this paper is to study the impacts of the DD, UD and KBR observations on GRACE-FO POD and gravity recovery. The precise orbit accuracy is assessed using several tests, which include analysis of orbital fits, satellite laser ranging residuals, KBR range residuals and orbit comparisons. The gravity recovery is validated by comparing different gravity solutions through coefficient-wise comparison, degree difference variances and water height variations over the whole Earth and selected area and river basins.

Long term and safe relative orbit design for heterogeneous spacecraft clusters

The problem of obtaining long term relative orbit configurations for spacecraft clusters with realistic operational considerations such as safety, station keeping and inter-spacecraft distances is addressed. Two different approaches are developed for station keeping and safety objectives. In the first approach, relative orbit configurations, or relative TLEs, are found minimizing deviations from reference mean orbit which would maximize the station-keeping objective. In second one, relative configurations are found from a reference initial condition by minimizing probability of collision, hence maximizing the safety objective, between the spacecraft in the cluster which are propagated numerically through a high precision orbit propagator. For the design optimization, a derivative free algorithm is proposed. Effectiveness of the approaches is demonstrated through simulations. Using this design framework, several configurations can be found by exploring the limits of the clusters in terms of spacecraft number, distance bounds and probabilities of collision for long time intervals.