Optimal range observability maneuvers of a spacecraft formation using angles-only navigation

IntroductionARGE constellations of spacecraft are set to become increasingly common.Orbit determination (OD) can be a non-trivial task for a large constellation operator, and efficient operations of such constellations is a crucial challenge.This Note aims at addressing one critical issue of large constellations, that is, the relative orbit determination (OD).Relative OD provides basic state information for subsequent spacecraft control and operation.In addition, determining relative orbits can be used to aid certain scientific and operational objectives, such as the gravity recovery in [1] and Earth observation in [2].A significant volume of work [3][4][5][6][7] has focused on relative OD of two spacecraft, this is essentially an estimation problem based on inter-satellite ranging measurements.The prior work [8] has shown that inter-satellite ranging measurements are not enough to determine all orbit elements.Hill and Born [9-10] detail the obtainable orbit elements using inter-satellite range measurements, and that the full orbit element set can only be obtained through consideration of multi-body dynamics.Qin [11] further analyzed the sensitivity of relative orbit element determination to orbital geometric configurations, showing that the relative OD performance in some special configurations is degraded, making some orbit elements underivable.To the best of the authors knowledge, no prior work has considered relative OD between non-directly connected spacecraft.The challenge of relative OD of unconnected spacecraft needs to be addressed so that every spacecraft can obtain its relative orbit, and the relative OD for a constellation can be completely derived.Without direct measurements, relative OD can neither be processed as an estimation problem nor a simple linear geometric problem.

I N RECENT years, there has been considerable interest in spacecraft formation flying, due to the advantages of autonomy, flexibility, and relatively low cost, which is mostly required in the future space missions. One important application is autonomous rendezvous and proximity operation to a cooperative or noncooperative target in space [1–4], where two spacecraft should be ensured to fly in close formation in order to accomplish inspection, servicing, or capture at the specified time [5]. To implement autonomous rendezvous, it is essential to study the relative motion of one spacecraft (referred to as the chaser) with respect to another reference spacecraft (referred to as the target). The relative motion problems have been treated in many references. For instance, early in 1960, the first study on the relative motion of satellite clusters was performed by Clohessy and Wiltshire [6]. The authors presented the linearized equations of relative motion under circular-orbit assumption, well known as Clohessy–Wiltshire equations. Thereafter, Lawden [7] and Tschauner and Hempel [8] first derived the relative motion equations for eccentric orbits. The effects of the reference orbit eccentricity, the gravity perturbation J2, drag, and solar radiation pressure on the relativemotion are shown in [9–13]. Additionally, in these studies, along with developing the relative motion dynamics, the relative trajectory design is also presented to specify six initial conditions for relative position and velocity components, also seen in [14–17]. However, further studies should be developed on two issues. On one hand, these studies mostly used relative position and velocity components to describe the analytic solution and to design the relative orbit. However, it is difficult to intuitively know the size, location, and orientation of the relative trajectory with respect to the target orbit from the relative position and velocity components. Lovell and Tragesser [18] first introduced a set of relative orbit elements to replace the position and velocity components through a simple coordinate transformation. Nevertheless, especially for the description of the orientation of relative trajectory plane, the use of the relative orbit elements is not intuitive enough. Hence, we introduce one new relative orbit element to describe intuitively the orientation of the relative trajectory plane. On the other hand, most of these studies seldom provided the practical considerations such as the requirements of the vehicle’s solar power generation and communications subsystems and thus cannot exploit the relationship of the relative orbit elements and the specified mission design. Reference [19] first investigates the relative orbit design while considering the relative navigation pointing and sun pointing constraints, especially optimizing the solar collection. However, the author illustrates the problem mainly based on one simple example and its simulation. Instead, the objective of our research is to provide thorough theoretical derivation for a general case. In this Note, flyaround orbit design during autonomous rendezvous for a cooperative target or noncooperative target is considered. The rest of this Note is organized as follows. In Sec. II, the relative dynamics is characterized and the relative orbit elements are introduced to describe intuitively the relative trajectory. Section III provides the chaser’s desired attitude constraints to satisfy relative navigation pointing and sun pointing requirements. Then the flyaround orbit design is mainly investigated to specify the initial conditions of relative orbit elements, especially the initial phase angle on the consideration of maximizing the solar power collection of the chaser. Conclusions and future work are given in Sec. IV.

The need to preserve commercial and scientific relevant orbits in the low Earth belt asks for the active removal of inoperative satellites, which lay on slowly decaying orbits and typically present a moderate eccentricity value. Proximity operations around non-cooperative targets require the capability to execute prompt inbound/receding trajectories as well as a certain level of autonomy to react in an operationally safe fashion. The guidance and control strategies developed for docking cooperative craft can hardly be used in this case for operational reasons. As a result, the study of control solutions applicable to close-range proximity operations and suitable for spaceborne implementation is an active research field.

Useful relative motion description method for Perturbations analysis in satellite 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.

While the description of orbiting spacecraft relative motion is usually done in the rotating Hill frame due to analytical first-order solutions and the intuitive shape of the relative orbits, it is disadvantageous for mission design requirements that are fixed in the inertial frame. This includes distributed space telescopes aligned with inertial targets as well as formations and servicing operations with inertially fixed keep-in/out zones, e.g. constraints imposed by the Sun direction. This paper studies the analytical first-order inertial frame solutions of the relative motion of orbiting spacecraft and derives geometrically meaningful relative orbit elements and invariants of motion for inertial frame relative orbits. It is found that the relative motion for a circular chief orbit corresponds to the motion of an epitrochoid. For elliptic chief orbits, the inertial frame relative orbits are stretched and distorted compared to the epitrochoid curve for circular chief orbits, but similar relative orbit elements are defined as well. Finally, the variational equations of the inertial relative orbit elements are developed and their use is demonstrated through an asymptotically stabilizing continuous feedback control law.

Relative orbital elements provide a geometric interpretation of the motion of a deputy spacecraft about a chief spacecraft. The formulation yields an intuitive understanding of how the relative motion evolves with time, and by incorporating velocity changes in the local-vertical, local-horizontal component directions, the change in relative motion due to impulsive maneuvers can be evaluated. This paper utilizes a relative orbital element formulation that characterizes relative motion where the chief spacecraft is assumed to be in a circular orbit. Expressions are developed for changes to the relative orbital elements as a function of the impulsive maneuver components in each coordinate direction. A general maneuver strategy is developed for targeting a set of relative orbital elements, and this strategy is applied to scenarios that are relevant for close proximity operations, including establishing a stationary relative orbit, natural motion circumnavigation, and station-keeping in a leading or trailing orbit.

This paper focuses on the aerospace application of a single beam laser rangefinder (LRF) for 3D imaging, shape detection, and reconstruction in the context of a space-based space situational awareness (SSA) mission scenario. The primary limitation to 3D imaging from LRF point clouds is the one-dimensional nature of the single beam measurements. A method that combines relative orbital motion and scanning attitude motion to generate point clouds has been developed and the design and characterization of multiple relative motion and attitude maneuver profiles are presented. The target resident space object (RSO) has the shape of a generic telecommunications satellite. The shape and attitude of the RSO are unknown to the chaser satellite however, it is assumed that the RSO is un-cooperative and has fixed inertial pointing. All sensors in the metrology chain are assumed ideal. A previous study by the authors used pure Keplerian motion to perform a similar 3D imaging mission at an asteroid. A new baseline for proximity operations maneuvers for LRF scanning, based on a waypoint adaptation of the Hill-Clohessy-Wiltshire (HCW) equations is examined. Propellant expenditure for each waypoint profile is discussed and combinations of relative motion and attitude maneuvers that minimize the propellant used to achieve a minimum required point cloud density are studied. Both LRF strike-point coverage and point cloud density are maximized; the capability for 3D shape registration and reconstruction from point clouds generated with a single beam LRF without catalog comparison is proven. Next, a method of using edge detection algorithms to process a point cloud into a 3D modeled image containing reconstructed shapes is presented. Weighted accuracy of edge reconstruction with respect to the true model is used to calculate a qualitative “metric” that evaluates effectiveness of coverage. Both edge recognition algorithms and the metric are independent of point cloud density, therefore they are utilized to compare the quality of point clouds generated by various attitude and waypoint command profiles. The RSO model incorporates diverse irregular protruding shapes, such as open sensor covers, instrument pods and solar arrays, to test the limits of the algorithms. This analysis is used to mathematically prove that point clouds generated by a single-beam LRF can achieve sufficient edge recognition accuracy for SSA applications, with meaningful shape information extractable even from sparse point clouds. For all command profiles, reconstruction of RSO shapes from the point clouds generated with the proposed method are compared to the truth model and conclusions are drawn regarding their fidelity.

This paper expands on previous studies by the authors into 3D imaging with a single-beam laser rangefinder (LRF) by implementing real-time attitude maneuvers of a chaser satellite flying in relative orbit around a resident space object (RSO). Point clouds generated with an LRF are much sparser than those generated with an imaging LIDAR, making it difficult to autonomously distinguish between gaps in coverage and truly empty space. Furthermore, if both the attitude and the shape of the target RSO are unknown, it is particularly difficult to register a collection of LRF strike points together and detect gaps in strike point coverage in realtime. This paper presents the incorporation of a narrow field of-view (NFOV) camera that detects the strike point on the RSO and supplements LRF distance measurements with image data. This data is used to generate attitude command profiles that efficiently fill LRF coverage gaps and generate high density point clouds, thus maximizing coverage of an unknown RSO. Results obtained so far point the way to a real-time implementation of the algorithm. A method to detect and close gaps in LRF strike point coverage is presented first. Coverage gap detection is achieved using Voronoi diagrams, where Voronoi cells are centered at the LRF strike points. A three-part algorithm is used that 1) creates a 3D panoramic map from “stitched” NFOV camera images; 2) correlates the areas of sparse LRF coverage to the map; and 3) generates attitude commands to close the coverage gaps. The map provides a consistent and reliable method to register positions of strike points relative to each other and to the NFOV image of the RSO without a priori knowledge of the RSO attitude. Using this algorithm, gaps and sparse areas in LRF coverage are covered with strike points, allowing for the generation of a higher-resolution point cloud than that obtained with preprogrammed attitude profiles. Attitude maneuvers can now be designed on-line in real-time such that they satisfy the constraints of the chaser spacecraft attitude determination and control system. Finally, the effectiveness of the camera-aided generation of attitude profiles is analyzed by using a weighted edge reconstruction metric, and comparing results to those generated with pre-programmed attitude maneuvers. The effect of on-line maneuver generation on the overall decrease of time and propellant expenditure to generate an adequate point cloud is also discussed. The analysis bears particular relevance to low-budget, nano-satellite demonstration missions for space-based space situational awareness (SSA).

Advanced multisatellite missions based on formation-flying and on-orbit servicing concepts require the capability to arbitrarily reconfigure the relative motion in an autonomous, fuel efficient, and flexible manner. Realistic flight scenarios impose maneuvering time constraints driven by the satellite bus, by the payload, or by collision avoidance needs. In addition, mission control center planning and operations tasks demand determinism and predictability of the propulsion system activities. Based on these considerations and on the experience gained from the most recent autonomous formation-flying demonstrations in near-circular orbit, this paper addresses and reviews multi-impulsive solution schemes for formation reconfiguration in the relative orbit elements space. In contrast to the available literature, which focuses on case-by-case or problem-specific solutions, this work seeks the systematic search and characterization of impulsive maneuvers of operational relevance. The inversion of the equations of relative motion parameterized using relative orbital elements enables the straightforward computation of analytical or numerical solutions and provides direct insight into the delta- cost and the most convenient maneuver locations. The resulting general methodology is not only able to refind and requalify all particular solutions known in literature or flown in space, but enables the identification of novel fuel-efficient maneuvering schemes for future onboard implementation.

This paper discusses the application of a single beam laser rangefinder (LRF) to point cloud generation, shape detection, and shape reconstruction for a space-based space situational awareness (SSA) mission. The LRF is part of the payload of a chaser satellite tasked to image a resident space object (RSO). The one-dimensional (1D) nature of LRF returns significantly increases the complexity of the imaging task. To maximize coverage, a method to autonomously detect and fill gaps in sparse point cloud coverage using a narrow field of view (NFOV) camera in conjunction with the LRF is presented. First, relative orbital motion and scanning attitude motion are combined to generate a baseline 3D point cloud of the RSO. The effectiveness of pregenerated command profiles is analyzed by using a weighted edge reconstruction metric that scores how well a point cloud characterizes RSO shape. The design and characterization of multiple relative motion and attitude maneuver profiles, as well as the time and propellant cost of each profile, are presented with the assumption that the entire metrology chain is error free. Next, a three-part algorithm is used that 1) creates a 3D panoramic map from stitched NFOV camera images, 2) correlates the areas of sparse LRF coverage to the map, and 3) generates attitude commands to close the coverage. This provides a consistent and reliable method to register positions of strike points relative to each other and to the NFOV image of the RSO with a priori knowledge of the RSO attitude. Gaps and sparse areas in LRF coverage are covered with strike points; the result is a point cloud of significantly higher resolution than that obtained with preprogrammed attitude profiles. The analysis bears particular relevance to power-constrained nanosatellite missions for space-based SSA for whom a multibeam LRF payload is not feasible. Maneuvers can now be designed on-line in real time; results presented validate the utility of a single-beam LRF as a tool for 3D imaging of RSO shapes.

PurposeThe purpose of this paper is to evaluate the safety of formation flying satellites, and propose a method for practical collision monitoring and collision avoidance manoeuvre.Design/methodology/approachA general formation description method based on relative orbital elements is proposed, and a collision probability calculation model is established. The formula for the minimum relative distance in the crosstrack plane is derived, and the influence of J2 perturbation on formation safety is analyzed. Subsequently, the optimal collision avoidance manoeuvre problem is solved using the framework of linear programming algorithms.FindingsThe relative orbital elements are illustrative of formation description and are easy to use for perturbation analysis. The relative initial phase angle between the in‐plane and cross‐track plane motions has considerable effect on the formation safety. Simulations confirm the flexibility and effectiveness of the linear programming‐based collision avoidance manoeuvre method.Practical implicationsThe proposed collision probability method can be applied in collision monitoring for the proximity operations of spacecraft. The presented minimum distance calculation formula in the cross‐track plane can be used in safe configuration design. Additionally, the linear programming method is suitable for formation control, in which the initial and terminal states are provided.Originality/valueThe relative orbital elements are used to calculate collision probability and analyze formation safety. The linear programming algorithms are extended for collision avoidance, an approach that is simple, effective, and more suitable for on‐board implementation.

Formation design in elliptical orbit using relative orbit elements

This paper establishes a methodology for obtaining the general solution to the spacecraft relative motion problem by utilizing the Cartesian configuration space in conjunction with classical orbital elements. The geometry of the relative motion configuration space is analyzed, and the relative motion invariant manifold is determined. Most importantly, the geometric structure of the relative motion problem is used to derive useful metrics for quantification of the minimum, maximum, and mean distance between spacecraft for commensurable and noncommensurable mean motions. A number of analytic solutions as well as useful examples are provided, illustrating the calculated bounds. A few particular cases that yield simple solutions are given. Nomenclature a = semimajor axis E = eccentric anomaly E = follower orbit e = eccentricity F = follower perifocal frame f = true anomaly I = inertial frame i = inclination Jk = Bessel function L = leader-fixed frame M = mean anomaly n = mean motion n0 = fundamental frequency R = leader position vector R = relative motion invariant manifold r = follower position vector W = distance function α = normalized semimajor axis μ = gravitational constant ρ = relative position vector � = right ascension of the ascending node ω = argument of periapsis ω = angular velocity vector |·| = vector norm �·� = signal norm Superscripts � = leader ∗ = relative orbital element

Analytical guidance for spacecraft relative motion under constant thrust using relative orbit elements