Decomposition analysis of spacecraft relative motion with different inter-satellite ranges

In this paper, the analytical closed-form solution of the Tschauner-Hempel equations that are used for rendezvous in elliptic orbits are studied and exploited. The solution is based on a state transition matrix that is completely explicit in time. Autonomous guidance algorithms to approach, to flyaround, and to depart from a target vehicle in elliptic orbits are exploited based on that solution. The algorithms are general and able to translate the chaser vehicle in any direction, decelerate while approaching the target vehicle, and accelerate when moving away. These guidance algorithms are based on the glideslope used in the past for rendezvous and proximity operations of the Space Shuttle with other vehicles. The algorithms are applied to the problem of terminal rendezvous near any Keplerian elliptic orbit in a Newtonian gravitational field showing the effects of target orbit eccentricity on the relative motion. These algorithms extend and generalize previously published solutions. Numerical simulations confirm the brevity and accuracy of the general solutions developed in the current paper.

Rendezvous maneuvers represent one of the most complex maneuvers in spaceflight. To approximate the relative motion between two objects during short time periods, linearized motion models, such as the well-known Clohessy-Wiltshire model, are commonly employed. To address the inaccuracy of these equations arising from the non-spherical gravitational potential field, Schweighart and Sedwick developed a linear relative motion model which accounts for the J 2 effect. Closed-form solutions to these equations including differential specific forces between the two spacecrafts have been widely applied by multiple researchers to study the methodology of differential lift and drag. While working with these solutions, however, it became apparent that small deviations between the numerically integrated and analytical solutions result if specific differential forces are applied in the tangential direction δ f y , which in the case of differential aerodynamic forces corresponds exactly to the main influence, namely differential drag. This insight was taken as an opportunity to determine corrected closed-form solutions via the computer algebra system Maxima (Maxima, 2023), a software package for performing computer algebra calculations. In this technical note, these solutions are presented, their correctness is demonstrated and subsequently the effect of the motion in the phase plane due to perturbing forces is discussed. This note is intended to contribute to the field of research by increasing the accuracy of future studies. • Corrected closed-form solutions to the Schweighart–Sedwick relative motion model. • Solutions obtained using the computer algebra system Maxima . • Solutions presented in a general and a decomposed form. • Discussion of the effect of differential forces on the phase plane. • Explicit elaboration of the deviating terms.

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

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

A systematic approach to modeling the relative motion of artificial satellites in the presence of perturbations is presented. The relative motion is described using relative position and velocities as states. The modeling here is restricted to low Earth orbit (LEO) satellites and therefore includes the differential J <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> and drag effects. In this paper we expand on the modeling approach that makes use of the Reference Satellite Variables for the chief's orbit using simple Newtonian mechanics to systematically derive the exact nonlinear relative motion model with differential J <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> and drag. These equations are exact for eccentric reference orbits as well as equatorial. This intuitive modeling approach shall establish a framework to incorporate other kinds of differential perturbations for higher fidelity models based on the significance of application. Simulation results of the developed nonlinear relative motion model show the effect of differential J <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> and drag captured by the equations for a LEO leader-follower formation with large intersatellite distances. The propagation errors of the model are studied for varying initial conditions and reference orbits. A subsequent analysis gives further insight into how the model developed is particularly free from singularities in the special case of J <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> and drag disturbances alone.

An integral sliding mode observer (ISMO)-based method for estimating the structure and motion (SaM) of an object in general motion is proposed using a monocular dynamic (moving) camera. As the unknown range and object velocity can be considered as disturbances and should be quickly estimated, it is necessary to maintain the robustness against these disturbances from the start by eliminating the reaching mode. Therefore, the ISMO-based method is proposed on the basis of a relative camera-object motion model. By formulating the relative motion model with three-dimensional measurable state variables, three unknown components among the range, unknown object velocity components, and unknown camera velocity components can be estimated by the proposed method in the following cases: i) the camera is in dynamic motion and the object is in semigeneral motion [i.e., initially static (stationary) and then in general (static or dynamic) motion], ii) the camera is in dynamic motion and the object is in general motion within a constrained space, and iii) both the object and camera are in general motion within a less constrained space when the range information is available. Simulation and experimental results demonstrate that the range and object velocity can be estimated with a satisfactory transient response using a monocular dynamic camera.

I N RECENT years, many proposed satellite missions have undergone a paradigm shift away from large, expensive spacecraft toward the use of smaller satellites flying in formation. Although formation flying offers a number of advantages, such as robustness of the mission to failure and potential cost savings, the benefits are partially offset by the increased complexity of the guidance, navigation, and control systems and by V-limited mission durations. The development of advantageous relative dynamics between the constituent satellites could help extend mission durations by minimizing fuel usage. In particular, periodic or quasiperiodic relative orbits would reduce the control effort necessary to maintain the formation and significantly prolong the mission’s lifetime. Attempts to identify naturally periodic relative orbits can be found throughout the current literature. Vaddi et al. [1] modified initial conditions from the Hill–Clohessy–Wiltshire (HCW) equations to enforce bounded relative motion in the presence of small eccentricity and second-order differential gravity terms. Kasdin and Kolemen [2] solved the Hamilton–Jacobi equation in terms of epicyclic orbital elements to derive bounded, periodic orbits in the presence of higher-order gravity and certain perturbations. Schaub and Alfriend [3] identified the conditions for J2-invariant relative orbits bymatching the secular drift rates of themean orbital elements of the chief and deputy satellites. In most of these analytical cases, however, the authors established precise periodic motion using approximated dynamical models. When higher fidelity dynamics and realistic orbital parameters are used, these methods break down and exhibit relative orbit drift or a high sensitivity to initial condition errors. Several numerical approaches to the problem of relative orbit periodicity have also been attempted. Sabatini et al. [4] and Izzo and Sabatini [5] used a refined genetic algorithm to optimize relative trajectories for maximum periodicity in the presence of J2 perturbations. Quasi-periodic orbits were obtained for two sets of “magic” inclinations (49.11 and 63.43 deg), at which resonance between the formation’s inand out-of-plane motion results in projected circular orbit (PCO) formations with very small orbital drift. Damaren [6] formulated an iterative shooting approach based on the Newton method to close the relative orbit and achieve “almost” periodic initial conditions. This technique requires a low-level state-feedback control loop to track a trajectory based on the HCW equations, but is robust to initial condition errors and exhibits very low drift characteristics in circular orbits. A similar approach was taken by Becerra et al. [7], who applied a nonlinear Hamiltonian model to the Newton method to obtain quasi-periodic relative initial conditions of the deputy in the presence of the J2 perturbation. A linear quadratic regulator (LQR) controller was used to track reference trajectories that were developed from sinusoidal functions of time and based on a Keplerian orbit. As yet, a method of numerically designing reference trajectories that match or closely match the natural perturbed relative motion of the deputy satellite in a generic orbit, and that are robust to initial condition errors, has not been developed. Such trajectories, with a properly designed low-authority controller, could be tracked for minimal V requirements, thus enabling formation flying missions to be significantly extended. The intent of this study is to design such trajectories by continuing the development of the methods used to search for almost-periodic orbits, begun in [6], and applying them to the J2-invariant trajectories in [3]. The resulting relative orbits will approach true periodicity by capitalizing on advantageous initial conditions, numerically designed reference trajectories, and optimal control strategies.

PurposeThe purpose of this paper is to propose an intuitive and effective cluster flight orbit design method for fractionated spacecraft.Design/methodology/approachBased on the concept of fractionated spacecraft, orbit design requirements for cluster flight in the case of fractionated spacecraft are proposed, and categorized into three requirements: stabilization requirement, passive safety requirement, and the maximum inter‐satellite distance requirement. These design requirements are then reformulated in terms of relative eccentricity and inclination vectors (E/I vectors) using a relative motion model based on relative orbital elements (ROEs). By using ROEs theory, the cluster flight orbit design issue is modelled as the distribution of relative E/I vectors for each member satellite in the cluster, and solved by combining three different heuristic search methods and one nonlinear programming (NLP) method.FindingsThe simulation results show that the NLP method is valid and efficient in solving the cluster flight orbit design problem and that for some cluster flight scenarios, the heuristic search methods can be adopted to give feasible solutions without the NLP method.Research limitations/implicationsThe cluster flight scenario in this paper is limited because the cluster should be in the near‐circular low earth orbit (LEO), and the relative distance between the member satellites should be small enough to satisfy the relative motion linearization assumption.Practical implicationsThe cluster flight orbit design method proposed in this paper can be applied by fractionated spacecraft mission designers to propose potential cluster flight orbit solutions.Originality/valueIn this paper, the relative E/I vectors method is adopted to propose an intuitive and effective cluster flight orbit design method for fractionated spacecraft.

An impulsive feedback control is developed to establish specie c relative orbits for spacecraft formation e ying. The relative orbit tracking errors are expressed in terms of mean orbit elements. The feedback control, based on Gauss’ s variational equations of motion, allows specie c orbit elements or orbit element sets to be controlled with minimal impact on the remaining osculating orbit elements. This is advantageous when J2-invariant orbits are to be controlled, where only the argument of perigee and mean anomaly will drift apart at equal and opposite rates. The advantage of this impulsive feedback control, compared to optimal control solutions, is that it can operate with little computational effort and in a near-optimal manner, whilerequiring only a marginal penalty in fuel cost. When applied to the spacecraft formation e ying problem, this control could also be used to perform general orbit corrections. Formulas are developed providing accurate estimates of the sensitivities of the mean semimajor axis and mean eccentricity with respect to the osculating inclination angle. With these sensitivities, the tracking error in semimajor axis, eccentricity, and inclination angle can be canceled within one orbit.

New models of Pacific absolute plate motion relative to hot spots and models of relative plate motion involving the Pacific plate all agree there was a significant change in the late Neogene (Chron 3A, ∼5.89 Ma), reflecting a more northerly absolute motion than previously determined. As Pacific absolute plate motion became slightly more northerly, left‐stepping transform segments came under compression. Some left‐stepping segments became microplates with clockwise rotation; others show clear evidence of compressional deformation. Conversely, right‐stepping transforms came under tension, and many developed intratransform spreading centers or show similar evidence for transform magmatism. Several large left‐stepping transform offsets represent portions of the Nazca plate protruding into the Pacific and as such act as obstacles to the more northerly Pacific absolute plate motion. We suggest these obstructions act to enhance the generally tensile equatorial Pacific stress regime caused by distant slab pull. As a consequence, the greater French Polynesia region has experienced diffuse volcanism that increased considerably following the Chron 3A plate motion change. We propose that since the Hawaii‐Emperor Bend time, the distant Pacific slab pull and the friction between the large, buoyant Ontong Java plateau and the northern margin of the Australia plate have produced episodes of increased tensile stresses in the equatorial Pacific. We believe these stress excursions are responsible for much of the intraplate volcanism observed in a wide triangular region from Samoa to the East Pacific and Pacific‐Antarctic Rises.

In this paper we develop the nonlinear motion equations in terms of the true anomaly varying Tschauner–Hempel equations relative to a notional orbiting particle in a Keplerian orbit, relatively close to an orbiting primary satellite to estimate the position of a spacecraft. A second orbiting body in Earth orbit relatively close to the first is similarly modelled. The dynamic relative motion models of the satellite and the second orbiting body, both of which are modelled in terms of independent relative motion equations, include several perturbing effects, such as the asymmetry of the Earth gravitational field resulting in the Earth's oblateness effect and the third body accelerations due to the Moon and the Sun. Linear control laws are synthesised for the primary satellite using the transition matrix so it can rendezvous with the second orbiting body. The control laws are then implemented using the state estimates obtained earlier to validate the feedback controller. Thus, we demonstrate the application of a Linear Quadratic Nonlinear Gaussian (LQNG) design methodology to the satellite rendezvous control design problem and validate it.

In this paper, the required control with the J 2∼J 4 perturbation is derived for the spacecraft hovering. The J 2∼J 4 perturbation is introduced into spacecraft relative motion model by Gauss’s variation equations, and the orbit elements affected by the J 2∼J 4 perturbation are obtained. Combining the nonlinear relative motion model of the spacecraft with the obtained orbit elements, the analytic expression of the hovering control is completely derived. The numerical verification results indicate that the derived control considering the J 2∼J 4 perturbation can ensure the high accuracy for the spacecraft hovering.

State and disturbance estimation for test masses of drag-free satellites based on self-recurrent wavelet neural network

This paper investigates the air–ground cooperative time-varying formation-tracking control problem of a heterogeneous cluster system composed of an unmanned ground vehicle (UGV) and an unmanned aerial vehicle (UAV). Initially, the structure of the UAV–UGV formation-control system is analyzed from the perspective of a cooperative combat system. Next, based on the motion relationship between the UAV–UGV in a relative coordinate system, the relative motion model between them is established, which can clearly reveal the physical meaning of the relative motion process in the UAV–UGV system. Then, under the premise that the control system of the UAG is closed-loop stable, the motion state of the UGV is modeled as an input perturbation. Finally, using a linear quadratic optimal control theory, a UAV–UGV formation-maintenance controller is designed to track the reference trajectory of the UGV based on the UAV–UGV relative motion model. The simulation results demonstrate that the proposed controller can overcome input perturbations, model-constant perturbations, and linearization biases. Moreover, it can achieve fast and stable adjustment and maintenance control of the desired UAV–UGV formation proposed by the cooperative combat mission planning system.

Gravitational and third-body perturbations can be modeled with sufficient precision for most applications in low Earth orbit. However, owing to severe uncertainty sources and modeling limitations, computational models of satellite aerodynamics and solar radiation pressure are bound to be biased. Aiming at orbital propagation consistent with observed satellite orbital dynamics, real-time estimation of these perturbations is desired. In this paper, a particle filter for the recursive inference and prediction of nongravitational forces is developed. Specifically, after assuming a parametric model for the desired perturbations, the joint probability distribution of the parameters is inferred by using a prescribed number of weighted particles, each consisting of one set of orbital elements and one set of parameters. The particle evolution is carried out by means of an underlying orbital propagator, and the Bayes rule is used to recursively update weights by comparing propagated orbital elements with satellite observations. The proposed formulation uses mean orbital elements as the only available measurements. This feature makes the algorithm a potentially valuable resource for space situational awareness applications, such as space debris trajectories prediction from two-line elements, or for onboard force estimation from Global Positioning System data. High-fidelity simulations show that nongravitational perturbations can be estimated with 20% accuracy.