Spacecraft rendezvous by differential drag under uncertainties

At low Earth orbits, differentials in the drag forces between spacecraft can be used for controlling their relative motion in the orbital plane. Current methods for determining the drag force may result in errors due to inaccuracies in the density models and drag coefficients. In this work, a methodology for relative maneuvering of spacecraft based on differential drag, accounting for uncertainties in the drag model, is proposed. A dynamical model composed of the mean semimajor axis and the argument of latitude is used for describing long-range maneuvers. For this model, a linear quadratic regulator is implemented, accounting for the uncertainties in the drag force. The actuation is the pitch angle of the satellites, considering saturation. The control scheme guarantees asymptotic stability of the system up to a certain magnitude of the state vector, which is determined by the uncertainties. Numerical simulations show that the method exhibits consistent robustness to accomplish the maneuvers, even in the presence of realistic modeling of density fields, drag coefficients, the corotation of the atmosphere, and zonal harmonics up to J(8)

Limited-duty-cycle Satellite Formation Control via Differential Drag

As CubeSat formation flying missions relying on differential drag control become increasingly common, additional missions based on this control must be studied. A mission planning tool is investigated to control the relative spacing of a CubeSat formation where differential drag is the sole control mechanism. System performance is investigated under varying perturbations and a range of system parameters, including limiting the control duty cycle. Optimal solutions based on using a pseudo spectral numerical solver, GPOPS-II, to minimize maneuver time. This study includes the development of a mission planning tool to work with the modeled CubeSat mission to calculate optimal maneuvers for its mission architecture. The effects of mission altitude, solar cycle, various maneuver sizes and formations, limited control, various computational methods, and error checkers were evaluated. The mission planning tool developed can properly execute all desired run parameters and options, though it suffers from computational complexity. Pseudo spectral methods executed in MatLab were determined to be poorly suited to the problem due to memory requirements involved. Limited duty cycle control can be applied with differential drag with varying effectiveness dependent on mission parameters

Satellite formation flying control of the relative trajectory shape and size using lorentz forces

Propellantless control approaches for small satellite formation flying represent a special interest and an important advantage for space industry nowadays. A formation flying control algorithm using the Lorentz force for Low­Earth Orbits to achieve a trajectory with required shape and size is proposed in this dissertation. The Lorentz force is produced as the result of interaction between the Earth’s magnetic field and an electrically charged spacecraft. Achieving the required trajectories represents a challenge since the control is the variation of the satellite’s charge value. This control mechanism simplicity cannot provide full controllability. A Lyapunov­based control is developed for elimination of the initial relative drift after launch and it is aimed for reaching a required relative trajectory with predefined shape and size. The control algorithm is constructed to correct different parameters of the relative trajectory at different relative positions. The required amplitudes for close relative trajectories for in­plane and out­of­plane motion as well as the relative drift and shift of elliptical relative orbits are controllable using Lorentz force. Due to the absence of full controllability, the algorithm is incapable to correct the in­plane and out­of­plane motion phases, once these parameters are defined by the deployment conditions and therefore arbitrary. The proposed control allows the convergence to the trajectory with required shape and size. Centralized and decentralized control approaches are implemented and their performance is studied. The centralized approach considers two satellites formation formed by an electrically neutral leader satellite moving on a circular LEO and a follower which actively controls its orbital motion by changing its charge in order to remain in close vicinity of the leader. Formation flying consisting of more than two satellites with charge­changing capability can also be controlled by the proposed algorithm using a decentralized approach. This work also considers the control of satellite swarm trajectories in a sphere­shaped formation. Numerical simulation of the relative motion is used to study performance of the control algorithm. It implements the model of the geomagnetic field as a tilted dipole. The repulsive collision avoidance control is proposed for the case when the system elements are inside a dangerous proximity area. The convergence time and final trajectory accuracy are evaluated for different simulation parameters and conditions.Métodos de controlo para formações de voo de satélites de pequenas dimensões que não recorram ao uso de combustível representam, atualmente, um interesse especial e uma importante vantagem para a indústria espacial. Nesta dissertação é proposto um algoritmo de controlo que, recorrendo à força de Lorentz em orbitas terrestres baixas (LEO), é capaz de alcançar trajetórias com o respetivo o formato e o tamanho desejados. A força de Lorentz resulta de uma interação entre o campo magnético terrestre e o satélite eletricamente carregado. Alcançar as trajetórias solicitadas revela­se como sendo um desafio visto que o único método de controlo é a variação da carga interna do satélite. Este mecanismo de controlo revela­se como sendo incapaz de conferir controlabilidade total ao dispositivo. Um controlo baseado no método de Lyapunov é desenvolvido com o objetivo de eliminar a deriva inicial do satélite após o lançamento orbital e é destinado a atingir o tamanho e formato predefinidos da trajetória relativa objetivo. O algoritmo de controlo é construído de forma a corrigir os diferentes parâmetros da trajetória relativa em diferentes posições relativas. Usando a força de Lorentz é possível atingir tanto as amplitudes objetivo, considerando ambos os movimentos dentro e for do plano da trajetória, mas também a deriva e o deslocamento relativos da trajetória. Devido à falta de controlabilidade total, o algoritmo desenvolvido é incapaz de corrigir completamente os movimentos dentro e fora do plano da trajetória, visto que estes parâmetros são definidos na sua totalidade pelas condições de lançamento e, como tal, arbitrários. O algoritmo de controlo proposto possibilita a convergência dos valores para o formato e tamanho da trajetória desejada. Ambas as estratégias de controlo centralizadas e descentralizadas são aplicadas e a respetiva performance estudadas. No caso da estratégia centralizada, é considerado um voo em formação composto por dois satélites, onde o Líder se revela como sendo eletricamente neutro enquanto, e prescrevendo uma trajetória terrestre baixa circular, enquanto que o segundo, eletricamente ativo, é capaz de alterar o seu posicionamento relativo requerido, permutando a sua carga interna. Uma formação de voo considerando um número superior a dois satélites, com capacidades de carregamento elétrico, é também controlável considerando o algoritmo proposto. Este trabalho tem também em consideração o controlo da trajetória de um swarm de satélites num formato esférico. Simulações numéricas são usadas como método de análise da performance do algoritmo desenvolvido. Durante o processo de análise é implementado o modelo do dipolo inclinado como forma de simular o campo magnético terrestre. É também aplicado um algoritmo responsável por evitar situações de colisão eminente para casos em que a convergência de movimento dos satélites entra em zonas de proximidade critica. O tempo de convergência e a precisão da trajetória final são avaliadas para diferentes parâmetros e condições de simulação

Optimization techniques for satellites proximity maneuvers

The main topic of this dissertation is the control optimization problem for satellites Rendezvous and Docking. Saving resources is almost as important as the mission safeness and effectiveness. Three different numerical approaches are developed. The first two techniques deal with realtime and sub-optimal control, generating a reliable control sequence for a chaser spacecraft which eventually docks to a target. The first approach uses dynamic programming to quickly generate a sub-optimal control sequence on a predetermined path to be followed by one of the two vehicles involved into the docking operations. The second method presents a fast direct optimization technique, which was previously validated on real aircraft for trajectory optimization. The third approach aims to take into account the limitations of space qualified hardware, in particular thrusters. The new technique fuses the use of a set of low thrust on-off engines with impulsive-high-thrust engines. The hybrid method here developed combines and customizes different techniques. The relative motion in the above mentioned control strategies is represented by a linear dynamic model. As secondary topic of this dissertation, the use of a genetic algorithm optimizer to find possible conditions under which spacecraft relative motion can be periodic, or at least bounded, is presented. This analysis takes into account the J2 gravity perturbation and some drag effects. The importance of the obtained results directly apply to the problem of formation keeping, as natural dynamics can be exploited to reduce the amount of active control preventing the spacecrafts to drift apart along tim

Optimization techniques for satellites proximity maneuvers

The main topic of this dissertation is the control optimization problem for satellites Rendezvous and Docking. Saving resources is almost as important as the mission safeness and effectiveness. Three different numerical approaches are developed. The first two techniques deal with realtime and sub-optimal control, generating a reliable control sequence for a chaser spacecraft which eventually docks to a target. The first approach uses dynamic programming to quickly generate a sub-optimal control sequence on a predetermined path to be followed by one of the two vehicles involved into the docking operations. The second method presents a fast direct optimization technique, which was previously validated on real aircraft for trajectory optimization. The third approach aims to take into account the limitations of space qualified hardware, in particular thrusters. The new technique fuses the use of a set of low thrust on-off engines with impulsive-high-thrust engines. The hybrid method here developed combines and customizes different techniques. The relative motion in the above mentioned control strategies is represented by a linear dynamic model. As secondary topic of this dissertation, the use of a genetic algorithm optimizer to find possible conditions under which spacecraft relative motion can be periodic, or at least bounded, is presented. This analysis takes into account the J2 gravity perturbation and some drag effects. The importance of the obtained results directly apply to the problem of formation keeping, as natural dynamics can be exploited to reduce the amount of active control preventing the spacecrafts to drift apart along tim

Solar radiation pressure effects on very high-eccentric formation flying

A real alternative to Lagrange point very low perturbed orbits, for universe observation missions, is high eccentric Earth orbits. Combination of high eccentricity and very large semi-major axis leads to orbits with an important part of flight time far from Earth and its perturbations. Modeling this particular relative motion is the scoop of this paper. Main perturbation in HEO orbits are solar radiation pressure (SRP) and lunisolar effects, but formations are mainly affected by SRP effects. The modellization of its effects is done in two ways. First we introduce the SRP effects in the equations of the relative acceleration. Second, we obtain explicit analytical expressions of the temporal evolution of the relative motion. Resulting expressions enable very fast computations. These models are used to study HEO missions. We focus on two different problems: estimation of thrust for station keeping and evaluation of collision risk. We also consider the influence of the difference of ratio surface/mass between satellites

Formation Flight of Earth Satellites on Low-Eccentricity KAM Tori

The problem of Earth satellite constellation and formation flight is investigated in the context of Kolmogorov-Arnold-Moser (KAM) theory. KAM tori are constructed utilizing Wiesel’s Low-Eccentricity Earth Satellite Theory, allowing numerical representation of the perturbed tori describing Earth orbits acted upon by geopotential perturbations as sets of Fourier series. A maneuvering strategy using the local linearization of the KAM tangent space is developed and applied, demonstrating the ability to maneuver onto and within desired torus surfaces. Constellation and formation design and maintenance on KAM tori are discussed, along with stability and maneuver error concerns. It is shown that placement of satellites on KAM tori results in virtually no secular relative motion in the full geopotential to within computational precision. The effects of maneuver magnitude errors are quantified in terms of a singular value decomposition of the modal system for several orbits of interest, introducing a statistical distribution in terms of torus angle drift rates due to mismatched energies. This distribution is then used to create expectations of the steady-state station-keeping costs, showing that these costs are driven by operational and spacecraft limitations, and not by limitations of the dynamics formulation. A non-optimal continuous control strategy for formations based on Control Lyapunov Functions is also outlined and demonstrated in the context of formation reconfiguration

Guidance and navigation for electromagnetic formation flight orbit modification

Electromagnetic formation flight (EMFF) is a recent concept, aiming to control relative motions of formation flying satellites using magnetic interactions. Each satellite is equipped with a magnetic dipole. The formation degree of cooperation,depending on the ability of each spacecraft to control its dipole and its attitude, has a great impact on the methods used to perform the formation GNC. This paper describes results obtained in the case of semi-cooperative EMFF composed of a chaser and a target, in the field of navigation and guidance. Preliminary studies indicate that the target relative position and attitude can be determined while measuring the magnetic field at the chaser location, and the acceleration of this chaser. Focus is also made on the guidance for the whole formation orbit transfer, if only the chaser has thrust capacity: theory shows that geometrical configurations exist for which the formation is in an equilibrium state

Optimal Control of a Circulare Satellite Formation Subject to Gravitational Perturbations

Satellite formations, otherwise known in the space community as satellite clusters or distributed satellite systems, have been studied extensively over the last 10 to 15 years. For use in remote sensing applications, formations consisting of smaller, simpler satellites provide numerous advantages over individual satellites. The image resolution capabilities of small-satellite formations constitute a significant technological leap in the ability to synthesize critical information. This research utilizes the nonlinear satellite dynamics, including gravitational perturbations, to search for the optimal fuel cost for maintaining a circular formation. The system dynamics were developed in an earth-centered inertial coordinate frame using the methods of Hamiltonian dynamics. Continuous dynamic optimization theory was used to minimize fuel requirements, resulting in a continuous thrust, open-loop control law. The uncontrolled reference trajectory off which the formation is based was restricted to a circular, inclined orbit. Given initial conditions which match the mean motion of every member of the formation, it is shown that 1-km circular formation configurations can be maintained for control costs on the order of 40-50 m/s/year at an altitude of 400 km. Additionally, further fuel savings are possible with modifications to orbit altitude, formation radius, and variations in the defined performance index