Rodolpho Moraes

Uma teoria para estudar o movimento orbital de satélites artificiais sobre efeitos do arrasto atmosférico e da pressão de radiação solar direta - considerando a sombra da Terra e alguns termos do geopotencial - é desenvolvida analiticamente. A sombra da Terra é modelada utilizando a função sombra (psi), como introduzida por Ferraz Mello: psi igual zero quando o satélite está na região de sombra e igual a um quando é iluminado pelo Sol. As componentes do arrasto são dadas por Vilhena de Moraes baseado no modelo atmosférico TD-88. O método de Hori para sistemas não-canônicos é aplicado para resolver as equações demovimento. Um software para manipulação algébrica é fundamental para fazer os cálculos necessários. Efeitos seculares e periódicos que influenciam no movimento orbital de satélites artificiais são analisados. Édada ênfase aos termos de acoplamento que surgem na soluçãodo sistema de equações diferenciais. Utilizando dados orbitais do satélite CBERS-1 é feito um estudo para ana...

ABSTRACT A study evaluating the influence due to the lunar gravitational potential, modeled by spherical harmonics, on the gravity acceleration is accomplished according to the model presented in Konopliv (2001). This model provides the components x, y and z for the gravity acceleration at each moment of time along the artificial satellite orbit and it enables to consider the spherical harmonic degree and order up to100. Through a comparison between the gravity acceleration from a central field and the gravity acceleration provided by Konopliv's model, it is obtained the disturbing velocity increment applied to the vehicle. Then, through the inverse problem, the Keplerian elements of perturbed orbit of the satellite are calculated allowing the orbital motion analysis. Transfer maneuvers and orbital correction of lunar satellites are simulated considering the disturbance due to non-uniform gravitational potential of the Moon, utilizing continuous thrust and trajectory control in closed loop. The simulations are performed using the Spacecraft Trajectory Simulator-STRS, Rocco (2008), which evaluate the behavior of the orbital elements, fuel consumption and thrust applied to the satellite over the time.

The present paper aims to study the cumulative collision probability of a target that crosses a cloud of particles that has the orbital parameters of each individual element modified by a close approach with the Earth. Clouds of this type are formed when natural or man-made bodies explode for some reason. After an explosion like that, the individual particles have different trajectories. The clouds are specified by a non-uniform distribution of semi-major axis and eccentricity of their particles which are assumed to pass close to the Earth, making a close approach that modifies the trajectory of every single particle that belongs to the cloud. This study makes simulations considering separately two different clouds based on the patched conics model to obtain the new trajectories of each particle and to analyze the density of the whole cloud. Then it is possible to map the new distribution of the orbital elements of the fragments that constituted the cloud, using the distribution as initial conditions. After calculating the spatial density for the whole cloud, it is possible to obtain the cumulative collision probability for one space vehicle that crosses the cloud. These pieces of information are important when planning satellite missions where a spacecraft passes close to a cloud of this type, because we can determine values to study the risks of collision and the possible maneuvers that need to be made to avoid them.

ABSTRACT The purpose of this work is to evaluate the nonlinear unscented Kalman filter (UKF) for the satellite orbit determination problem, using GPS measurements. The assessment is based on the robustness of the filter. The main subjects for the evaluation are convergence speed and dynamical model complexity. Such assessment is based on comparing the UKF results with the extended Kalman filter (EKF) results in the solution of the same problem. Based on the analysis of such criteria, the advantages and drawbacks of the implementations are presented. In this work, the orbit of an artificial satellite is determined using real data from the Global Positioning System (GPS) receivers. In this orbit determination problem the focus is to analyze UKF convergence behavior using different sampling rates for the GPS signals, where scattering of measurements will be taken into account. A second aim is to evaluate how the dynamical model complexity affects the performance of the estimators in such adverse situation. Therefore, a performance comparison between EKF and UKF is justified. In this work, the standard differential equations describing the orbital motion and the GPS measurements equations used in the EKF algorithm need to be placed in a suitable form. They are adapted for the unscented transformation application, so that the UKF algorithm is also used for estimating the orbital state. After solving the real time satellite orbit determination problem using actual GPS measurements, through both the EKF and the UKF algorithms, the results obtained are compared in computational terms such as complexity, convergence, and accuracy.

The problem of orbit determination consists essentially of estimating parameter values that completely specify the body trajectory in the space, processing a set of information (measure-ments) from this body. Such observations can be collected through a conventional tracking network on Earth or through sensors like GPS. The Global Positioning System (GPS) is a powerful and low cost way to allow

Nowadays, there are several studies for missions that will place a satellite around Europa. There are many important aspects that deserve to be studied in this natural satellite of Jupiter. It makes the study of orbits around Europa a particular important part of the mission, since a good choice for them will reduce the costs related to station-keeping and then increasing the duration of the mission. In some previous studies, a search for frozen orbit around Europa was presented based in average techniques. The present research has the objective of using the new concept of stability of orbits with respect to station-keeping maneuvers that is available in the literature to study circular orbits around Europa. This concept is based in the integral of the perturbing forces over the time. This value can estimate the total variation of velocity required by the station-keeping propulsion system to compensate the perturbations suffered by the spacecraft. The value of this integral is a characteristic of the perturbations considered and the orbit chosen for the spacecraft. Numerical simulations are made showing the costs of station-keeping for circular orbits around Europa are shown as a function of the eccentricity and semi-major axis of the orbits.

ABSTRACT The purpose of this work is to present a complete first order analytical solution, which includes short periodic terms, for the problem of optimal low-thrust limited power trajectories with large amplitude transfers (no rendezvous) between coplanar orbits with small eccentricities in Newtonian central gravity field. The study of these transfers is particularly interesting because the orbits found in practice often have a small eccentricity and the problem of transferring a vehicle from a low earth orbit to a high earth orbit is frequently found. Besides, the analysis has been motivated by the renewed interest in the use of low-thrust propulsion systems in space missions verified in the last two decades. Several researchers have obtained numerical and sometimes analytical solutions for a number of specific initial orbits and specific thrust profiles. Averaging methods are also used in such researches. Firstly, the optimization problem associated to the space transfer problem is formulated as a Mayer problem of optimal control with Cartesian elements - position and velocity vectors - as state variables. After applying the Pontryagin Maximum Principle, successive Mathieu transformations are performed and suitable sets of orbital elements are introduced. The short periodic terms are eliminated from the maximum Hamiltonian function through an infinitesimal canonical transformation built through Hori method - a perturbation canonical method based on Lie series. The new Hamiltonian function, which results from the infinitesimal canonical transformation, describes the extremal trajectories for long duration maneuvers. Closed-form analytical solutions are obtained for the new canonical system by solving the Hamilton-Jacobi equation through the separation of variables technique. By applying the transformation equations of the algorithm of Hori method, a first order analytical solution for the problem is obtained in non-singular orbital elements. For long duration maneuvers, the existence of conjugate points is investigated through the Jacobi condition and the envelope of extremals is obtained considering different configurations of initial and final orbits. An iterative algorithm based on the first order analytical solution is described for solving the two-point boundary value problem of going from an initial orbit to a final orbit. Numerical results for some missions are compared to the results obtained using numerical techniques such as method of neighboring extremals or gradient method.

A semi-analytical method is presented to study the system of differential equations governing the rotational motion of an artificial satellite. Gravity gradient and non gravitational torques are considered. Operations with trigonometric series were performed using an algebraic manipulator. Andoyer's variables are used to describe the rotational motion. The osculating elements are transformed analytically into a mean set of elements. As

the purpose of this work is to present the orbital eccentricities and inclinations characteristics for some real artificial satellites, some of them already inactive, whose mean motions are commensurable with the Earth's rotation period. The correspondent geopotential coefficients for each considered resonance are also presented.

Aplicações recentes de satélites artificiais, principalmente aquelas com finalidades geodinâmicas ou altimétricas, requerem órbitas determinadas com bastante precisão. Em particular os satélites do sistema GPS, que têm sido envolvidos direta ou indiretamente em tais problemas, necessitam de ter suas órbitas muito bem conhecidas. As órbitas dos satélites GPS tem uma peculiaridade: o período orbital está em comensurabilidade 2: 1, aproximada, com o período de rotação da Terra. A existência de ressonâncias faz com que métodos usuais de teoria de perturbações não possam ser usados para se estudar órbitas com esta característica. No presente trabalho são apresentados dois processos para se estudar tal problema. Para tanto o sistema dinâmico que descreve o movimento orbital de satélites artificiais, perturbado por forças que derivem ou não de um potencial, incluindo ressonância, é inicialmente colocado em forma canônica estendida. Um dos processos apresentado é baseado na teoria de Lie-Ho...

Aplicações recentes de satélites artificiais com finalidades geodinâmicas requerem órbitas determinadas com bastante precisão. Em particular marés terrestres influenciam o potencial terrestre causando perturbações adicionais no movimento de satélites artificiais, as quais tem sido medidas por diversos processos. A atração exercida pela lua e pelo sol sobre a terra produz deslocamentos elásticos em seu interior e uma protuberância em sua superfície. O resultado é uma pequena variação na distribuição da massa na terra, consequentemente no geopotencial. As perturbações nos elementos orbitais de satélites artificiais terrestres devidas a maré terrestre podem ser estudadas a partir das equações de Lagrange, considerando-se um conveniente potencial. Por outro lado, como tem sido feito pelo IERS, as mudanças induzidas pela maré terrestre no geopotencial podem ser convenientemente modeladas como variações nos coeficientes Cnm e Snm do geopotencial. As duas teorias ainda não foram comparados...

Space missions to visit the natural satellite of Jupiter, Europa, constitute an important topic in space activities today, because missions to this moon are under study now. Several considerations have to be made for these missions. The present paper searches for less perturbed circular orbits around Europa. This search is made based on the total effects of the perturbing forces over the time, evaluated by the integral of those forces over the time. This value depends on the dynamical model and on the orbit of the spacecraft. The perturbing forces considered are the third-body perturbation that comes from Jupiter and theJ2,J3, andC22terms of the gravitational potential of Europa. Several numerical studies are performed and the results show the locations of the less perturbed orbits. Using those results, it is possible to find near-circular frozen orbits with smaller amplitudes of variations of the orbital elements.

ABSTRACT A comparison between the extended Kalman filter (EKF) and the nonlinear sigma point Kalman filter (SPKF) for a real time satellite orbit determination problem, using GPS measurements is presented. Such comparison is based on testing the filters robustness for degraded initial conditions. The main subjects for the comparison between the estimators are convergence speed and computational implementation complexity. Based on the analysis of such criteria, the advantages and drawbacks of each estimator are presented. In this work, the orbit of an artificial satellite is determined using real data from a space borne Global Positioning System (GPS) receiver. This is a fully nonlinear problem, with respect to both the dynamics and measurements equations, in which the disturbing forces are not easily modeled. The problem of orbit determination consists essentially of estimating values that completely specify the body trajectory in the space, processing a set of measurements related to this body. In this orbit determination problem the focus is to analyze each filter convergence behavior in situations where the initial conditions are inaccurate, introducing since small up to larger errors in the initial accurate position conditions. Concomitantly another aim is to know how such inaccuracies affect the estimators performance.

The concept of frozen orbit has been applied in space missions mainly for orbital tracking and control purposes. This type of orbit is important for orbit design because it is characterized by keeping the argument of perigee and eccentricity constant on average, so that, for a given latitude, the satellite always passes at the same altitude, benefiting the users through this regularity. Here, the system of nonlinear differential equations describing the motion is studied, and the effects of geopotential and atmospheric drag perturbations on frozen orbits are taken into account. Explicit analytical expressions for secular and long period perturbations terms are obtained for the eccentricity and the argument of perigee. The classical equations of Brouwer and Brouwer and Hori theories are used. Nonsingular variables approach is used, which allows obtaining more precise previsions for CBERS (China Brazil Earth Resources Satellite) satellites family and similar satellites (SPOT, Landsat,...

ABSTRACT The purpose of this work is to evaluate the nonlinear unscented Kalman filter (UKF) for the satellite orbit determination problem, using GPS measurements. The assessment is based on the robustness of the filter. The main subjects for the evaluation are convergence speed and dynamical model complexity. Such assessment is based on comparing the UKF results with the extended Kalman filter (EKF) results in the solution of the same problem. Based on the analysis of such criteria, the advantages and drawbacks of the implementations are presented. In this work, the orbit of an artificial satellite is determined using real data from the Global Positioning System (GPS) receivers. In this orbit determination problem the focus is to analyze UKF convergence behavior using different sampling rates for the GPS signals, where scattering of measurements will be taken into account. A second aim is to evaluate how the dynamical model complexity affects the performance of the estimators in such adverse situation. Therefore, a performance comparison between EKF and UKF is justified. In this work, the standard differential equations describing the orbital motion and the GPS measurements equations used in the EKF algorithm need to be placed in a suitable form. They are adapted for the unscented transformation application, so that the UKF algorithm is also used for estimating the orbital state. After solving the real time satellite orbit determination problem using actual GPS measurements, through both the EKF and the UKF algorithms, the results obtained are compared in computational terms such as complexity, convergence, and accuracy.

ABSTRACT The purpose of this work is to present a complete first order analytical solution, which includes short periodic terms, for the problem of optimal low-thrust limited power trajectories with large amplitude transfers (no rendezvous) between coplanar orbits with small eccentricities in Newtonian central gravity field. The study of these transfers is particularly interesting because the orbits found in practice often have a small eccentricity and the problem of transferring a vehicle from a low earth orbit to a high earth orbit is frequently found. Besides, the analysis has been motivated by the renewed interest in the use of low-thrust propulsion systems in space missions verified in the last two decades. Several researchers have obtained numerical and sometimes analytical solutions for a number of specific initial orbits and specific thrust profiles. Averaging methods are also used in such researches. Firstly, the optimization problem associated to the space transfer problem is formulated as a Mayer problem of optimal control with Cartesian elements - position and velocity vectors - as state variables. After applying the Pontryagin Maximum Principle, successive Mathieu transformations are performed and suitable sets of orbital elements are introduced. The short periodic terms are eliminated from the maximum Hamiltonian function through an infinitesimal canonical transformation built through Hori method - a perturbation canonical method based on Lie series. The new Hamiltonian function, which results from the infinitesimal canonical transformation, describes the extremal trajectories for long duration maneuvers. Closed-form analytical solutions are obtained for the new canonical system by solving the Hamilton-Jacobi equation through the separation of variables technique. By applying the transformation equations of the algorithm of Hori method, a first order analytical solution for the problem is obtained in non-singular orbital elements. For long duration maneuvers, the existence of conjugate points is investigated through the Jacobi condition and the envelope of extremals is obtained considering different configurations of initial and final orbits. An iterative algorithm based on the first order analytical solution is described for solving the two-point boundary value problem of going from an initial orbit to a final orbit. Numerical results for some missions are compared to the results obtained using numerical techniques such as method of neighboring extremals or gradient method.

ABSTRACT Frozen orbits are studied, including perturbations due to J5 harmonic and atmospheric drag. Non-singular variables are introduced according to non-singular theory used at INPE Satellite Tracking and Control Center (CRC). Using CBERS-1 (China Brazil Earth Resources Satellite) satellite's data, results are compared with those used nowadays at the INPE CRC. The keplerian elements studied are the eccentricity, the argument of perigee and the semi-major axis. Terms up to the J5 order for the geopotential model are considered. The atmospheric drag model used is a classical one and provides a good indicative for the order of the drag perturbations on the orbital elements.

This work presents a semi-analytical and numerical study of the perturbation caused in a spacecraft by a third-body using a double averaged analytical model with the disturbing function expanded in Legendre polynomials up to the second order. The important reason for this procedure is to eliminate terms due to the short periodic motion of the spacecraft and to show smooth curves for the evolution of the mean orbital elements for a long-time period. The aim of this study is to calculate the effect of lunar perturbations on the orbits of spacecrafts that are traveling around the Earth. An analysis of the stability of near-circular orbits is made, and a study to know under which conditions this orbit remains near circular completes this analysis. A study of the equatorial orbits is also performed.

A low-cost computer procedure to determine the orbit of an artificial satellite by using short arc data from an onboard GPS receiver is proposed. Pseudoranges are used as measurements to estimate the orbit via recursive least squares method. The algorithm applies orthogonal Givens rotations for solving recursive and sequential orbit determination problems. To assess the procedure, it was applied to the TOPEX/POSEIDON satellite for data batches of one orbital period (approximately two hours), and force modelling, due to the full JGM-2 gravity field model, was considered. When compared with the reference Precision Orbit Ephemeris (POE) of JPL/NASA, the results have indicated that precision better than 9 m is easily obtained, even when short batches of data are used.