Dynamics of a Gyrostat Satellite with the Vector of Gyrostatic Moment along the Principal Plane of Inertia

Artificial satellites are one of the most crucial components of modern life. The study of attitude control and stabilization of satellite is necessary to ensure a successful operation. There are two types of stabilization schemes: the passive methods and active methods. In this dissertation is investigated the dynamics of a gyrostat satellite, subjected to a semi-passive method of stabilization, namely the gravitational torque and the gyroscopic proprieties of rotating rotors, along a circular orbit. In a particular case, when the gyrostatic moment vector is along one of satellite’s principal central planes of inertia. To solve the problem is proposed a mathematical analytical-numerical method for determining all equilibrium positions of the gyrostat satellite in the orbital coordinate system in function of dimensionless gyrostatic moment vector components (???? ??=1,2,3) and the dimensionless inertial parameter ??. The conditions of existence of the equilibrium solutions are obtained. Sufficient conditions of stability for each group of equilibrium solutions are derived from the analysis of the generalized integral energy used as a Lyapunov’s function. The study of the evolution of equilibria bifurcation of the gyrostat is carried out in function of parameter ?? in detail. Also, the evolution of equilibrium solutions in function of spacecraft angles is analyzed and it is verified the existence of small regions of 12 and 16 equilibrium positions referred in [14] and [20]. This work shows that the number of equilibria of a gyrostat satellite, in this particular case, does not exceeds 24 and does not go below 8. The study of the equilibria bifurcation shows that there are small regions of 12 equilibrium positions that approach each other for infinite ??3 and never vanish, these regions seems to have a relation with the regions referred by Santos in [14] and Santos et. al. [20]. The study of the evolution of stability for every equilibrium solution in function ?? and ??3, shows that the number of stable equilibria varies between 2 and 6.Satélites artificias são uns dos componentes cruciais da vida moderna. O estudo do controlo da atitude e estabilização de um satélite é necessário para assegurar uma missão bem-sucedida. Existem dois tipos de métodos de estabilização: os métodos passivos e os métodos ativos. Nesta dissertação é investigado a dinâmica de um satélite tipo giróstato, sujeito a um método semi-passivo de estabilização, nomeadamente o momento gravítico e as propriedades giroscópicas de rotores, ao longo de uma órbita circular. No caso particular, quando o vetor de momento girostático está ao longo de um dos principais planos de inércia do satélite. Para resolver este problema é proposto um modelo matemático numérico-analítico para determinar todos as posições de equilíbrio de um satélite giróstato, em um sistema coordenado orbital em função das componentes adimensionais do vetor de momento girostático (???? ??=1,2,3) e do parâmetro inercial adimensional ??. As condições de existência das soluções de equilíbrio são obtidas. As condições suficientes de estabilidade para cada grupo de soluções de equilíbrio são derivadas, a partir da análise do integral de energia generalizado como uma função de Lyapunov. O estudo da evolução da bifurcação do equilíbrio foi realizado em detalhe em função do parâmetro ??. Também, a evolução das soluções de equilíbrio em função dos ângulos do satélite é analisada e é verificado a existência de pequenas regiões de 12 e 16 posições de equilíbrio referidas em [14] e [20]. Este trabalho mostra que o número de posições de equilíbrio de um satélite tipo giróstato, neste caso particular, não ultrapassa 24 e não é inferior a 8. O estudo da bifurcação do equilíbrio revela a existência de regiões de 12 posições de equilíbrio que se aproximam, para valores infinitos de ??3 e que nunca desaparecem, estas regiões sugerem ter uma relação com as regiões referidas por Santos [14] e Santos et al.[20]. O estudo da evolução da estabilidade para cada solução de equilíbrio em função de ?? e ??3 revela que o número de posições de equilíbrio estáveis varia entre 2 e 6

Dynamics of a gyrostat satellite with the vector of gyrostatic moment tangent to the orbital plane

In this paper, a gyrostat satellite in a circular orbit with its gyrostatic moment tangent to the orbital plane and collinear with the orbital speed is studied regarding its equilibria, bifurcation of equilibria, and asymptotic stability conditions. In the general case, where any gyrostat angular momentum is aligned with any of the orbital coordinate frames, interesting results arose regarding its equilibria bifurcation regarding conditions near to the ones presented in this paper, namely equilibria regions outside their main regions near to the orbital plane tangent. For equilibria and bifurcation of equilibria, a symbolic-numerical method is used to obtain the polynomial equations in function of non-dimensional parameters whose roots are equivalent to the number of equilibria positions. For the asymptotic stability, the results are tested using the Lyapunov stability theory scheme

An introduction to Lie group integrators -- basics, new developments and applications

We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion of discrete gradient methods is generalised to Lie groups

Attitude dynamics and shape control of reflectivity modulated gossamer spacecraft

The utilisation of space provides many opportunities to deliver pioneering innovations during the 21st century. One of these opportunities is the gossamer spacecraft, an emerging technology to achieve very low mass, large area and low stowage volume. Examples include large ultra-lightweight membrane reflectors and distributed tethered formations. Gossamer spacecraft offer the potential to deliver innovative new science and applications missions to aid our growing globalised societies: high-performing communications antennae, scientific telescopes and space-based solar power collectors. However, the ability to control such large structures in space is essential for their successful operation. To this aim, this thesis investigates a novel means to control large gossamer spacecraft by exploiting modulated solar radiation pressure (SRP), thus by modifying the nominal light pressure acting on the structure in space. Various concepts have been proposed in the past to control the attitude of a gossamer spacecraft, employing complex mechanical systems or thrusters. Furthermore, methods to control the surface shape of a large membrane reflector using, for example, piezoelectric actuators, are being developed. Since on-board control systems need to be high-performance, reliable and importantly lightweight, this thesis investigates the use of thin-film reflectivity control devices across the spacecraft surface. Controlling the reflectivity modulates the Sun's light pressure acting on a thin membrane thus controlling its shape. In addition, body forces and torques become available to control the attitude of such a large structure 'optically', without using traditional mechanical systems. The concept is demonstrated first by controlling a two-mass tethered formation in a Sun-centred orbit, showing that the spacecraft attitude can be stabilised around new equilibria created by controlling the surface reflectivity of the masses. Subsequently, the concept is applied to control the attitude of a large membrane reflector, which confirms the viability of reflectivity modulation by generating variable optical torques in the membrane plane. In particular, the nominal SRP forces are modified by introducing different surface reflectivity distributions across the membrane. It is shown that through these optical torques, the reflector can be steered, for example, to a Sunpointing attitude from an arbitrary initial displacement. The analysis also considers the variation of the SRP force magnitude with changing light incidence angle towards the Sun during the manoeuvre, thereby presenting solutions to a challenging attitude control problem. Furthermore, by adopting a highly-integrated multi-functional design approach, the concept of reflectivity modulation is also employed to control the surface shape of a large membrane reflector. First, the nominal (non-parabolic) deflection shapes due to uniform SRP across the surface are presented. Subsequently, a closed-form solution for the reflectivity function across the membrane required to create a true parabolic deflection shape is derived. In order to improve the quite large focal lengths of the deflected shapes that can be generated for a tensioned membrane, shape control of a slack suspended surface is also considered. The achievable (shorter) focal lengths support the feasibility of exploiting modulated SRP for controlled surface deflection. In summary, this thesis demonstrates the potential of using surface reflectivity modulation to control the attitude and morphology of large gossamer spacecraft without using complex mechanical systems or thrusters. Therefore, the concept of optical control represents a major step towards highly-integrated adaptive gossamer structures and supports the development of this promising key-technology to deliver advanced space applications.The utilisation of space provides many opportunities to deliver pioneering innovations during the 21st century. One of these opportunities is the gossamer spacecraft, an emerging technology to achieve very low mass, large area and low stowage volume. Examples include large ultra-lightweight membrane reflectors and distributed tethered formations. Gossamer spacecraft offer the potential to deliver innovative new science and applications missions to aid our growing globalised societies: high-performing communications antennae, scientific telescopes and space-based solar power collectors. However, the ability to control such large structures in space is essential for their successful operation. To this aim, this thesis investigates a novel means to control large gossamer spacecraft by exploiting modulated solar radiation pressure (SRP), thus by modifying the nominal light pressure acting on the structure in space. Various concepts have been proposed in the past to control the attitude of a gossamer spacecraft, employing complex mechanical systems or thrusters. Furthermore, methods to control the surface shape of a large membrane reflector using, for example, piezoelectric actuators, are being developed. Since on-board control systems need to be high-performance, reliable and importantly lightweight, this thesis investigates the use of thin-film reflectivity control devices across the spacecraft surface. Controlling the reflectivity modulates the Sun's light pressure acting on a thin membrane thus controlling its shape. In addition, body forces and torques become available to control the attitude of such a large structure 'optically', without using traditional mechanical systems. The concept is demonstrated first by controlling a two-mass tethered formation in a Sun-centred orbit, showing that the spacecraft attitude can be stabilised around new equilibria created by controlling the surface reflectivity of the masses. Subsequently, the concept is applied to control the attitude of a large membrane reflector, which confirms the viability of reflectivity modulation by generating variable optical torques in the membrane plane. In particular, the nominal SRP forces are modified by introducing different surface reflectivity distributions across the membrane. It is shown that through these optical torques, the reflector can be steered, for example, to a Sunpointing attitude from an arbitrary initial displacement. The analysis also considers the variation of the SRP force magnitude with changing light incidence angle towards the Sun during the manoeuvre, thereby presenting solutions to a challenging attitude control problem. Furthermore, by adopting a highly-integrated multi-functional design approach, the concept of reflectivity modulation is also employed to control the surface shape of a large membrane reflector. First, the nominal (non-parabolic) deflection shapes due to uniform SRP across the surface are presented. Subsequently, a closed-form solution for the reflectivity function across the membrane required to create a true parabolic deflection shape is derived. In order to improve the quite large focal lengths of the deflected shapes that can be generated for a tensioned membrane, shape control of a slack suspended surface is also considered. The achievable (shorter) focal lengths support the feasibility of exploiting modulated SRP for controlled surface deflection. In summary, this thesis demonstrates the potential of using surface reflectivity modulation to control the attitude and morphology of large gossamer spacecraft without using complex mechanical systems or thrusters. Therefore, the concept of optical control represents a major step towards highly-integrated adaptive gossamer structures and supports the development of this promising key-technology to deliver advanced space applications

Torque equilibrium attitude control for Skylab reentry

All the available torque equilibrium attitudes (most were useless from the standpoint of lack of electrical power) and the equilibrium seeking method are presented, as well as the actual successful application during the 3 weeks prior to Skylab reentry

Condensation Calculations in Planetary Science and Cosmochemistry

Cool a piece of the Sun to 1000 K at one millibar pressure to yield a mineral assemblage consistent with those found in the most primitive meteorites. This is an equilibrium or fractional condensation experiment simulated by calculations using equations of state for hundreds of gaseous molecules, condensed mineral solids, and silicate liquids, the products of a century of experimental measurements and theoretical studies. Such calculations have revolutionized our understanding of the chemistry of the cosmos. The mid-20th Century realization that meteorites are fossil records of the early Solar System made chemistry central to understanding planetary origins. Thus "condensation", the distribution of elements and isotopes between vapor and condensed solids and/or liquids at or approaching chemical equilibrium, deeply informs discussion of how meteor/comet compositions bear on planets. Condensation calculations have been applied to disks around young stars, to the mineral "rain" of mineral grains expected to form in cool dwarf star atmospheres, to the expanding envelopes of giant stars, to the vapor plumes that form in planetary impacts, and to the chemically and isotopically distinct "shells" computed and observed to exist in supernovae. As with all sophisticated calculations, there are inherent caveats, subtleties, and computational difficulties. Local chemistry has yet to be consistently integrated into dynamical astrophysical simulations so that effects like the blocking of radiation by grains, absorption and reemission of light by grains, and buffering of heat by grain evaporation/condensation feed back into the physics at each node of a gridded calculation over time. A deeper integration of thermochemistry with physical models makes the prospect of a general protoplanetary disk model as hopeful now as a general circulation model for global climate was in the early 1970's.Comment: Review for Oxford Encyclopedi

The Relative Degree of Controllability of Solar Sails in Artificial Lagrangian Orbits

The problem of orbital controllability of solar sails in artificial Lagrangian orbits (ALOs) has received little attention from academics in the field of control. As a result, an important property for missions involving ALOs has not yet been investigated; namely, the degree of controllability (DOC). This thesis examines the relative degree of orbital controllability of solar sails in artificial Lagrangian orbits using the singular values of the controllability Gramian. The relative degree of controllability is determined by comparing the condition number of the controllability Gramian corresponding to individual ALOs. The finite time controllability Gramian is computed for the unstable system, and the magnitude of the individual singular values are used to determine the relative potential for control. Regions of potential for control are revealed for the Earth-Sun system; particularly near the Earth. No conclusions are made regarding absolute controllability, nor is an exact measure of controllability computed for arbitrary ALOs