This dissertation addresses the problem of parameter and state estimation of nonlinear dynamical systems and its applications for satellites in Low Earth Orbits. The main focus in Bayesian filtering methods is to recursively estimate the state a posteriori probability density function conditioned on available measurements. Exact optimal solution to the nonlinear Bayesian filtering problem is intractable as it requires knowledge of infinite number of parameters. Bayes' probability distribution can be approximated by mixture of orthogonal expansion of probability density function in terms of higher order moments of the distribution. In general, better series approximations to Bayes' distribution can be achieved using higher order moment terms. However, use of such density function increases computational complexity especially for multivariate systems. Mixture of orthogonally expanded probability density functions based on lower order moment terms is suggested to approximate the Bayes' probability density function. The main novelty of this thesis is development of new Bayes' filtering algorithms based on single and mixture series using a Monte Carlo simulation approach. Furthermore, based on an earlier work by Culver [1] for an exact solution to Bayesian filtering based on Taylor series and third order orthogonal expansion of probability density function, a new filtering algorithm utilizing a mixture of orthogonal expansion for such density function is derived. In this new extension, methods to compute parameters of such finite mixture distributions are developed for optimal filtering performance. The results have shown better performances over other filtering methods such as Extended Kalman Filter and Particle Filter under sparse measurement availability. For qualitative and quantitative performance the filters have been simulated for orbit determination of a satellite through radar measurements / Global Positioning System and optical navigation for a lunar orbiter. This provides a new unified view on use of orthogonally expanded probability density functions for nonlinear Bayesian filtering based on Taylor series and Monte Carlo simulations under sparse measurements. Another new contribution of this work is analysis on impact of process noise in mathematical models of nonlinear dynamical systems. Analytical solutions for nonlinear differential equations of motion have a different level of time varying process noise. Analysis of the process noise for Low Earth Orbital models is carried out using the Gauss Legendre Differential Correction method. Furthermore, a new parameter estimation algorithm for Epicyclic orbits by Hashida and Palmer [2], based on linear least squares has been developed. The foremost contribution of this thesis is the concept of nonlinear Bayesian estimation based on mixture of orthogonal expansions to improve estimation accuracy under sparse measurements. •.

This work presents the first known energy storage and attitude control subsystem (ESACS) for small satellites, proving this technology to be viable, applicable to more complex, demanding space missions, and laden with substantial benefits, such as agile slewing, robust singularity avoidance, increased lifetime, mass savings, and favourable peak power density. In capturing the key features of this novel system, it investigates the design sizing, feasibility, mission utility, experimentation, and performance benefits for using variable-speed control moment gyroscopes (VSCMGs) to store and drain energy while controlling satellite orientation. First, a novel optimal ESACS sizing algorithm is developed for a practical, miniature spotlight synthetic aperture RADAR (SAR) space mission. When given a set of small satellite agility and energy storage requirements, the design is cast as a constrained nonlinear programming problem using a performance index constructed from subsystem design margins including the attitude torque, peak power, energy capacity, and subsystem mass margins and solved using a reduced-order, gradient-based solver software code. The resulting method permits an efficient, structured approach to designing an optimally sized ESACS while enabling comparison of new technology performance to an existing system in order to identify the advantages and disadvantages of such new technology. The process shown generates point designs which are then compared via a design scoring process. Then, realistic usable energy capacity is studied, yielding a more practical system capable of meeting the desired requirements albeit with reduced mass savings benefits from theoretical levels. This factor, although presented in the early 1970s, is often overlooked in the literature. Next, a novel ESACS gimbal steering law is derived to permit independent gimbal and wheel control of VSCMGs with continued singularity avoidance, a situation that allows direct incorporation of an ESACS into an existing small satellite energy storage (ES) subsystem. This law rejects the disturbances generated during independent ES wheel control which can be significant if the power is stored and drained rapidly, demanding high wheel deceleration/ acceleration. Meanwhile, the separation of control renders simultaneous control law singularity avoidance through coordinated wheel torquing and gimballing impossible, thus a conventional CMG gimbal singularity avoidance steering algorithm was also added to this new law. As it permits directly interfacing this small satellite ESACS into a conventional satellite, this novel, composite gimbal steering law is more immediately practical than pre-existing simultaneous steering laws.

Momentum wheels are frequently used for the attitude control of satellites. Most existing momentum wheels support their spinning rotor using ball or roller bearings. These mechanical bearings are difficult to lubricate in the vacuum of space; they are prone to wear and so have a limited lifespan that can limit the useful lifespan of the satellite. Jitter in the bearings can generate microvibrations on the satellite that can affect its image quality; the noise level on the satellite platform is becoming increasingly important as the resolution of the cameras on small satellites approaches one metre ground resolution. Replacing the mechanical bearings with active magnetic bearings, where the spinning rotor is suspended by electromagnets potentially offers many benefits. There will be no contact between moving parts, eliminating friction, stiction and wear. Because the rotor's position is actively controlled it can be controlled precisely generating very low levels of noise. This thesis introduces a new concept - the 3Dwheel. The number of degrees of freedom that can be actively controlled in a magnetic bearing can be chosen. An engineering model of the 3Dwheel, a magnetically levitated momentum wheel with five degrees of freedom actively controlled, that has been designed, built and successfully tested is presented here. This design allows the rotor to be tilted generating a gyroscopic output torque; one 3Dwheel can therefore generate a torque about all three principal axes of the spacecraft. The electromagnets allow the wheel to be tilted with a high rate generating an output torque with a large magnitude and a bandwidth much greater than existing actuators. In this thesis the theory behind magnetism is used to model and investigate the design of a magnetic bearing. From this, the design of the 3Dwheel is presented and explained. This process can be used to help design other magnetically levitated momentum wheel designs. The engineering model of the 3Dwheel was successfully levitated in the laboratory by four different controllers. The position of the rotor can be maintained within three standard deviations of the desired position within 11.6 us». It has been tilted at a rate of 0.556 rads"", so generating a torque of 0.68 Nm while spinning at 5000 rpm. This resolution allows the 3Dwheel to have very low output noise levels. The 3Dwheel's bandwidth has been demonstrated to be two orders of magnitude greater than a conventional momentum wheel's. Various techniques for improving the performance of controllers and allowing stable levitation of the rotor at spin rates exceeding the controller's bandwidth are presented. Simulations of the 3Dwheel fitted to a small satellite prove that from its demonstrated ability in the laboratory a single 3Dwheel is capable of providing 3-axis attitude control of a small satellite.

The term microvibrations generally refers to accelerations in the order of micro-gs and which manifest in a bandwidth from a few Hz up to say 500-1000 Hz. The need to accurately characterise this small disturbances acting on-board modern satellites, thus allowing the design of dedicated minimisation and control systems, is nowadays a major concern for the success of some space missions. The main issues related to microvibrations are the feasibility to analytically describe the microvibration sources using a series of analysis tools and test experiments and the prediction of how the dynamics of the microvibration sources couple with those of the satellite structure. In this thesis, a methodology to facilitate the modelling of these phenomena is described. Two aspects are investigated: the characterisation of the microvibration sources with a semi-empirical procedure which allows derivation of the dynamic mass properties of the source, also including the gyroscopic effect, with a significantly simpler test configuration and lower computational effort compared to traditional approaches; and the modelling of the coupled dynamics when the source is mounted on a representative supporting structure of a spacecraft, including the passive and active effects of the source, which allows prediction of the structure response at any location. The methodology has been defined conducting an extensive study, both experimental and numerical, on a reaction wheel assembly, as this is usually identified as the main contributory factor among all microvibration sources. The contributions to the state-of-the-art made during this work include: i) the development of a cantilever configured reaction wheel analytical model able to reproduce all the configurations in which the mechanism may operate and inclusive of the gyroscopic effect; ii) the reformulation of the coupling theory which allows retrieving the dynamic mass of a microvibration source over a wide range of frequencies and speeds, by means of the experimental data obtained from measurements of the forces generated when the source is rigidly secured on a dynamometric platform and measurements of the accelerations at the source mounting interface in a freefree suspended boundary condition; iii) a practical example of coupling between a reaction wheel and a honeycomb structural panel, where the coupled loads and the panel response have been estimated using the mathematical model and compared with test results, obtained during the physical microvibration testing of the structural panel, showing a good level of agreement when the gyroscopic effect is also taken into account.

In this thesis we present two related aspects for studying the performance of an Earth Observation service provided by small satellites. The first part of the thesis explores the problem of predicting precise timings for when a LEO satellite will be above a given target on the ground. We present a new algorithm for LEO satellite orbit prediction - FPSCA (Fast Prediction of Satellite Closest Approach) which is suitable for satellite nadir tracking with respect to specific ground targets for imaging and communications. We demonstrate that this method is at least 220 times faster than the conventional approach. The new algorithm not only takes account of secular perturbations but also atmospheric drag and periodic variations along the orbit. The algorithm has been designed to run on the satellite so that the satellite can continuously determine an imaging schedule with asynchronous requests from multiple ground users. It also provides a capability for long-term orbit prediction to high level of accuracy. The second part of this thesis introduces the application of queueing theory to model the behaviour of small satellite providing a multi-user imaging service, of user defined ground target. This service has been analysed in terms of two queues - an image capture queue and an image download queue. As an extension of M/M/1 queue, we develop a new model which can be used to successfully analyse the image capture queue with a peak probability for a steady state number of requests to always reside in the queue. We also generalise M/M/1 queue to M/G/1 where the general output distribution is expressed in terms of orthogonal basis functions, the first of which degenerates to M/M/1. We prove that such queues are invertible, so that given the probability distribution of queue lengths we can solve for the service rate distribution required. Finally we show that the image download queue can be modelled in terms of a batch queue. Key words: orbit determination, imaging prediction, epicycle motion, small satellite, low Earth orbit, rise and set time, imaging service, queueing theory.

Actuator failures onboard satellites have caused severe and even disastrous consequences on several space missions. In this thesis, the problem of the attitude control of a LEO satellite, subject to actuator failures, is addressed. The three axis stabilisation of a satellite with the two remaining control torques on the principal axes, is a challenging problem since the control system is nonholonomic. Such a system has been proven by Brockett to be non-stabilisable using smooth (continuous and time invariant) control laws. Different non-smooth stabilizing control laws for the underactuated attitude control of a satellite are investigated here using pairs of thrusters, and also using reaction wheels. Using two pairs of thrusters, known singular or time varying approaches are applied with a systematic study of the effects of the torque saturation, PWM, singularity avoidance, noise, external disturbances, sampling and angular velocity tracking that intervene in a realistic case. Using two reaction wheels, a novel control strategy based on a singular nonlinear control approach, is mathematically proven and demonstrated by simulation. The 3-axis stability is proven using Rodriguez parameters and then using quaternions. The study of the symmetrical satellite case using thrusters, and the investigation of the effect of a non-zero total momentum using wheels, are done separately. Practical difficulties of the underactuated attitude control of small satellites using two pairs of on/off thrusters are pointed out. Conversely, using two reaction wheels, the possibility of decisive 3-axis manoeuvres is demonstrated (under realistic assumptions). Indeed, using two wheels, the 3-axis stabilisation is achieved with acceptable torque levels and very satisfactory performance. The activation of the non-smooth controller must be done under small momentum conditions. A complete control strategy, (in case of a high initial bias) including a detumbling phase with magnetorquing, and avoiding the non-smooth controller to start from a singularity, is presented. Following the encouraging results from the SSTL's UoSAT-12 simulator, (accounting for noises and external disturbance torques) in-orbit testing of an underactuated control strategy using two wheels has been successfully achieved on UoSAT-12 (by restricting the attitude to sun tracking due to power consumption problems on UoSAT-12). Another in orbit experiment on UK-DMC, for nadir pointing, has been even more successful. Practical results therefore confirm the possibility of using only two control torques for the 3-axis stabilisation of a satellite. One of many interesting consequences of these results is that a fully redundant 3-axis control can be practically envisaged using a 3-wheel configuration.

Formation control algorithms for unmanned vehicles are proposed and developed based on the concept of tensegrity structures to solve the cooperative control problems. The tensegrity concept, which is enhanced by virtualising the strings and bars of the tensegrity grid, is used to establish the relationship between isolated vehicles that forms a formation topology and to develop mathematical algorithms in characterising the interaction control forces among the vehicles. The simple form of tensegrity mathematics significantly simplified the design of the formation control algorithm and its computation procedure while optimising the corresponding formation topology. The formation is proved to be exponentially stable by the Lyapunov stability theory, if all the vehicles are driven to their equilibrium under the regulation of interaction control forces among them. The control force is dynamically redefined as a non linear tension force based on tensegrity's virtual string which demonstrates better performance in compensating the external disturbance forces to maintain a firm formation while allowing flexible shape transformation to achieve obstacle avoidance. A decentralized scheme is also proposed to simplify the mathematical analysis of the complex formation control problem, in addition to solving the scalable formation problem by decomposing the formation system into an ensemble of vehicle subsystems. By defining the applied interaction force as a bounded energy exogenous input to the subsystem, a H∞ robust controller, which is formulated by solving a LMI optimisation problem, stabilises each subsystem in the formation by regulating its applied tension force. This H∞ controller provides robustness to the scalable multi-vehicle formation system and improves its overall performance by attenuating the formation error propagation. This algorithm is extended for the formation flight of multiple Aerosonde UAVs to verify the effectiveness of the proposed algorithm using 3D formation stability results. The LMI-based H∞ autopilot was developed to regulate the related induced forces and thus stabilises all the individual UAVs in the formation. Simulation results demonstrated the stability and effectiveness of the proposed tensegrity-based formation control strategy for a diverse range of multi-UAV formation achieving, maintaining, transforming and manoeuvring scenarios.

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The ROSAT spacecraft is due for launch in June 1990. On board will be an X-Ray Telescope (XRT) and an XUV Wide Field Camera (WFC). The XRT will perform the first ever imaging all sky survey in the 0.1 keV- 2.0 keV wave-band. This will be complemented by the first all sky survey in the 0.062 keV - 0.21 keV wave-band using the WFC. A scientific simulator has been developed to assist the WFC mission in the following areas: 1. Production of representative telemetry to develop and test the ROSAT WFC ground system software. 2. Determination of optimum parameters for several WFC firmware and hardware features. 3. Evaluation of background radiation effects on operational and astrophysical aspects of the WFC mission. The modelling and simulation of the performance aspects of the WFC telescope and its interaction with the predicted environment is described here. The complete model has been implemented as a user orientated software package and can be run to produce scientific telemetry and other simulated output that interface with the ROSAT WFC ground system software. The results of the simulator provide essential foresight into the following: The efficacy of image analysis software. The efficiency of in-orbit WFC low energy (50 keV) electron rejection. Prediction of all sky exposure coverage, the background environment, and minimum source sensitivity.