In this dissertation, we discuss the Distributed Spacecraft Attitude Control System Simulator (DSACSS) testbed developed at Virginia Polytechnic Institute and State University for the purpose of investigating various control techniques for single and multiple spacecraft. DSACSS is comprised of two independent hardware-in-the-loop simulators and one software spacecraft simulator. The two hardware-in-the-loop spacecraft simulators have similar subsystems as flight-ready spacecraft (e.g. command and data handling; communications; attitude determination and control; power; payload; and guidance and navigation). The DSACSS framework is a flexible testbed for investigating a variety of spacecraft control techniques, especially control scenarios involving coupled attitude and orbital motion. The attitude hardware simulators along with numerical simulations assist in the development and evaluation of Lyapunov based asymptotically stable, nonlinear attitude controllers with three reaction wheels as the control device. The angular rate controller successfully tracks a time varying attitude trajectory. The Modified Rodrigues Parmater (MRP) attitude controller results in successfully tracking the angular rates and MRP attitude vector for a time-varying attitude trajectory. The attitude controllers successfully track the reference attitude in real-time with hardware similar to flight-ready spacecraft. Numerical simulations and the attitude hardware simulators assist in the development and evaluation of a robust, asymptotically stable, nonlinear attitude controller with three reaction wheels as the actuator for attitude control. The MRPs are chosen to represent the attitude in the development of the controller. The robust spacecraft attitude controller successfully tracks a time-varying reference attitude trajectory while bounding system uncertainties. The results of a Global Positioning System (GPS) hardware-in-the-loop simulation of two spacecraft flying in formation are presented. The simulations involve a chief spacecraft in a low Earth orbit (LEO), while a deputy spacecraft maintains an orbit position relative to the chief spacecraft. In order to maintain the formation an orbit correction maneuver (OCM) for the deputy spacecraft is required. The control of the OCM is accomplished using a classical orbital element (COE) feedback controller and simulating continual impulsive thrusting for the deputy spacecraft. The COE controller requires the relative position of the six orbital elements. The deputy communicates with the chief spacecraft to obtain the current orbit position of the chief spacecraft, which is determined by a numerical orbit propagator. The position of the deputy spacecraft is determined from a GPS receiver that is connected to a GPS hardware-in-the-loop simulator. The GPS simulator creates a radio frequency (RF) signal based on a simulated trajectory, which results in the GPS receiver calculating the navigation solution for the simulated trajectory. From the relative positions of the spacecraft the COE controller calculates the OCM for the deputy spacecraft. The formation flying simulation successfully demonstrates the closed-loop hardware-in-the-loop GPS simulator. This dissertation focuses on the development of the DSACSS facility including the development and implementation of a closed-loop GPS simulator and evaluation of nonlinear feedback attitude and orbit control laws using real-time hardware-in-the-loop simulators. / Ph. D.

<p>A time-discrete formulation of the variational principle of mechanics is used to provide a consistent theoretical framework for the construction and analysis of low order integration methods. These are applied to mechanical systems subject to mixed constraints and dry frictional contacts and impacts---machines. The framework includes physics motivated constraint regularization and stabilization schemes. This is done by adding potential energy and Rayleigh dissipation terms in the Lagrangian formulation used throughout. These terms explicitly depend on the value of the Lagrange multipliers enforcing constraints. Having finite energy, the multipliers are thus massless ghost particles. The main numerical stepping method produced with the framework is called SPOOK.</p><p>Variational integrators preserve physical invariants globally, exactly in some cases, approximately but within fixed global bounds for others. This allows to product realistic physical trajectories even with the low order methods. These are needed in the solution of nonsmooth problems such as dry frictional contacts and in addition, they are computationally inexpensive. The combination of strong stability, low order, and the global preservation of invariants allows for large integration time steps, but without loosing accuracy on the important and visible physical quantities. SPOOK is thus well-suited for interactive simulations, such as those commonly used in virtual environment applications, because it is fast, stable, and faithful to the physics.</p><p>New results include a stable discretization of highly oscillatory terms of constraint regularization; a linearly stable constraint stabilization scheme based on ghost potential and Rayleigh dissipation terms; a single-step, strictly dissipative, approximate impact model; a quasi-linear complementarity formulation of dry friction that is isotropic and solvable for any nonnegative value of friction coefficients; an analysis of a splitting scheme to solve frictional contact complementarity problems; a stable, quaternion-based rigid body stepping scheme and a stable linear approximation thereof. SPOOK includes all these elements. It is linearly implicit and linearly stable, it requires the solution of either one linear system of equations of one mixed linear complementarity problem per regular time step, and two of the same when an impact condition is detected. The changes in energy caused by constraints, impacts, and dry friction, are all shown to be strictly dissipative in comparison with the free system. Since all regularization and stabilization parameters are introduced in the physics, they map directly onto physical properties and thus allow modeling of a variety of phenomena, such as constraint compliance, for instance.</p><p>Tutorial material is included for continuous and discrete-time analytic mechanics, quaternion algebra, complementarity problems, rigid body dynamics, constraint kinematics, and special topics in numerical linear algebra needed in the solution of the stepping equations of SPOOK.</p><p>The qualitative and quantitative aspects of SPOOK are demonstrated by comparison with a variety of standard techniques on well known test cases which are analyzed in details. SPOOK compares favorably for all these examples. In particular, it handles ill-posed and degenerate problems seamlessly and systematically. An implementation suitable for large scale performance and accuracy testing is left for future work.</p>

A time-discrete formulation of the variational principle of mechanics is used to provide a consistent theoretical framework for the construction and analysis of low order integration methods. These are applied to mechanical systems subject to mixed constraints and dry frictional contacts and impacts---machines. The framework includes physics motivated constraint regularization and stabilization schemes. This is done by adding potential energy and Rayleigh dissipation terms in the Lagrangian formulation used throughout. These terms explicitly depend on the value of the Lagrange multipliers enforcing constraints. Having finite energy, the multipliers are thus massless ghost particles. The main numerical stepping method produced with the framework is called SPOOK. Variational integrators preserve physical invariants globally, exactly in some cases, approximately but within fixed global bounds for others. This allows to product realistic physical trajectories even with the low order methods. These are needed in the solution of nonsmooth problems such as dry frictional contacts and in addition, they are computationally inexpensive. The combination of strong stability, low order, and the global preservation of invariants allows for large integration time steps, but without loosing accuracy on the important and visible physical quantities. SPOOK is thus well-suited for interactive simulations, such as those commonly used in virtual environment applications, because it is fast, stable, and faithful to the physics. New results include a stable discretization of highly oscillatory terms of constraint regularization; a linearly stable constraint stabilization scheme based on ghost potential and Rayleigh dissipation terms; a single-step, strictly dissipative, approximate impact model; a quasi-linear complementarity formulation of dry friction that is isotropic and solvable for any nonnegative value of friction coefficients; an analysis of a splitting scheme to solve frictional contact complementarity problems; a stable, quaternion-based rigid body stepping scheme and a stable linear approximation thereof. SPOOK includes all these elements. It is linearly implicit and linearly stable, it requires the solution of either one linear system of equations of one mixed linear complementarity problem per regular time step, and two of the same when an impact condition is detected. The changes in energy caused by constraints, impacts, and dry friction, are all shown to be strictly dissipative in comparison with the free system. Since all regularization and stabilization parameters are introduced in the physics, they map directly onto physical properties and thus allow modeling of a variety of phenomena, such as constraint compliance, for instance. Tutorial material is included for continuous and discrete-time analytic mechanics, quaternion algebra, complementarity problems, rigid body dynamics, constraint kinematics, and special topics in numerical linear algebra needed in the solution of the stepping equations of SPOOK. The qualitative and quantitative aspects of SPOOK are demonstrated by comparison with a variety of standard techniques on well known test cases which are analyzed in details. SPOOK compares favorably for all these examples. In particular, it handles ill-posed and degenerate problems seamlessly and systematically. An implementation suitable for large scale performance and accuracy testing is left for future work.

Particle suspensions occur in many situations in nature and industry. In this master’s thesis, the motion of a single rigid spheroidal particle immersed in Stokes flow is studied numerically using a boundary integral method and a new specialized quadrature method known as quadrature by expansion (QBX). This method allows the spheroid to be massless or inertial, and placed in any kind of underlying Stokesian flow.   A parameter study of the QBX method is presented, together with validation cases for spheroids in linear shear flow and quadratic flow. The QBX method is able to compute the force and torque on the spheroid as well as the resulting rigid body motion with small errors in a short time, typically less than one second per time step on a regular desktop computer. Novel results are presented for the motion of an inertial spheroid in quadratic flow, where in contrast to linear shear flow the shear rate is not constant. It is found that particle inertia induces a translational drift towards regions in the fluid with higher shear rate. / Partikelsuspensioner förekommer i många sammanhang i naturen och industrin. I denna masteruppsats studeras rörelsen hos en enstaka stel sfäroidisk partikel i Stokesflöde numeriskt med hjälp av en randintegralmetod och en ny specialiserad kvadraturmetod som kallas quadrature by expansion (QBX). Metoden fungerar för masslösa eller tröga sfäroider, som kan placeras i ett godtyckligt underliggande Stokesflöde.   En parameterstudie av QBX-metoden presenteras, tillsammans med valideringsfall för sfäroider i linjärt skjuvflöde och kvadratiskt flöde. QBX-metoden kan beräkna kraften och momentet på sfäroiden samt den resulterande stelkroppsrörelsen med små fel på kort tid, typiskt mindre än en sekund per tidssteg på en vanlig persondator. Nya resultat presenteras för rörelsen hos en trög sfäroid i kvadratiskt flöde, där skjuvningen till skillnad från linjärt skjuvflöde inte är konstant. Det visar sig att partikeltröghet medför en drift i sidled mot områden i fluiden med högre skjuvning.