This dissertation develops a methodology for automated trajectory control of a spacecraft about a non-maneuvering target. The methodology utilizes relative orbital elements (ROEs), combined with guidance laws based upon artificial potential functions (APFs), to perform automated trajectory planning and maneuver design. The investigation provides a definitive reference on the definition and use of ROEs for relative proximity operations. The detailed derivation of ROEs is provided, along with transformations between ROEs and relative Cartesian state elements, characteristics of unforced motion in terms of ROEs, and the effect of impulsive maneuvers on ROEs. Operationally-useful guidance algorithms utilizing ROEs are developed and demonstrated. These ROE-based algorithms for rendezvous, circumnavigation and station-keeping provide a toolkit for relative proximity operations mission planning. A new approach for APF formulation using ROEs as the target variables is developed. While previous approaches allowed targeting of a specified relative position, the present approach allows the targeting of relative orbit geometries. The approach capitalizes upon the orbital dynamics represented through the ROEs, and retains the computational simplicity offered by the APFs. Formulations for the APF targeting of individual ROEs, as well as simultaneous targeting of a set of ROEs, are established. An approach for combining ROE targeting using APFs with obstacle avoidance is presented. The trajectory guidance algorithm performance is evaluated using a flight-like guidance, navigation and control simulation environment, including orbital perturbations. Algorithm performance is established through a set of operationally relevant scenarios. The guidance algorithms are shown to be capable of correcting for environmental disturbances, while meeting the targeted relative orbits in an automated fashion.

This dissertation presents an experimental and analytical study of the Vision Based Navigation system (VisNav). VisNav is a novel intelligent optical sensor system invented by Texas A&M University recently for autonomous proximity operations. This dissertation is focused on system calibration techniques and navigation algorithms. This dissertation is composed of four parts. First, the fundamental hardware and software design configuration of the VisNav system is introduced. Second, system calibration techniques are discussed that should enable an accurate VisNav system application, as well as characterization of errors. Third, a new six degree-of-freedom navigation algorithm based on the Gaussian Least Squares Differential Correction is presented that provides a geometrical best position and attitude estimates through batch iterations. Finally, a dynamic state estimation algorithm utilizing the Extended Kalman Filter (EKF) is developed that recursively estimates position, attitude, linear velocities, and angular rates. Moreover, an approach for integration of VisNav measurements with those made by an Inertial Measuring Unit (IMU) is derived. This novel VisNav/IMU integration technique is shown to significantly improve the navigation accuracy and guarantee the robustness of the navigation system in the event of occasional dropout of VisNav data.

This dissertation presents an experimental and analytical study of the Vision Based Navigation system (VisNav). VisNav is a novel intelligent optical sensor system invented by Texas A&M University recently for autonomous proximity operations. This dissertation is focused on system calibration techniques and navigation algorithms. This dissertation is composed of four parts. First, the fundamental hardware and software design configuration of the VisNav system is introduced. Second, system calibration techniques are discussed that should enable an accurate VisNav system application, as well as characterization of errors. Third, a new six degree-of-freedom navigation algorithm based on the Gaussian Least Squares Differential Correction is presented that provides a geometrical best position and attitude estimates through batch iterations. Finally, a dynamic state estimation algorithm utilizing the Extended Kalman Filter (EKF) is developed that recursively estimates position, attitude, linear velocities, and angular rates. Moreover, an approach for integration of VisNav measurements with those made by an Inertial Measuring Unit (IMU) is derived. This novel VisNav/IMU integration technique is shown to significantly improve the navigation accuracy and guarantee the robustness of the navigation system in the event of occasional dropout of VisNav data.

The topic of this thesis is on-board estimation of spacecraft collision probability for orbital rendezvous and proximity operations. All of the examples shown in this work assume that the satellite dynamics are described by the Clohessy-Wiltshire equations, and that the spacecraft are spherical. Several collision probability metrics are discussed and compared. Each metric can be placed into one of three categories. The first category provides an estimate of the instantaneous probability of collision, and places an upper bound on the total probability of collision. The second category provides an estimate of total collision probability directly. The last category uses Monte Carlo analysis and a novel Pseudo Monte Carlo analysis algorithm to determine total collision probability. The metrics are compared and their accuracy is determined for a variety of on-orbit conditions. Lastly, a method is proposed in which the metrics are arranged in a hierarchy such that those metrics that can be computed quickest are calculated first. As the proposed algorithm progresses the metrics become more costly to compute, but yield more accurate estimates of collision probability. Each metric is compared to a threshold value. If it exceeds the limits determined by mission constraints, the algorithm computes a more accurate estimate by calculating the next metric in the series. If the threshold is not reached, it is assumed there is a tolerable collision risk and the algorithm is terminated. In this way the algorithm is capable of adapting to the level of collision probability, and can be sufficiently accurate without needless calculations being performed. This work shows that collision probability can be systematically estimated.

Spacecraft relative motion modeling and control promises to enable or augment a wide range of missions for scientific research, military applications, and space situational awareness. This dissertation focuses on the development of novel, optimization-based, control design for some representative relative-motion-enabled missions. Spacecraft relative motion refers to two (or more) satellites in nearly identical orbits. We examine control design for relative configurations on the scale of meters (for the purposes of proximity operations) as well as on the scale of tens of kilometers (representative of science gathering missions). Realistic control design for satellites is limited by accurate modeling of the relative orbital perturbations as well as the highly constrained nature of most space systems. We present solutions to several types of optimal orbital maneuvers using a variety of different, realistic assumptions based on the maneuver objectives. Initially, we assume a perfectly circular orbit with a perfectly spherical Earth and analytically solve the under-actuated, minimum-energy, optimal transfer using techniques from optimal control and linear operator theory. The resulting open-loop control law is guaranteed to be a global optimum. Then, recognizing that very few, if any, orbits are truly circular, the optimal transfer problem is generalized to the elliptical linear and nonlinear systems which describe the relative motion. Solution of the minimum energy transfer for both the linear and nonlinear systems reveals that the resulting trajectories are nearly identical, implying that the nonlinearity has little effect on the relative motion. A continuous-time, nonlinear, sliding mode controller which tracks the linear trajectory in the presence of a higher fidelity orbit model shows that the closed-loop system is both asymptotically stable and robust to disturbances and un-modeled dynamics. Next, a novel method of computing discrete-time, multi-revolution, finite-thrust, fuel-optimal, relative orbit transfers near an elliptical, perturbed orbit is presented. The optimal control problem is based on the classical, continuous-time, fuel-optimization problem from calculus of variations, and we present the discrete-time analogue of this problem using a transcription-based method. The resulting linear program guarantees a global optimum in terms of fuel consumption, and we validate the results using classical impulsive orbit transfer theory. The new method is shown to converge to classical impulsive orbit transfer theory in the limit that the duration of the zero-order hold discretization approaches zero and the time horizon extends to infinity. Then the fuel/time optimal control problem is solved using a hybrid approach which uses a linear program to solve the fuel optimization, and a genetic algorithm to find the minimizing time-of-flight. The method developed in this work allows mission planners to determine the feasibility for realistic spacecraft and motion models. Proximity operations for robotic inspection have the potential to aid manned and unmanned systems in space situational awareness and contingency planning in the event of emergency. A potential limiting factor is the large number of constraints imposed on the inspector vehicle due to collision avoidance constraints and limited power and computational resources. We examine this problem and present a solution to the coupled orbit and attitude control problem using model predictive control. This control technique allows state and control constraints to be encoded as a mathematical program which is solved on-line. We present a new thruster constraint which models the minimum-impulse bit as a semi-continuous variable, resulting in a mixed-integer program. The new model, while computationally more expensive, is shown to be more fuel-efficient than a sub-optimal approximation. The result is a fuel efficient, trajectory tracking, model predictive controller with a linear-quadratic attitude regulator which tracks along a pre-computed ``safe'' trajectory in the presence of un-modeled dynamics on a higher fidelity orbital and attitude model. / Ph. D.

The term proximity operations has been widely used in recent years to describe a wide range of space missions that require a spacecraft to remain close to another space object. Such missions include, for example, the inspection, health monitoring, surveillance, servicing, and refueling of a space asset by another spacecraft. One of the biggest challenges in autonomous space proximity operations, either cooperative or uncooperative, is the need to autonomously and accurately track time-varying relative position and attitude references, i.e., pose references, with respect to a moving target, in order to avoid on-orbit collisions and achieve the overall mission goals. In addition, if the target spacecraft is uncooperative, the Guidance, Navigation, and Control (GNC) system of the chaser spacecraft must not rely on any help from the target spacecraft. In this case, vision-based sensors, such as cameras, are typically used to measure the relative pose between the spacecraft. Although vision-based sensors have several attractive properties, they introduce new challenges, such as no direct linear and angular velocity measurements, slow update rates, and high measurement noise. This dissertation investigates the problem of autonomously controlling and estimating the pose of a chaser spacecraft with respect to a moving target spacecraft, possibly uncooperative. Since this problem is inherently hard, the standard approach in the literature is to split the attitude-tracking problem from the position-tracking problem. Whereas the attitude-tracking problem is relatively simple, since the rotational motion is independent from the translational motion, the position-tracking problem is more complicated, as the translational motion depends on the rotational motion. Hence, whereas strong theoretical results exist for the attitude problem, the position problem typically requires additional assumptions. An alternative, more general approach to the pose control and estimation problems is to consider the fully coupled 6-DOF motion. However, fewer results exist that directly address this higher dimensional problem. The main contribution of this dissertation is to show that dual quaternions can be used to extend the theoretical results that exist for the attitude motion into analogous results for the combined position and attitude motion. Moreover, this dissertation shows that this can be accomplished by (almost) just replacing quaternions by dual quaternions in the original derivations. This is because dual quaternions are built on and are an extension of classical quaternions. Dual quaternions provide a compact representation of the pose of a frame with respect to another frame. Using this approach, three new results are presented in this dissertation. First, a pose-tracking controller that does not require relative linear and angular velocity measurements is derived with vision-based sensors in mind. Compared to existing literature, the proposed velocity-free pose-tracking controller guarantees that the pose of the chaser spacecraft will converge to the desired pose independently of the initial state, even if the reference motion is not sufficiently exciting. In addition, the convergence region does not depend on the gains of the controller. Second, a Dual Quaternion Multiplicative Extended Kalman Filter (DQ-MEKF) is developed from the highly successful Quaternion MEKF (Q-MEKF) as an alternative way to achieve pose-tracking without velocity measurements. Existing dual quaternion EKFs are additive, not multiplicative, and have two additional states. The DQ-MEKF is experimentally validated and compared with two conventional EKFs on the 5-DOF platform of the Autonomous Spacecraft Testing of Robotic Operations in Space (ASTROS) facility at the School of Aerospace Engineering at Georgia Tech. Finally, the velocity-free pose-tracking controller is compared qualitatively and quantitatively to a pose-tracking controller that uses the velocity estimates produced by the DQ-MEKF through a realistic proximity operations simulation. Third, a pose-tracking controller that does not require the mass and inertia matrix of the chaser satellite is suggested. This inertia-free controller takes into account the gravitational acceleration, the gravity-gradient torque, the perturbing acceleration due to Earth's oblateness, and constant -- but otherwise unknown -- disturbance forces and torques. Sufficient conditions on the reference pose are also given that guarantee the identification of the mass and inertia matrix of the satellite. Compared to the existing literature, this controller has only as many states as unknown elements and it does not require a priori known upper bounds on any states or parameters. Finally, the inertia-free pose-tracking controller and the DQ-MEKF are tested on a high-fidelity simulation of the 5-DOF platform of the ASTROS facility and also experimentally validated on the actual platform. The equations of motion of the 5-DOF platform, on which the high-fidelity simulation is based, are derived for three distinct cases: a 3-DOF case, a 5-DOF case, and a (2+1)-DOF case. Four real-time experiments were run on the platform. In the first, a sinusoidal reference attitude with respect to the inertial frame is tracked using VSCMGs. In the second, a constant reference attitude is maintained with respect to a target object using VSCMGs and measurements from a camera. In the third, the same sinusoidal reference attitude with respect to the inertial frame tracked in the first experiment is now tracked using cold-gas thrusters. Finally, in the fourth and last experiment, a time-varying 5-DOF reference pose with respect to the inertial frame is tracked using cold-gas thrusters.

Tradition spacecraft proximity operations require large and expensive on-board sensors and significant ground support. Relative angle measurements can be obtained from small, simple, and inexpensive on-board sensors, but have not traditionally been used for proximity operation because of difficulty generating rang information. In this thesis it is shown that useful relative range data can be generated provided that the spacecraft is experiencing a small continuous thrust such as would be provided by a low-thrust propulsion system.

Angles-only navigation is simple, robust, and well proven in many applications. However, it is sometimes ill-conditioned for orbital rendezvous and proximity operations because, without a direct range measurement, the distance to approaching satellites must be estimated by firing thrusters and observing the change in the target's bearing. Nevertheless, the simplicity of angles-only navigation gives it great appeal. The viability of this technique for relative navigation is examined by building a high-fidelity simulation and evaluating the sensitivity of the system to sensor errors. The relative performances of square-root filtering methods, including Potter, Carlson, and UD factorization filters, are compared to the conventional and Joseph formulations. Filter performance is evaluated during closed-loop "station keeping" operations in simulation.

Due to mass and volume constraints on planetary missions, the development of control techniques that do not require fuel are of big interest. For those planets that have a dense enough atmosphere, aerodynamic drag can play an important role. The use of atmospheric differential drag for formation keeping was first proposed by Carolina L. Leonard in 1986, and has been proven to work in Earth atmosphere by many missions. Moreover, atmospheric drag has been used in the Mars atmosphere as aerobraking technique to decelerate landing vehicles, and to circularize the orbit of the spacecraft. Still, no literature was available related to formation flying on Mars. To analyze the use of differential drag on the Mars atmosphere, the researcher accessed the two high resolution models available: NASA’s Mars-GRAM and ESA’s Mars Climate Database. These models allowed the simulation of conditions that a spacecraft would experience while in orbit around the planet. To explore the feasibility, the researcher first conducted a study where Mars atmosphere density was compared to Earth atmosphere, determining its applicability. Then, a simulation using MATLAB® was conducted, using a Keplerian two-body problem including the effects of Mars zonal harmonics (i.e. J2) and drag perturbations. Two 6U CubeSat were used in the simulation with deployable drag plates of different sizes, giving the possibility of having five differential drag scenarios as means of formation control. The conclusions showed that, although with some limitations, the use of differential drag as means of autonomous formation flying and proximity operations control is feasible using proven techniques previously validated in Low Earth Orbit. Lyapunov control was selected as the control strategy, where three different methods were evaluated and compared.