**Delta-V Roadmap for Earth-Moon / Cis-Lunar Orbits**

A road-map for orbital transfers (delta-v, TOF, phasing, etc.) in the Earth-Moon system does not currently exist in any universally useful capacity, especially with regards to Lagrangian orbits. The goal of this project is to create a catalog of orbital transfers in the Earth-Moon R3BP for use in mission design.

Faculty: David Spencer, Robert Melton

Students: Davide Conte, Guanwei He, Matthew Shaw

**High-Fidelity Astrodynamics Library**

We’re currently developing a complete astrodynamics library for use by current and future ARGoPS members. The library will include high-fidelity propagators with full perturbation models, interplanetary trajectory solvers, maneuver optimization tools, orbit determination filters, and more.

Faculty: David Spencer, Robert Melton

Students: Jason Reiter, Andrew Goodyear

**Non-Discrete Collision Avoidance**

Presently, conjunction assessment is accomplished by identifying individual conjunctions, performing risk assessment activities on each, and, for those conjunctions that are determined to be substantially risky, determining what sorts of mitigation activities (typically a satellite maneuver) might be appropriate to remediate the risk. This approach is feasible so long as the number of discrete events that require extended analysis be relatively small. If, however, the space catalogue should increase greatly in size (due either to debris-producing events or an increase in sensitivity in space sensors), then the number of discrete events will grow to a level that will make individual adjudication of conjunctions impossible; and some other method for conjunction assessment will be necessary. The goal of this project is to create a tool for optimizing collision avoidance maneuvers under such circumstances.

Faculty: David Spencer, Robert Melton

Students: Jason Reiter, Andrew Goodyear, Ghanghoon Paik, Mollik Nayyar

**Optimal Low-Thrust Geostationary Transfer Orbit Using a Hybrid Pseudospectral Collocation Method**

The focus of this thesis is a two-phase method for the solution of trajectory optimization problems where a low-thrust ion engine is used for a many-revolution transfer from a low-Earth orbit to a geostationary orbit. The problem is formulated using a perturbation model in the modified equinoctial element set. Perturbations considered are that of a low-thrust ion propulsion source and a truncated spherical harmonic model for Earth’s gravity. Extra consideration for Earth’s umbral and penumbral regions is also presented. The formulation for both phase one as an indirect optimal control problem and phase two as a direct optimal control problem is provided as well as an explanation of the Legendre-Gauss-Radau pseudospectral method used to find numerical solutions for the chosen test case. The test case used within this thesis is based on a test case provided in the recent literature by Betts [1]. A simulation of the phase one problem, with an analysis on how to obtain the control curves is provided. All simulations were performed and solved in MATLAB using a nonlinear optimizer created by for this thesis and loosely based on code from Reference [2]. The formulation for the phase one problem has produced encouraging results for future research testing the derived control laws for spacecraft steering provided. The phase two direct optimization problem provides additional future work.

Faculty: David Spencer

Students: Andrew Goodyear

**Parametric Trade Study of Multiple Libration Point Orbits in the Circular Restricted Four-body Problem**

Studying an asteroid up close has been up until more recently nothing more than wishful thinking. With NASA’s Asteroid Redirect Mission in the planning stages, the prospect of bringing an asteroid back to the vicinity of the Earth is tantalizing. Once an asteroid has been retrieved and brought back to the Earth–Moon system and placed into orbit for study, human crews will visit it and study it up close. This thesis explores the orbital dynamics of an asteroid in orbit around the Earth–Moon libration point, EML1. The dynamics of the motions for a spacecraft in close proximity to an asteroid are found using the circular restricted four-body problem (CR4BP). Treating the problem as the superposition of two circular restricted threebody problems (CR3BP), the asteroid becomes an additional gravitational perturbation to a spacecraft close to the asteroid. Two sets of coupled equations of motion, one for the asteroid and one for the spacecraft are derived and solved simultaneously. A trade study to examine the near-term behavior of a spacecraft’s orbit relative to the asteroid is conducted via a series of simulations utilizing a variety of variables such as the asteroid’s location and mass relative to the spacecraft, the size of orbit, and the varying of initial conditions. The characteristics being studied are the stability of the spacecraft’s orbit over a short duration as well as the range between the spacecraft and its target asteroid over the mission duration. In addition, the minimum safe stand-off distance between both objects is determined to ensure no collisions or orbital instability. The ultimate goal is to obtain various datasets to deduce the most stable conditions for placing an asteroid and to determine where to fly a spacecraft in formation with the asteroid in orbit about the EML1 libration point. The results show the viability of a couple orbits as well as the prevalence of impacts. How the data can be utilized for future missions was also studied.

Faculty: David Spencer

Students: Peter Scarcella

**Semi-analytical Solutions for Proximity Operations in the Three-Body Problem**

This research aims at characterizing the relative motion of spacecraft in periodic orbits in the restricted three-body problem. Proximity operations maneuvers, such as constellation-keeping and rendezvous, are approximated using a semi-analytical approach, i.e. by combining the analytical approximation of the nominal periodic orbit of the targeted spacecraft, constellation, or orbital location with the numerical integration of its related state transition matrix. The absolute propellant-optimal result is then found through an optimization method and compared to nearby local minima for a given set of time constraints in order to find a maneuver that allows for flexibility regarding departure and arrival time along with contingency plans. The proposed approximation is validated against the use of the more computationally expensive full nonlinear equations of motion of the three-body problem and the “area of applicability” of the proposed method is defined based on a metric (currently TBD). Sample results for the Earth-Moon, Mars-Phobos, and Mars-Deimos systems are presented for cis-lunar and cis-Martian orbits of interest, including halo orbits and distant retrograde orbits. The implementation of the proposed method in pre-phase A mission design is also demonstrated in a sample full Earth to Mars mission.

Faculty: David Spencer

Students: Davide Conte

**Spacecraft Stealth through Optimization of Orbit-Perturbing Maneuvers: A Game Theory Approach**

This research involves the development of two tools. An orbit determination system will be designed to track maneuvering objects through the use of advanced filters and pattern recognition. A second tool will be designed to perturb the orbit of a satellite attempting to avoid detection by the aforementioned orbit determination system. Statistical models will be used to simulate environmental conditions, satellite characteristics, and other variables in such a scenario and the two entities will be pitted against each other using game theory approaches. The goal is for the orbit perturbing tool to be able to successfully evade even the most capable tracking systems.

Faculty: David Spencer

Students: Jason Reiter

**Design of Interplanetary Trajectories with Multiple Synergetic Gravitational Assist Maneuvers via Heuristic Optimization**

The design capacity for synergetic gravity assists (powered flyby’s) changes the type of possible optimal trajectories to distant planets. Heuristic optimization methods – particle Swarm Optimization (PSO) and Differential Evolution (DE) – are used to determine optimal mission trajectories from Earth orbit to planets of interest, subject to synergetic gravity assist maneuver(s) in between. Research methods are tested on known missions such as Voyager 1, Voyager 2, and Cassini to verify against actual mission data. The methods of this research can then be applied to designing new interplanetary space missions.

Faculty: Robert Melton

Students: Matthew Shaw

**Method Analysis of a Time-Optimal, Constrained Satellite Reorientation Maneuver**

I will be investigating the computational efficiency and accuracy of different stochastic optimizers, such as Particle Swarm Optimization and the Artificial Bee Colonization, for the inverse dynamics solution to a constrained satellite reorientation maneuver. For this implementation, Chebyshev polynomials are being used to model the components of our Euler angles. This consideration enforces that the sum of the squares of our Euler angles is always one, a necessary physical constraint, in addition to allowing for simple derivative formulations. The unknowns for our optimizations will be the coefficients of these Chebyshev polynomials, which can be used to express our torques and angular momenta over time. For testing, the algorithms will be developed using MATLAB, with the intent on expanding into C or C++ for faster computation times.

Faculty: Robert Melton

Students: Michael Nino

**Design of Nonlinear Observers and Optimal Feedback Control Laws**

The problem of finding optimal feedback control laws for nonlinear dynamical systems remains a significant challenge, even in the case when perfect state information is available. This is because, the feedback solution is typically derived from the so-called value function, which is the solution to a nonlinear partial differential equation known as the *Hamilton Jacobi Bellman (HJB) equation* . The chief impediment in the task of envisioning a systematic solution process for these PDEs is associated with the fact that the number of spatial variables is equal to the state dimension, which is twice the number of degrees of freedom of a mechanical system. The main focus of my research is to develop a computationally efficient unified approach to solve linear and nonlinear PDEs with several independent variables.

Faculty: Puneet Singla

Students: Mehrdad Mirzaei

**Higher Order Polynomial Series Expansion Solution to the Uncertain Lambert Problem**

The goal of this research is to find solutions to the Lambert Problem in which the initial and final position of a spacecraft are subjected to generic probability distribution functions (pdf’s). The pdf assigns uncertainty to the position of the spacecraft similar to the uncertainty in real world satellite operations, where the position of the spacecraft can only be measured to finite accuracy. Finding the solution to the Lambert Problem under these conditions can allow deeper quantitative analysis in areas such as spacecraft collision probability, maneuver detection, debris tracking and data association. In traditional analyses of the Uncertain Lambert Problem, only the first order term in the Taylor Series Expansion (aka state transition matrix) is used to compute the deviation of velocity from the nominal solution due to uncertainty. Neglecting terms other than this first order STM causes error in calculated velocity due to non-linearities to rapidly accumulate as the true satellite orbit moves away from the nominal orbit. The novelty of this work comes in the application of a newly developed method known as Conjugate Unscented Transformation (CUT) to the reduce the heavy computational load associated with evaluating the higher dimensional sensitivity matrices. This approach allows higher fidelity solutions over much wider ranges of uncertainty at no additional computational expense.

Faculty: Puneet Singla

Students: Zachary Hall

**Kalman Filtering in Regularized Coordinates: Applications to Astrodynamics**

This research aims to investigate the benefits of conducting orbit estimation in regularized coordinates. Regularizations are coordinate transformations which eliminate singularities in dynamic systems by mapping time to a new independent variable. In orbital dynamics, this new independent variable is typically an anomaly (true, universal, etc.). These coordinate transformations are appealing for orbital mechanics problems because both the motion and uncertainty propagation become mostly linear (at the expense of increased dimensionality). Regularization can allow for more accurate and robust estimation, especially in the long term, relative to traditional estimation in the Cartesian space. Transformations to be considered include Burdet and Kustaanheimo-Stiefel; orbit determination in these coordinate spaces will be conducted using extended and unscented Kalman filters which must be properly constrained to accommodate the redundant state variables.

__Faculty__: Puneet Singla

__Students__: David Ciliberto