Cesar Ocampo

Indirect Optimization of Spiral Trajectories

Indirect optimization is used to compute minimum propellant spiral escapes and captures. A two-step estimation process generates accurate estimates of the Lagrange multipliers. The first step, an adjoint control transformation, converts the thrust unit direction vector to the actual multipliers that control the trajectory. Next, curve fits are matched with the values of the initial multipliers from spirals found with the adjoint control transformation and used to extrapolate the multipliers for longer spirals. Spherical rather than Cartesian coordinates are used because the spherical multipliers evolve in a well-behaved fashion, allowing the accurate extrapolation of the multipliers. Long-duration Earth escapes are presented for spirals as long as 150 days. Solutions are also presented with control limits on the thrust or specific impulse. Additionally, a transformation is developed that converts the spherical multipliers to the corresponding Cartesian multipliers and vice versa. Another transformation is developed to convert the optimal multipliers for two-dimensional, planar equatorial spirals into the optimal multipliers for planar-inclined orbits in either Cartesian or spherical coordinates in which the initial orbit may have arbitrary values for inclination, right ascension of the ascending node, and argument of periapsis.

Optimization of Roundtrip, Time-Constrained, Finite Burn Trajectories via an Indirect Method

An indirect trajectory optimization method is used to compute optimal, time-constrained, roundtrip, finite burn trajectories between any two orbits around a common central body. This method involves solving the optimal control problem as a multipoint boundary value problem with two discontinuities in the controls corresponding to the arrival at and the departure from the target. Solutions are provided that minimize the propellant, given either an initial or final mass, while constraining the stay time at the target to be greater than or equal to a specified minimum value and while constraining the total roundtrip time to be less than or equal to a specified maximum value. The results are applied to human-crewed, one-year Earth-Mars roundtrip missions with a minimum two-month stay at Mars and four year Earth-Jupiter missions with a minimum one-year Jovian stay. These missions utilize high-power, nuclear electric propulsion with either a constant or variable specific impulse engine. This theoretical formulation was used to find quick, efficient, converged solutions that are shown to be at least as optimal or slightly more so compared to another optimization method that is hybrid in nature but still uses continuous control where the thrust is along the primer vector.

Finite Burn Maneuver Modeling for a Generalized Spacecraft Trajectory Design and Optimization System

Abstract: The modeling, design, and optimization of finite burn maneuvers for a generalized trajectory design and optimization system is presented. A generalized trajectory design and optimization system is a system that uses a single unified framework that facilitates the modeling and optimization of complex spacecraft trajectories that may operate in complex gravitational force fields, use multiple propulsion systems, and involve multiple spacecraft. The modeling and optimization issues associated with the use of controlled engine burn maneuvers of finite thrust magnitude and duration are presented in the context of designing and optimizing a wide class of finite thrust trajectories. Optimal control theory is used examine the optimization of these maneuvers in arbitrary force fields that are generally position, velocity, mass, and are time dependent. The associated numerical methods used to obtain these solutions involve either, the solution to a system of nonlinear equations, an explicit parameter optimization method, or a hybrid parameter optimization that combines certain aspects of both. The theoretical and numerical methods presented here have been implemented in copernicus, a prototype trajectory design and optimization system under development at the University of Texas at Austin.

Indirect Optimization of Three-Dimensional Finite-Burning Interplanetary Transfers Including Spiral Dynamics

The indirect optimization problem for a three-dimensional transfer from low Earth orbit to low Mars orbit is solved. A step-by-step process developed for a two-dimensional model and techniques for accurately estimating the unknown costates for three-dimensional escape and capture spirals are used. Minimum-propellant trajectories for finite-burning engines are calculated. Solutions are considered with and without control limits on specific impulse and compared with previous research. Unlike other research, the entire trajectory, including the Martian capture sequence, is integrated in an Earth-referenced frame. Additionally, the capture sequence is not found by iteratively lowering the final targeted lowMars orbit, but the desired final orbit is directly targeted with no successive iterations of increasingly smaller lowMars orbits. As in the two-dimensional case, more fuel-efficient trajectories are found for the same mission objectives and constraints published in other research, emphasizing the importance of this technique.Whereas previous research only achieved final Martian orbits of 6 Mars radiiDUM (20,382 km), the new approach finds solutions for final Martian circular orbits of 1.47–2.00 DUM (5000–6794 km).

Initial Trajectory Model for a Multi-Maneuver Moon-to-Earth Abort Sequence

To support the mission design and trajectory design problems associated with the moon-to-Earth trajectories for the crew exploration vehicle, a starting trajectory model that serves as the first iterate for a complete targeting and optimization procedure that takes a spacecraft from any closed lunar parking orbit to the Earth entry interface state for any date is developed. The motivation for this work is to examine the any-time abort capability required for human moon missions. The results presented here are limited to impulsive maneuvers. An analytical procedure is developed that constructs a multi-impulse escape trajectory from the moon propagated forward in time and a backward propagated trajectory from the Earth with a mismatch in position and velocity near the sphere of influence of the moon. The position and velocity discontinuities at the mismatch point are small enough to lie within the convergence envelope of a simple gradient based differential correction procedure that can, at a minimum, generate a feasible solution. This solution can then be analyzed further and serve as an initial estimate for an optimization procedure. The efficiency of the method is illustrated by solving any-time abort transfer problems typical for a human mission.

Automated Generation of Symmetric Lunar Free-Return Trajectories

A procedure for free-return trajectory generation in a simplified Earth-moon system is presented. With two-body and circular restricted three-body models, the algorithm constructs an initial guess of the translunar injection state and time of flight. Once the initial trajectory is found, the Jacobian of the constraints is derived analytically using linear perturbation theory, and a square system of nonlinear equations is solved numerically to target Earth entry interface conditions leading to feasible free-return trajectories. No trial and error is required to generate the initial guess. Possible free returns include departures from both posigrade and retrograde Earth orbits coupled with circumlunar or cislunar flight, in and out of the Earth-moon plane.

Variational Equations for a Generalized Spacecraft Trajectory Model

Linear perturbation theory is used to develop the variational equations needed to determine the sensitivities of the state at some final time with respect to all of the independent variables associated with a spacecraft trajectory model that is general enough for most applications of interest. The state vector is an augmented vector that includes the position, velocity, mass, and all other control-related variables, such as thrust magnitude and direction. The force model is general and the trajectory can have any number of impulsive and/or finite burn maneuvers. The gradient expressions depend, in part, on the system state transition matrix associated with the given state and its corresponding equations of motion. As an example, the procedure developed is applied to the numerical optimization of a multi-impulse escape trajectory from the moon.

Analytical Gradients for Gravity Assist Trajectories Using Constant Specific Impulse Engines

A procedure for calculating the analytical derivatives required to optimize long duration constant specific impulse finite burns and multiple gravity assist trajectories is presented. The analytical derivatives are calculated using the state transition matrix associated with the complete set of the Euler–Lagrange equations of the optimal control problem on each trajectory segment. Another transition matrix maps perturbations across any discontinuities in the state due to a zero sphere of influence patched conic flyby or discontinuities in the equations of motion that occur when the engine turns on or off. As applications, the method is used to find optimal Earth to Saturn trajectories. The state transition matrix derivatives are shown to find optimal trajectories from sets of initial conditions where finite difference derivatives fail to converge.

Transfers to Earth centered orbits via lunar gravity assist

Abstract The use of lunar gravity assist to transfer payloads to Earth centered orbits is investigated. Such payloads include communications spacecraft bound for geostationary orbits or spacecraft bound for high inclination Earth orbits. Though payloads need to be injected on translunar trajectories with apogees at the lunar distance, the use of lunar gravity to change the inclination of the orbit provides (for a range of launch site latitudes) a substantial increase in the payload mass delivered or a reduction in the required characteristic velocity required to achieve orbit. By using the restricted three body problem as an approximate model for the motion of a spacecraft in the Earth–Moon system, a perturbation procedure and an optimization algorithm are used to construct nominal solutions.

Trajectory Analysis for the Lunar Flyby Rescue of AsiaSat-3/HGS-1

Abstract: On May 13, 1998, the Hughes Global Services 1 Spacecraft (HGS‐1, originally known as AsiaSat 3) became the first commercial spacecraft to fly by the Moon on a trajectory to reposition it into a useful geosynchronous orbit. This was necessary due to the failure of the last stage of the launch vehicle that left it in a high inclination, eccentric, and unusable orbit. The spacecraft did not have enough propellant to perform the maneuvers required to place it into its intended geostationary orbit via a standard transfer trajectory. However, it did have enough propellant to place it on a trajectory that flew by the Moon twice to finally achieve a useful low inclination geosynchronous orbit. In addition to being the first commercial operation in the vicinity of the Moon, it was the last successful lunar mission of the twentieth century. We discuss of the events leading up to the start of the rescue operation that included contributions from external organizations. We also describe the analytic estimates used to construct the trajectory and provide an overview of the details of the actual mission.