Direct transcription of low-thrust trajectories with finite trajectory elements

This paper presents a novel approach to the design of Low-Thrust trajectories, based on a first order approximated analytical solution of Gauss planetary equations. This analytical solution is shown to have a better accuracy than a second-order explicit numerical integrator and at a lower computational cost. Hence, it can be employed for the fast propagation of perturbed Keplerian motion when moderate accuracy is required. The analytical solution was integrated in a direct transcription method based on a decomposition of the trajectory into direct finite perturbative elements (DFPET). DFPET were applied to the solution of two-point boundary transfer problems. Furthermore the paper presents an example of the use of DFPET for the solution of a multiobjective trajectory optimisation problem in which both the total ∆V and transfer time are minimized with respect to departure and arrival dates. Two transfer problems were used as test cases: a direct transfer from Earth to Mars and a spiral from a low Earth orbit to the International Space Station

Guidance, flight mechanics and trajectory optimization. Volume 12 - Relative motion, guidance equations for terminal rendezvous

Equations of relative motion and guidance for orbital transfer and docking maneuvers in spacecraft rendezvou

Extended analytical formulas for the perturbed Keplerian motion under a constant control acceleration

This paper presents a set of analytical formulae for the perturbed Keplerian motion of a spacecraft under the effect of a constant control acceleration. The proposed set of formulae can treat control accelerations that are fixed in either a rotating or inertial reference frame. Moreover, the contribution of the (Formula presented.) zonal harmonic is included in the analytical formulae. It will be shown that the proposed analytical theory allows for the fast computation of long, multi-revolution spirals while maintaining good accuracy. The combined effect of different perturbations and of the shadow regions due to solar eclipse is also included. Furthermore, a simplified control parameterisation is introduced to optimise thrusting patterns with two thrust arcs and two cost arcs per revolution. This simple parameterisation is shown to ensure enough flexibility to describe complex low thrust spirals. The accuracy and speed of the proposed analytical formulae are compared against a full numerical integration with different integration schemes. An averaging technique is then proposed as an application of the analytical formulae. Finally, the paper presents an example of design of an optimal low-thrust spiral to transfer a spacecraft from an elliptical to a circular orbit around the Earth

Interplanetary mission design with applications to guidance and optimal control of aero-assisted trajectories

A method for finding optimal aerogravity-assist tours of the solar system is developed using indirect methods. Two cost functionals are used in the optimization; finding the minimum required maximum lift-to-drag ratio, with and without a convective heating-rate path constraint, and the path which provides the minimum total stagnation point convective heat load. It is found that using present or near-future thermal protection system materials will suffice for certain aerogravity assist trajectories at Mars. Minimum heat load optimal trajectories are found for aerocapture maneuvers at Uranus and Neptune. With a large radius, and short rotational periods, atmospheric rotation must be taken into account to accurately model the system dynamics. Investigation of the 2018 Inspiration Mars free-return opportunity is conducted. A broad search over 100 years of Mars free-return trajectories is catalogued, and a Pareto front analysis is employed to find the overall best trajectories in the timespan. The geometry is explored further with the use of a time-free ephemeris to see where minimal energy transfer arcs between Earth and Mars occur, and see if the 2018 opportunity is one such transfer. It turned out that both the 2017 and 2064 candidates found from the 100-year search were the closest to minimum energy, highlighting the rarity of the Inspiration Mars opportunity, and gives a motivating push to fly this mission

Interplanetary mission design with applications to guidance and optimal control of aero-assisted trajectories

A method for finding optimal aerogravity-assist tours of the solar system is developed using indirect methods. Two cost functionals are used in the optimization; finding the minimum required maximum lift-to-drag ratio, with and without a convective heating-rate path constraint, and the path which provides the minimum total stagnation point convective heat load. It is found that using present or near-future thermal protection system materials will suffice for certain aerogravity assist trajectories at Mars. Minimum heat load optimal trajectories are found for aerocapture maneuvers at Uranus and Neptune. With a large radius, and short rotational periods, atmospheric rotation must be taken into account to accurately model the system dynamics. Investigation of the 2018 Inspiration Mars free-return opportunity is conducted. A broad search over 100 years of Mars free-return trajectories is catalogued, and a Pareto front analysis is employed to find the overall best trajectories in the timespan. The geometry is explored further with the use of a time-free ephemeris to see where minimal energy transfer arcs between Earth and Mars occur, and see if the 2018 opportunity is one such transfer. It turned out that both the 2017 and 2064 candidates found from the 100-year search were the closest to minimum energy, highlighting the rarity of the Inspiration Mars opportunity, and gives a motivating push to fly this mission

Hybrid multi-objective trajectory optimization of low-thrust space mission design

Mención Internacional en el título de doctorThe overall goal of this dissertation is to develop multi-objective optimization algorithms for computing low-thrust trajectories. The thesis is motivated by the increasing number of space projects that will benefit from low-thrust propulsion technologies to gain unprecedented scientific, economic and social return. The low-cost design of such missions and the inclusion of concurrent engineering practices during the preliminary design phase demand advanced tools to rapidly explore different solutions and to benchmark them with respect to multiple conicting criteria. However, the determination of optimal low-thrust transfers is a challenging task and remains an active research field that seeks performance improvements. This work contributes to increase the efficiency of searching wide design spaces, reduce the amount of necessary human involvement, and enhance the capabilities to include complex operational constraints. To that end, the general low-thrust trajectory optimization problem is stated as a multi-objective Hybrid Optimal Control Problem. This formulation allows to simultaneously optimize discrete decisionmaking processes, discrete dynamics, and the continuous low-thrust steering law. Within this framework, a sequential two-step solution approach is devised for two different scenarios. The first problem considers the optimization of low-thrust multi-gravity assist trajectories. The proposed solution procedure starts by assuming a planar shape-based model for the interplanetary trajectory. A multi-objective heuristic algorithm combined with a gradient-based solver optimize the parameters de_ning the shape of the trajectory, the number and sequence of the gravity assists, the departure and arrival dates, and the launch excess velocity. In the second step, candidate solutions are deemed as initial guesses to solve the Nonlinear Programming Problem resulting from applying a direct collocation transcription scheme. In this step, the sequence of planetary gravity assists is known and provided by the heuristic search, dynamics is three-dimensional, and the steering law is not predefined. Operational constraints to comply with launch asymptote declination limits and fixed reorientation times during the transfer apply. The presented approach is tested on a rendezvous mission to Ceres, on a yby mission to Jupiter, and on a rendezvous mission to Pluto. Pareto-optimal solutions in terms of time of ight and propellant mass consumed (or alternatively delivered mass) are obtained. Results outperform those found in the literature in terms of optimality while showing the effectiveness of the proposed methodology to generate quick performance estimates. The second problem considers the simultaneous optimization of fully electric, fully chemical and combined chemical-electric orbit raising transfers between Earth's orbits is considered. In the first step of the solution approach, the control law of the electric engine is parameterized by a Lyapunov function. A multi-objective heuristic algorithm selects the optimal propulsion system, the transfer type, the low-thrust control history, as well as the number, orientation, and magnitude of the chemical firings. Earth's shadow, oblateness and Van-Allen radiation effects are included. In the second step, candidate solutions are deemed as initial guesses to solve the Nonlinear Programming Problem resulting from applying a direct collocation scheme. Operational constraints to avoid the GEO ring in combination to slew rate limits and slot phasing constraints are included. The proposed approach is applied to two transfer scenarios to GEO orbit. Pareto-optimal solutions trading of propellant mass, time of ight and solar-cell degradation are obtained. It is identified that the application of operational restrictions causes minor penalties in the objective function. Additionally, the analysis highlights the benefits that combined chemical-electric platforms may provide for future GEO satellites.El objetivo principal de esta trabajo es desarrollar algoritmos de optimización multi-objetivo para la obtención de trayectorias espaciales con motores de bajo empuje. La tesis está motivada por el creciente número de misiones que se van a beneficiar del uso de estas tecnologías para conseguir beneficios científicos, económicos y sociales sin precedentes. El diseño de bajo coste de dichas misiones ligado a los principios de ingeniería concurrente requieren herramientas computacionales avanzadas que exploren rápidamente distintas soluciones y las comparen entre sí respecto a varios criterios. Sin embargo, esta tarea permanece como un campo de investigación activo que busca continuamente mejoras de rendimiento durante el proceso. Este trabajo contribuye a aumentar la eficiencia cuando espacio de diseño es amplio, a reducir la participación humana requerida y a mejorar las capacidades para incluir restricciones operacionales complejas. Para este fin, el problema general de optimización de trayectorias de bajo empuje se presenta como un problema híbrido de control óptimo. Esta formulación permite optimizar al mismo tiempo procesos de toma de decisiones, dinámica discreta y la ley de control del motor. Dentro de este marco, se idea un algoritmo secuencial de dos pasos para dos escenarios diferentes. El primer problema considera la optimización de trayectorias de bajo empuje con múltiples y-bys. El proceso de solución propuesto comienza asumiendo un modelo plano y shape-based para la trayectoria interplanetaria. Un algoritmo de optimización heurístico y multi-objetivo combinado con un resolvedor basado en gradiente optimizan los parámetros de la espiral que definen la forma de la trayectoria, el número y la secuencia de las maniobras gravitacionales, las fechas de salida y llegada, y la velocidad de lanzamiento. En el segundo paso, las soluciones candidatas se usan como estimación inicial para resolver el problema de optimización no lineal que resulta de aplicar un método de transcripción directa. En este paso, las secuencia de y-bys es conocida y determinada por el paso anterior, la dinámica es tridimensional, y la ley de control no está prefinida. Además, se pueden aplicar restricciones operacionales relacionadas con las declinación de la asíntota de salida e imponer tiempos de reorientación fijos. Este enfoque es probado en misiones a Ceres, a Júpiter y a Plutón. Se obtienen soluciones óptimas de Pareto en función del tiempo de vuelo y la masa de combustible consumida (o la masa entregada). Los resultados obtenidos mejoran los disponibles en la literatura en términos de optimalidad, a la vez que reflejan la efectividad de la metodología a propuesta para generar estimaciones rápidas. El segundo problema considera la optimización simultanea de transferencias entre órbitas terrestres que usan propulsión eléctrica, química o una combinación de ambas. En el primer paso del método de solución, la ley de control del motor eléctrico se parametriza por una función de Lyapunov. Un algoritmo de optimización heurístico y multi-objetivo selecciona el sistema propulsivo óptimo, el tipo de transferencia, la ley de control del motor de bajo empuje, así como el número, la orientación y la magnitud de los impulsos químicos. Se incluyen los efectos de la sombra y de la no esfericidad de la Tierra, además de la radiación de Van-Allen. En el segundo paso, las soluciones candidatas se usan como estimación inicial para resolver el problema de optimización no lineal que resulta de aplicar un método de transcripción directa. El método de solución propuesto se aplica a dos transferencias a GEO diferentes. Se obtienen soluciones óptimas de Pareto con respecto a la masa de combustible, el tiempo de vuelo y la degradación de las células solares. Se identifican que la aplicación de las restricciones operacionales penaliza mínimamente la función objetivo. Además, los análisis presentados destacan los beneficios que la propulsión química y eléctrica combinada proporcionarían a los satélites en GEO.Programa de Doctorado en Mecánica de Fluidos por la Universidad Carlos III de Madrid; la Universidad de Jaén; la Universidad de Zaragoza; la Universidad Nacional de Educación a Distancia; la Universidad Politécnica de Madrid y la Universidad Rovira i Virgili.Presidente: Rafael Vázquez Valenzuela.- Secretario: Claudio Bombardelli.- Vocal: Bruce A. Conwa

Spacecraft Trajectory Optimization: A review of Models, Objectives, Approaches and Solutions

This article is a survey paper on solving spacecraft trajectory optimization problems. The solving process is decomposed into four key steps of mathematical modeling of the problem, defining the objective functions, development of an approach and obtaining the solution of the problem. Several subcategories for each step have been identified and described. Subsequently, important classifications and their characteristics have been discussed for solving the problems. Finally, a discussion on how to choose an element of each step for a given problem is provided.La Caixa, TIN2016-78365-

Global optimisation of multiple gravity assist trajectories

Multiple gravity assist (MGA) trajectories represent a particular class of space trajectories in which a spacecraft exploits the encounter with one or more celestial bodies to change its velocity vector; they have been essential to reach high Delta-v targets with low propellant consumption. The search for optimal transfer trajectories can be formulated as a mixed combinatorial-continuous global optimisation problem; however, it is known that the problem is difficult to solve, especially if deep space manoeuvres (DSM) are considered. This thesis addresses the automatic design of MGA trajectories through global search techniques, in answer to the requirements of having a large number of mission options in a short time, during the preliminary design phase. Two different approaches are presented. The first is a two-level approach: a number of feasible planetary sequences are initially generated; then, for each one, families of the MGA trajectories are built incrementally. The whole transfer is decomposed into sub-problems of smaller dimension and complexity, and the trajectory is progressively composed by solving one problem after the other. At each incremental step, a stochastic search identifies sets of feasible solutions: this region is preserved, while the rest of the search space is pruned out. The process iterates by adding one planet-to-planet leg at a time and pruning the unfeasible portion of the solution space. Therefore, when another leg is added to the trajectory, only the feasible set for the previous leg is considered and the search space is reduced. It is shown, through comparative tests, how the proposed incremental search performs an effective pruning of the search space, providing families of optimal solutions with a lower computational cost than a non-incremental approach. Known deterministic and stochastic methods are used for the comparison. The algorithm is applied to real MGA case studies, including the ESA missions BepiColombo and Laplace. The second approach performs an integrated search for the planetary sequence and the associated trajectories. The complete design of an MGA trajectory is formulated as an autonomous planning and scheduling problem. The resulting scheduled plan provides the planetary sequence for a MGA trajectory and a good estimation of the optimality of the associated trajectories. For each departure date, a full tree of possible transfers from departure to destination is generated. An algorithm inspired by Ant Colony Optimization (ACO) is devised to explore the space of possible plans. The ants explore the tree from departure to destination, adding one node at a time, using a probability function to select one of the feasible directions. Unlike standard ACO, a taboo-based heuristics prevents ants from re-exploring the same solutions. This approach is applied to the design of optimal transfers to Saturn (inspired by Cassini) and to Mercury, and it demonstrated to be very competitive against known traditional stochastic population-based techniques

Astrodynamics-optimization theory and guidance theory Research achievements review series no. 15-16

Review of research achievements in astrodynamics, optimization theory, and guidance theor

Computational optimal control of the terminal bunt manoeuvre

This work focuses on a study of missile guidance in the form of trajectory shaping of a generic cruise missile attacking a fixed target which must be struck from above. The problem is reinterpreted using optimal control theory resulting in two formulations: I) minimum time-integrated altitude and 2) minimum flight time. Each formulation entails nonlinear, two-dimensional missile flight dynamics, boundary conditions and path constraints. Since the thus obtained optimal control problems do not admit analytical solutions, a recourse to computational optimal control is made. The focus here is on informed use of the tools of computational optimal control, rather than their development. Each of the formulations is solved using a three-stage approach. In stage I, the problem is discretised, effectively transforming it into a nonlinear programming problem, and hence suitable for approximate solution with the FORTRAN packages DIRCOL and NUDOCCCS. The results of this direct approach are used to discern the structure of the optimal solution, i.e. type of constraints active, time of their activation, switching and jump points. This qualitative analysis, employing the results of stage I and optimal control theory, constitutes stage 2. Finally, in stage 3, the insight of stage 2 are made precise by rigorous mathemati cal formulation of the relevant two-point boundary value problems (TPBVPs), using the appropriate theorems of optimal control theory. The TPBVPs obtained from this indirect approach are then solved using the FORTRAN package BNDSCO and the results compared with the appropriate solutions of stage I. For each formulation (minimum altitude and minimum time) the influence of boundary conditions on the structure of the optimal solution and the performance index is investigated. The results are then interpreted from the operational and computational perspectives. Software implementation employing DIRCOL, NUDOCCCS and BNDSCO, which produced the results, is described and documented. Finally, some conclusions are drawn and recommendations made