A New Mixed Iterative Algorithm to Solve the Fuel-Optimal Linear Impulsive Rendezvous Problem

International audienceThe optimal fuel impulsive time-fixed rendezvous problem is reviewed. In a linear setting, it may be reformulated as a non convex polynomial optimization problem for a pre-specified fixed number of velocity increments. Relying on variational results previously published in the literature, an improved mixed iterative algorithm is defined to address the issue of optimization over the number of impulses. Revisiting the primer vector theory, it combines variational tests with sophisticated numerical tools from algebraic geometry to solve polynomial necessary and sufficient conditions of optimality. Numerical examples under circular and elliptic assumptions show that this algorithm is efficient and can be integrated into a rendezvous planning tool

Model Predictive Control for Spacecraft Rendezvous in Elliptical Orbits with On/Off Thrusters

IFAC Workshop on Advanced Control and Navigation for Autonomous Aerospace Vehicles. 08/06/2015. SevillaIn previous works, the authors have developed a trajectory planning algorithm for spacecraft rendezvous which computed optimal Pulse-Width Modulated (PWM) control signals, for circular and eccentric Keplerian orbits. The algorithm is initialized by solving the impulsive problem first and then, using explicit linearization and linear programming, the solution is refined until a (possibly local) optimal value is reached. However, trajectory planning cannot take into account orbital perturbations, disturbances or model errors. To overcome these issues, in this paper we develop a Model Predictive Control (MPC) algorithm based on the open-loop PWM planner and test it for elliptical target orbits with arbitrary eccentricity (using the linear time-varying Tschauner-Hempel model). The MPC is initialized by first solving the open-loop problem with the PWM trajectory planning algorithm. After that, at each time step, our MPC saves time recomputing the trajectory by applying the iterative linearization scheme of the trajectory planning algorithm to the solution obtained in the previous time step. The efficacy of the method is shown in a simulation study where it is compared to MPC computed used an impulsive-only approach

Trajectory Planning for Spacecraft Rendezvous in Elliptical Orbits with On / Off Thrusters

The 19th World Congress of the International Federation of Automatic Control 2014 Cape Town, SudáfricaIn a previous work, the authors developed a trajectory planning algorithm for spacecraft rendezvous which computed optimal Pulse-Width Modulated (PWM) control signals, assuming that the target was moving in a circular Keplerian orbit. In this paper we extend the algorithm to the case of an elliptical target orbit with arbitrary eccentricity. Since the orbit is elliptical, the linear time-varying Tschauner-Hempel model is used, whose exact solution is possible by using true (or eccentric) anomaly instead of time (which is directly related to both via Kepler's equation). Unlike in the circular case, computing the PWM solution itself requires numerical integration. However, explicit linearization around the computed solution turns out to be possible and is exploited for rapidly improving the solution using linear programming (LP) techniques. The algorithm is initialized by solving the impulsive problem first; the impulses are converted to PWM signals, which are used as an initial guess. Using the explicit linearization and LP, the solution is refined until a (possibly local) optimal value is reached. The efficacy of the method is shown in a simulation study where it is compared to the impulsive-only approach

Linearized Impulsive Fixed-Time Fuel-Optimal Space rendezvous: A New Numerical Approach

International audienceThis paper focuses on the fixed-time minimum-fuel rendezvous between close elliptic orbits of an active spacecraft with a passive target spacecraft, assuming a linear impulsive setting and a Keplerian relative motion. Following earlier works developed in the 1960s, the original optimal control problem is transformed into a semi-infinite convex optimization problem using a relaxation scheme and duality theory in normed linear spaces. A new numerical convergent algorithm based on discretization methods is designed to solve this problem. Its solution is then used in a general simple procedure dedicated to the computation of the optimal velocity increments and optimal impulses locations. It is also shown that the semi-infinite convex programming has an analytical solution for the out-of-plane rendezvous problem. Different realistic numerical examples illustrate these results

Field programmable gate array based predictive control system for spacecraft rendezvous in elliptical orbits

A field programmable gate array (FPGA)-based model predictive controller (MPC) for two phases of spacecraft rendezvous is presented. Linear time varying prediction models are used to accommodate elliptical orbits, and a variable prediction horizon is used to facilitate finite time completion of the longer-range man{\oe}uvres, whilst a fixed and receding prediction horizon is used for fine-grained tracking at close range. The resulting constrained optimisation problems are solved using a primal dual interior point algorithm. The majority of the computational demand is in solving a system of simultaneous linear equations at each iteration of this algorithm. To accelerate these operations, a custom circuit is implemented, using a combination of Mathworks HDL Coder and Xilinx System Generator for DSP, and used as a peripheral to a MicroBlaze soft core processor on the FPGA, on which the remainder of the system is implemented. Certain logic that can be hard-coded for fixed sized problems is implemented to be configurable online, in order to accommodate the varying problem sizes associated with the variable prediction horizon. The system is demonstrated in closed loop by linking the FPGA with a simulation of the spacecraft dynamics running in Simulink on a PC, using Ethernet. Timing comparisons indicate that the custom implementation is substantially faster than pure embedded software-based interior point methods running on the same MicroBlaze, and could be competitive with a pure custom hardware implementation.This work was supported by the Engineering and Physical Sciences Research Council Grant Number [EP/G030308/1] as well as industrial support from Xilinx, Mathworks, and the European Space Agency

Predictive control for spacecraft rendezvous in an elliptical orbit using an FPGA

A field programmable gate array (FPGA)-based predictive controller for a spacecraft rendezvous man{\oe}uvre is presented. A linear time varying prediction model is used to accommodate elliptical orbits, and a variable prediction horizon is used to facilitate finite time completion of man{\oe}uvres. The resulting constrained optimisation problems are solved using a primal dual interior point algorithm. The majority of the computational demand is in solving a set of linear equations at each iteration of this algorithm. To accelerate this operation, a custom circuit is implemented, using a combination of Mathworks HDL Coder and Xilinx System Generator for DSP, and used as a peripheral to a MicroBlaze soft core processor. The system is demonstrated in closed loop by linking the FPGA with a simulation of the plant dynamics running in Simulink on a PC, using Ethernet.This work was supported by the Engineering and Physical Sciences Research Council (Grant EP/G030308/1) as well as industrial support from Xilinx, Mathworks and the European Space Agency.European Control Conference 2013 (ECC13), July 17-19, Zurich, Switzerlan

Efficient meta-heuristics for spacecraft trajectory optimization

Meta-heuristics has a long tradition in computer science. During the past few years, different types of meta-heuristics, specially evolutionary algorithms got noticeable attention in dealing with real-world optimization problems. Recent advances in this field along with rapid development of high processing computers, make it possible to tackle various engineering optimization problems with relative ease, omitting the barrier of unknown global optimal solutions due to the complexity of the problems. Following this rapid advancements, scientific communities shifted their attention towards the development of novel algorithms and techniques to satisfy their need in optimization. Among different research areas, astrodynamics and space engineering witnessed many trends in evolutionary algorithms for various types of problems. By having a look at the amount of publications regarding the development of meta-heuristics in aerospace sciences, it can be seen that a high amount of efforts are dedicated to develop novel stochastic techniques and more specifically, innovative evolutionary algorithms on a variety of subjects. In the past decade, one of the challenging problems in space engineering, which is tackled mainly by novel evolutionary algorithms by the researchers in the aerospace community is spacecraft trajectory optimization. Spacecraft trajectory optimization problem can be simply described as the discovery of a space trajectory for satellites and space vehicles that satisfies some criteria. While a space vehicle travels in space to reach a destination, either around the Earth or any other celestial body, it is crucial to maintain or change its flight path precisely to reach the desired final destination. Such travels between space orbits, called orbital maneuvers, need to be accomplished, while minimizing some objectives such as fuel consumption or the transfer time. In the engineering point of view, spacecraft trajectory optimization can be described as a black-box optimization problem, which can be constrained or unconstrained, depending on the formulation of the problem. In order to clarify the main motivation of the research in this thesis, first, it is necessary to discuss the status of the current trends in the development of evolutionary algorithms and tackling spacecraft trajectory optimization problems. Over the past decade, numerous research are dedicated to these subjects, mainly from two groups of scientific communities. The first group is the space engineering community. Having an overall look into the publications confirms that the focus in the developed methods in this group is mainly regarding the mathematical modeling and numerical approaches in dealing with spacecraft trajectory optimization problems. The majority of the strategies interact with mixed concepts of semi-analytical methods, discretization, interpolation and approximation techniques. When it comes to optimization, usually traditional algorithms are utilized and less attention is paid to the algorithm development. In some cases, researchers tried to tune the algorithms and make them more efficient. However, their efforts are mainly based on try-and-error and repetitions rather than analyzing the landscape of the optimization problem. The second group is the computer science community. Unlike the first group, the majority of the efforts in the research from this group has been dedicated to algorithm development, rather than developing novel techniques and approaches in trajectory optimization such as interpolation and approximation techniques. Research in this group generally ends in very efficient and robust optimization algorithms with high performance. However, they failed to put their algorithms in challenge with complex real-world optimization problems, with novel ideas as their model and approach. Instead, usually the standard optimization benchmark problems are selected to verify the algorithm performance. In particular, when it comes to solve a spacecraft trajectory optimization problem, this group mainly treats the problem as a black-box with not much concentration on the mathematical model or the approximation techniques. Taking into account the two aforementioned research perspectives, it can be seen that there is a missing link between these two schemes in dealing with spacecraft trajectory optimization problems. On one hand, we can see noticeable advances in mathematical models and approximation techniques on this subject, but with no efforts on the optimization algorithms. On the other hand, we have newly developed evolutionary algorithms for black-box optimization problems, which do not take advantage of novel approaches to increase the efficiency of the optimization process. In other words, there seems to be a missing connection between the characteristics of the problem in spacecraft trajectory optimization, which controls the shape of the solution domain, and the algorithm components, which controls the efficiency of the optimization process. This missing connection motivated us in developing efficient meta-heuristics for solving spacecraft trajectory optimization problems. By having the knowledge about the type of space mission, the features of the orbital maneuver, the mathematical modeling of the system dynamics, and the features of the employed approximation techniques, it is possible to adapt the performance of the algorithms. Knowing these features of the spacecraft trajectory optimization problem, the shape of the solution domain can be realized. In other words, it is possible to see how sensitive the problem is relative to each of its feature. This information can be used to develop efficient optimization algorithms with adaptive mechanisms, which take advantage of the features of the problem to conduct the optimization process toward better solutions. Such flexible adaptiveness, makes the algorithm robust to any changes of the space mission features. Therefore, within the perspective of space system design, the developed algorithms will be useful tools for obtaining optimal or near-optimal transfer trajectories within the conceptual and preliminary design of a spacecraft for a space mission. Having this motivation, the main goal in this research was the development of efficient meta-heuristics for spacecraft trajectory optimization. Regarding the type of the problem, we focused on space rendezvous problems, which covers the majority of orbital maneuvers, including long-range and short-range space rendezvous. Also, regarding the meta-heuristics, we concentrated mainly on evolutionary algorithms based on probabilistic modeling and hybridization. Following the research, two algorithms have been developed. First, a hybrid self adaptive evolutionary algorithm has been developed for multi-impulse long-range space rendezvous problems. The algorithm is a hybrid method, combined with auto-tuning techniques and an individual refinement procedure based on probabilistic distribution. Then, for the short-range space rendezvous trajectory optimization problems, an estimation of distribution algorithm with feasibility conserving mechanisms for constrained continuous optimization is developed. The proposed mechanisms implement seeding, learning and mapping methods within the optimization process. They include mixtures of probabilistic models, outlier detection algorithms and some heuristic techniques within the mapping process. Parallel to the development of algorithms, a simulation software is also developed as a complementary application. This tool is designed for visualization of the obtained results from the experiments in this research. It has been used mainly to obtain high-quality illustrations while simulating the trajectory of the spacecraft within the orbital maneuvers.La Caixa TIN2016-78365R PID2019-1064536A-I00 Basque Government consolidated groups 2019-2021 IT1244-1

Efficient meta-heuristics for spacecraft trajectory optimization

190 p.Uno de los problemas más difíciles de la ingeniería espacial es la optimización de la trayectoria de las naves espaciales. Dicha optimización puede formularse como un problema de optimización que dependiendo del tipo de trayectoria, puede contener además restricciones de diversa índole. El principal objetivo de esta tesis fue el desarrollo de algoritmos metaheurísticos eficientes para la optimización de la trayectoria de las naves espaciales. Concretamente, nos hemos centrado en plantear soluciones a maniobras de naves espaciales que contemplan cambios de orbitas de largo y coto alcance. En lo que respecta a la investigación llevada a cabo, inicialmente se ha realizado una revisión de estado del arte sobre optimización de cambios de orbitas de naves espaciales. Según el estudio realizado, la optimización de trayectorias para el cambio de orbitas cuenta con cuatro claves, que incluyen la modelización matemática del problema, la definición de las funciones objetivo, el diseño del enfoque a utilizar y la obtención de la solución del problema. Una vez realizada la revisión del estado del arte, se han desarrollado dos algoritmos metaheurísticos. En primer lugar, se ha desarrollado un algoritmo evolutivo híbrido auto-adaptativo para problemas de cambio de orbitas de largo alcance y multi-impulso. El algoritmo es un método híbrido, combinado con técnicas de autoajuste y un procedimiento derefinamiento individual basado en el uso de distribuciones de probabilidad. Posteriormente, en lo que respecta a los problemas de optimización de trayectoria de los encuentros espaciales de corto alcance, se desarrolla un algoritmo de estimación de distribuciones con mecanismos de conservación de viabilidad. Los mecanismos propuestos aplican métodos innovadores de inicialización, aprendizaje y mapeo dentro del proceso de optimización. Incluyen mixturas de modelos probabilísticos, algoritmos de detección de soluciones atípicas y algunas técnicas heurísticas dentro del proceso de mapeo. Paralelamente al desarrollo de los algoritmos, se ha desarrollado un software de simulación para la visualización de los resultados obtenidos en el cambio de orbitas de las naves espaciales