Ossama Abdelkhalik

Hidden Genes Genetic Algorithm for Multi-Gravity-Assist Trajectories Optimization

The problem of optimal design of a multi-gravity-assist space trajectory, with a free number of deep space maneuvers, poses a multimodal cost function. In the general form of the problem, the number of design variables is solution dependent. This paper presents a genetic-based method developed to handle global optimization problems where the number of design variables vary from one solution to another. A fixed length for the design variables is assigned for all solutions. Independent variables of each solution are divided into effective and ineffective segments. Ineffectivesegments(hiddengenes)areexcludedincostfunctionevaluations.Full-lengthsolutionsundergostandard geneticoperations.Thisnewmethodisappliedtoseveralinterplanetarytrajectorydesignproblems.Thismethodhas the capability to determine the number of swing-bys, the planets to swing by, launch and arrival dates, and the number of deep space maneuvers, as well as their locations, magnitudes, and directions, in an optimal sense. The results presented in this paper show that solutions obtained using this tool match known solutions for complex case studies. Nomenclature A = continuous design variable B = discrete design variable C = transformation matrix F = fitness (cost function) f = flight direction h = pericenter altitude, km

N-Impulse Orbit Transfer Using Genetic Algorithms

T HE orbit transfer problems using impulsive thrusters have attracted researchers for a long time [1]. One of the objectives in these problems is to find the optimal fuel orbit transfer between two orbits, generally inclined eccentric orbits. The optimal two-impulse orbit transfer problem poses multiple local optima, and classical optimization methods find only local optimum solution. McCue [2] solved the problem of optimal two-impulse orbit transfer using a combination between numerical search and steepest descent optimization procedures. Jezewaski and Rozendall [3] developed an iterative method to calculate local minima solutions for the nimpulse fixed time rendezvous problems. Genetic algorithms (GAs) have been used in the literature to search for the global optimal orbit maneuver. Reichert [4] addressed the optimum two-impulse orbit transfer problem for coplanar orbits only. The accuracy obtained using this formulation is not good unless a narrow range, around the optimal value, for each design variable is known in advance [4]. Given narrow ranges for the design variables, the solution obtained using this formulation does not guarantee that the satellite will be inserted exactly into final orbit, but rather there is a small error unless the GA finds exactly the global optimal solution. Kim and Spencer [5] introduced a different formulation to the two-impulse orbit transfer problem by using six design variables for coplanar orbits. This formulation also does not guarantee the satellite is placed exactly in the final orbit. In this note, a new formulation to the problem is introduced. This formulation is general for noncoplanar elliptical orbits. It can also implement any number of thrust impulses. For the case of twoimpulse maneuver, this formulation requires only three design variables for any noncoplanar orbit transfer. The solution obtained by this formulation is guaranteed to insert the satellite in the final orbit exactly, even if the GAs did not converge to the global optimal solution. This formulation requires solving Lambert’s problem to find the parameters of the transfer orbit for a given set of the three design variables. The next section describes the orbit maneuver algorithm. The two-impulse transfer is considered a special case and is presented separately. Validation to this formulation is performed by solving several case studies to which the optimal solution is known.

Dynamic-Size Multiple Populations Genetic Algorithm for Multigravity-Assist Trajectory Optimization

The problem of the optimal design of a multigravity-assist space trajectory, with a free number of deep space maneuvers, in its general form poses a multimodal objective function in which design space size is variable. This paperpresents a genetic-basedmethoddeveloped to handle global, variable-size, design space optimization problems where the number of design variables varies from one solution to another. Subpopulations of fixed-size design spaces are randomly initialized. Standard genetic operations are carried out for a stage of generations. A new population is then created by reproduction from all members in all subpopulations based on their relative fitnesses. The resulting subpopulations have different sizes from their initial sizes in general. The process repeats, leading to an increase in the size of subpopulations of more fit solutions and a decrease in the size of subpopulations of less fit solutions. This method has the capability to determine the number of swing-bys, the planets to swing by, launch and arrival dates, and the number of deep space maneuvers as well as their locations, magnitudes, and directions in an optimal sense. This newmethod is applied to several interplanetary trajectory design problems. The results presented in this paper show that solutions obtained using this tool match known solutions for complex case studies.

Shape-Based Approximation of Constrained Low-Thrust Space Trajectories Using Fourier Series

Space mission trajectory design using low-thrust capabilities is becoming increasingly popular. However, trajectory optimization is a very challenging and time-consuming task. In this paper, we build upon existing shapebased techniques to present an alternative Fourier series approximation for rapid low-thrust-rendezvous/orbitraising trajectory construction with thrust-acceleration constraint-handling capability. The new flexible representation, along with the constraint-handling capability, makes this method a suitable candidate for feasibility assessment of a whole range of trajectories within the given system propulsive budget. In addition, the solutions present opportunities for direct optimization techniques. A key point in the proposed method is its ability to solve problemswith a greater number of free parameters than in shape-basedmethods. Several case studies are presented: simpleEarth–Mars rendezvous, rendezvous/orbit raising for lowEarth orbit to geostationary orbit, and two phasing problems. Results clearly depict the advantage of the proposed method in handling thrust constraints and its applicability to a wide range of problems.

Fast Initial Trajectory Design for Low-Thrust Restricted-Three-Body Problems

This paper presents an approach to generate initial trajectories in a three-body dynamical framework, assuming the use of a low-thrust propulsion system. Finite Fourier series were previously implemented successfully in approximating two-dimensional continuous-thrust trajectories in two-body dynamic models. In this paper, Finite Fourier series are implemented in a dual-level-solver strategy for low-thrust trajectory approximation, in the presence of thrust acceleration constraints, in the three-body dynamic model. This approximation enables the feasibility assessment of low-thrust trajectories, especially in the presence of thrust level constraint. The developed method demonstrates capability in generating trajectories using a fully automated procedure for various levels of thrust acceleration with a moderate to high number of revolutions. The suitability of using the approximated trajectories as initial guesses for high-fidelity solvers is demonstrated.

Hidden Genes Genetic Optimization for Variable-Size Design Space Problems

This paper introduces the biologically inspired concept of hidden genes genetic algorithms; they search for optimal solutions to global optimization problems of multimodal objective functions with a variable number of design variables. A fixed chromosome length is assumed for all solutions in the population. Each chromosome is divided into effective and ineffective segments. The effective segment includes the design variables for that solution. The ineffective segment includes only hidden genes. Hidden genes are excluded in objective function evaluations. The effect of the hidden genes on the convergence of the genetic algorithm is studied. Two test cases are presented.

Initial Orbit Design from Ground Track Points

M OTIVATED by the need for optimal orbit design algorithms for remote sensing spacemissions, the problemof orbit design given three ground sites on the ground track is addressed. Given three ground sites of interest, characterized by their right ascensions and latitudes, the objective is to calculate an orbit such that its ground track passes through the three sites within a given time frame. The problem has been formulated, and two solution algorithms have been developed. The first algorithm finds the exact orbit for which the ground track passes through the given ground sites; the second algorithm calculates an approximate solution. The approximate solution can be used as an initial guess in the first algorithm for an effective search for the exact solution. This note describes the case of three sites. In previous developments [1], a heuristic approach was proposed to search for a natural orbit that enables the spacecraft to visit the n number of sites naturally, without the use of propulsion. This solution, if obtained, is a passive solution, because no propulsion is needed. The method worked for a small number of sites of interest. The success of this method depends on the locations of the sites and the available time frame [2]. In the case of too many sites of interest, the set of sites will be split into subsets, in an optimal sense. Toward developing a splitting algorithm, this note addresses the case of only three ground sites. For three arbitrary ground sites, what are the solutions (orbits) that have ground tracks visiting the sites within a given time frame? Visiting a ground site does not necessarily require that the satellite pass directly above the ground site. However, in this development, the satellite is required to pass directly above the given ground sites. The reason is that, in the case of visiting only three ground sites, there is always at least one orbit that passes directly above the three sites; hence, considering only these orbits limits the number of solutions and simplifies the analysis. In future developments, cases with higher numbers of ground sites will be considered, and the sensor’s field of view will be taken into consideration to allow orbits with ground tracks not passing directly through the sites.

Spacecraft Formation Orbit Estimation Using WLPS-Based Localization

This paper studies the implementation of a novel wireless local positioning system (WLPS) for spacecraft formation flying to maintain high-performance spacecraft relative and absolute position estimation. A WLPS equipped with antenna arrays allows each spacecraft to measure the relative range and coordinate angle(s) of other spacecraft located in its coverage area. The dynamic base station and the transponder of WLPS enable spacecraft to localize each other in the formation. Because the signal travels roundtrip in WLPS, and due to the high spacecraft velocities, the signal transmission time delay reduces the localization performance. This work studies spacecraft formation positions estimation performance assuming that only WLPS is available onboard. The feasibility of estimating the spacecraft absolute position using only one-dimensional antenna array is also investigated. The effect of including GPS measurements in addition to WLPS is studied and compared to a GPS standalone system.

A novel space-based solar power collection via LEO satellite networks: Orbital management via wireless local positioning systems

A space-based solar power technology that uses networks of small Low Earth Orbit (LEO) satellites is proposed. Due to the relative motion of these satellites with respect to the earth, multiple power-collecting base stations (PCBS) are implemented on the Earth to allow effective energy collection. Compared to the traditional Solar Power Satellite (SPS), in the proposed technique, satellites fly at a lower altitude. This leads to lower path loss attenuation in transferring energy from space to the earth. In addition, they have lower launching cost, efficient power transmission to the Earth due to lower energy dispersion, and minimal environmental effects due to the low power transmission. Moreover, no “in space” assembly of large structures is required and their maintenance is cost-effective. This paper investigates the orbital management for a specific group of satellites that form a cluster and stay in formation at all times. A wireless local positioning system (WLPS) is incorporated to compute the relative distances between satellites for orbital management. The WLPS enables each satellite to measure the position of other satellites located in its coverage area. An Extended Kalman Filter (EKF) is implemented to allow high performance localization, and to maintain satellites in their proper orbit. The effect of the number of satellites in the formation on the relative positions estimation is studied.

Autonomous Planning of Multigravity-Assist Trajectories with Deep Space Maneuvers Using a Differential Evolution Approach

The biologically inspired concept of hidden genes has been recently introduced in genetic algorithms to solve optimization problems where the number of design variables is variable. In multigravity-assist trajectories, the hidden genes genetic algorithms demonstrated success in searching for the optimal number of swing-bys and the optimal number of deep space maneuvers. Previous investigations in the literature for multigravity-assist trajectory planning problems show that the standard differential evolution is more effective than the standard genetic algorithms. This paper extends the concept of hidden genes to differential evolution. The hidden genes differential evolution is implemented in optimizing multigravity-assist space trajectories. Case studies are conducted, and comparisons to the hidden genes genetic algorithms are presented in this paper.