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About this book

This book presents advanced case studies that address a range of important issues arising in space engineering. An overview of challenging operational scenarios is presented, with an in-depth exposition of related mathematical modeling, algorithmic and numerical solution aspects. The model development and optimization approaches discussed in the book can be extended also towards other application areas.

The topics discussed illustrate current research trends and challenges in space engineering as summarized by the following list:

• Next Generation Gravity Missions

• Continuous-Thrust Trajectories by Evolutionary Neurocontrol

• Nonparametric Importance Sampling for Launcher Stage Fallout

• Dynamic System Control Dispatch

• Optimal Launch Date of Interplanetary Missions

• Optimal Topological Design

• Evidence-Based Robust Optimization

• Interplanetary Trajectory Design by Machine Learning

• Real-Time Optimal Control

• Optimal Finite Thrust Orbital Transfers

• Planning and Scheduling of Multiple Satellite Missions

• Trajectory Performance Analysis

• Ascent Trajectory and Guidance Optimization

• Small Satellite Attitude Determination and Control

• Optimized Packings in Space Engineering

• Time-Optimal Transfers of All-Electric GEO Satellites

Researchers working on space engineering applications will find this work a valuable, practical source of information. Academics, graduate and post-graduate students working in aerospace, engineering, applied mathematics, operations research, and optimal control will find useful information regarding model development and solution techniques, in conjunction with real-world applications.

Table of Contents


Control Propellant Minimization for the Next Generation Gravity Mission

This chapter addresses the Next Generation Gravity Mission (NGGM), a candidate Earth observation mission of the European Space Agency (ESA), currently undergoing system and technology studies. NGGM is intended to continue the series of ESA missions measuring Earth gravity from space, successfully started with the Gravity field and Ocean Circulation Explorer (GOCE) satellite which flew between 2009 and 2013. Whereas GOCE measured static gravity by a three-axis gradiometer, NGGM will monitor the temporal variations of the gravity field due to mass (primarily water) transport in the Earth system with a concept pioneered by GRACE (Gravity Recovery and Climate Experiment), with improved sensitivity, thanks to laser tracking between satellite pairs. As a monitoring mission, NGGM shall be of a long duration, 11 years according to the current scientific requirements. In addition, the laser interferometer and accelerometer payloads impose demanding requirements such as suppression of the air drag disturbances, precise pointing, and angular rate control. The long lifetime and the control requirements can only be met by using electric thrusters with high specific impulse, hence low mass consumption. Nevertheless, propellant mass minimization remains a dominant task of the mission design. This objective requires proper selection of the thruster operating ranges, as well as an optimized thruster layout and thrust dispatching algorithms. The method applied to solve the thrust dispatching problem is the subject of another chapter in this volume. The present chapter illustrates the flow-down of mission and system requirements into the proposed spacecraft implementation and operation features, focusing on the thruster layout optimization problem. The proposed design is shown to meet the mission requirements, thus validating the methodology adopted as well as the results achieved. Further research avenues opened by the current work are outlined in the conclusions.
Alberto Anselmi, Stefano Cesare, Sabrina Dionisio, Giorgio Fasano, Luca Massotti

Global Optimization of Continuous-Thrust Trajectories Using Evolutionary Neurocontrol

Searching optimal continuous-thrust trajectories is usually a difficult and time-consuming task. The solution quality of traditional optimal-control methods depends strongly on an adequate initial guess because the solution is typically close to the initial guess, which may be far from the (unknown) global optimum. Evolutionary neurocontrol attacks continuous-thrust optimization problems from the perspective of artificial intelligence and machine learning, combining artificial neural networks and evolutionary algorithms. This chapter describes the method and shows some example results for single- and multi-phase continuous-thrust trajectory optimization problems to assess its performance. Evolutionary neurocontrol can explore the trajectory search space more exhaustively than a human expert can do with traditional optimal-control methods. Especially for difficult problems, it usually finds solutions that are closer to the global optimum. Another fundamental advantage is that continuous-thrust trajectories can be optimized without an initial guess and without expert supervision.
Bernd Dachwald, Andreas Ohndorf

Nonparametric Importance Sampling Techniques for Sensitivity Analysis and Reliability Assessment of a Launcher Stage Fallout

Space launcher complexity arises, on the one hand, from the coupling between several subsystems such as stages or boosters and other embedded systems, and on the other hand, from the physical phenomena endured during the flight. Optimal trajectory assessment is a key discipline since it is one of the cornerstones of the mission success. However, during the real flight, uncertainties can affect the different flight phases at different levels and be combined to lead to a failure state of the space vehicle trajectory. After their propelled phase, the different stages reach successively their separation altitudes and may fall back into the ocean. Such a dynamic phase is of major importance in terms of launcher safety since the consequence of a mistake in the prediction of the fallout zone can be dramatic in terms of human security and environmental impact. For that reason, the handling of uncertainties plays a crucial role in the comprehension and prediction of the global system behavior. Consequently, it is of major concern to take them into account during the reliability analysis. In this book chapter, two new sensitivity analysis techniques are considered to characterize the system uncertainties and optimize its reliability.
Pierre Derennes, Vincent Chabridon, Jérôme Morio, Mathieu Balesdent, Florian Simatos, Jean-Marc Bourinet, Nicolas Gayton

Dynamic System Control Dispatch: A Global Optimization Approach

This work originates from research related to an optimal control dispatch problem in space: the problem in question is presented in detail in another chapter of this volume by Anselmi et al. (Control propellant minimization for the next generation gravity mission. In: Fasano G, Pintér JD (eds) Modeling and optimization in space engineering – state of the art and new challenges. Springer, New York, 2019). Here we discuss the general issue of dispatching the control of a dynamic system through a number of actuators, presenting a novel model development and algorithmic solution approach. A control law, expressed in terms of total force and torque demand, represents the operational scenario. This gives rise to a very challenging optimization problem, concerning the actuator accommodation and utilization. Following the model formulation, a dedicated heuristic approach–involving nonlinear and mixed integer linear programming–is proposed. The numerical results presented illustrate the efficiency of the methodology adopted.
Giorgio Fasano

Choice of the Optimal Launch Date for Interplanetary Missions

Interplanetary missions are strictly dependent on the launch date. Mission planning requires a knowledge not only of the technological and cost constraints but also the study of the influence of the launch opportunity on the spacecraft performance and on the feasibility of the mission. Pork-chop plots are effective tools to design interplanetary missions, providing a graphical overview of the relationship between the fundamental parameters of the mission design, namely, the launch date, the duration and the energy requirements. In this way it is possible to evaluate the best timing to accomplish the mission under current constraints. Plots of a similar type can be drawn also for optimizing missions based on low thrust or on propellantless propulsion—like solar sails. The cost function described in these plots may be the square of the hyperbolic excess speed, the ΔV, the ratio between the initial mass and the payload mass or between the mass of the propulsion system and the payload or the cost function J used in the optimization of the trajectory. In case of two-way missions, it is possible to plot a cost function of the same type, by adding the values related to the forward and the backward journeys, as a function of the duration of the two legs of the travel, once the stay on the planet has been fixed. In this way it is possible to optimize also missions which are intrinsically two ways, like human exploration missions or sample return missions.
Giancarlo Genta, P. Federica Maffione

Optimal Topological Design of a Thermal Isolator for a Monopropellant Space Thruster

This work is focused on the study of the thermal-structural behavior of a thermal isolator device employed in a monopropellant thruster for space applications. Engines of this kind are widely used to perform attitude corrections in artificial satellites. Their operating principle is based on the catalytic decomposition of the fuel (hydrazine), producing gasification with a consequent heat generation. These gases are properly conducted to a nozzle to produce thrust. A couple of redundant solenoid on-off electro-valves, in a serial configuration, are used to control the fuel supply system. To avoid leak risk in this system, soft seals are also used. Duration and performance of this kind of engine rely on two main aspects. The first one is the number of cold ignitions. When the engine starts at low temperature conditions, the catalytic bed is subjected to a thermal transient (high gradient—hundreds of C/s) which generates a breakage of grains, causing low size particles to fill the inter-granular spaces, clogging the downstream gas flow. The second aspect to consider in the reduction of the life span is the loss of reliability in the soft seals used in the fuel supply system due to high temperature degradation. Such degradation can drive the module out of service, generate a catastrophic failure in the reactor, or lead to mission stoppage. A thermal isolator is used to protect the seals from a premature degradation due to thermal effects. Its structural design is optimized by using a novelty structural optimization methodology based on topological sensitivity analysis in this work. This analysis allows achieving the best structural configuration that minimizes the temperature field around the seals and also the isolator weight. Finally, a thermal and structural evaluation of the monopropellant thruster is presented in order to validate the structural strength and integrity. Inertial forces due to high G’s are considered in this analysis.
Sebastián Miguel Giusti, Augusto Alejandro Romero, Javier Eduardo Salomone

Evidence-Based Robust Optimization of Pulsed Laser Orbital Debris Removal Under Epistemic Uncertainty

An evidence-based robust optimization method for pulsed laser orbital debris removal (LODR) is presented. Epistemic type uncertainties due to limited knowledge are considered. The objective of the design optimization is set to minimize the debris lifetime while at the same time maximizing the corresponding belief value. The Dempster–Shafer theory of evidence (DST), which merges interval-based and probabilistic uncertainty modeling, is used to model and compute the uncertainty impacts. A Kriging based surrogate is used to reduce the cost due to the expensive numerical life prediction model. Effectiveness of the proposed method is illustrated by a set of benchmark problems. Based on the method, a numerical simulation of the removal of Iridium 33 with pulsed lasers is presented, and the most robust solutions with minimum lifetime under uncertainty are identified using the proposed method.
Liqiang Hou, Massimiliano Vasile, Zhaohui Hou

Machine Learning and Evolutionary Techniques in Interplanetary Trajectory Design

After providing a brief historical overview on the synergies between artificial intelligence research, in the areas of evolutionary computations and machine learning, and the optimal design of interplanetary trajectories, we propose and study the use of deep artificial neural networks to represent, on-board, the optimal guidance profile of an interplanetary mission. The results, limited to the chosen test case of an Earth–Mars orbital transfer, extend the findings made previously for landing scenarios and quadcopter dynamics, opening a new research area in interplanetary trajectory planning.
Dario Izzo, Christopher Iliffe Sprague, Dharmesh Vijay Tailor

Real-Time Optimal Control Using TransWORHP and WORHP Zen

In many industrial applications solutions of optimal control problems are used, where the need for low computational times outweighs the need for absolute optimality. For solutions of fully discretized optimal control problems we propose two methods to approximate the solutions of problems with modified parameter values in real-time by using sensitivity derivatives.
We use TransWORHP to transcribe an optimal control problem to a sparse nonlinear programming problem, which will be solved using our NLP solver WORHP. For this nominal solution sensitivity derivatives can be computed with respect to any system parameter using WORHP Zen. On NLP level, the sensitivity derivatives allow to perform correction steps for changes in the system parameters. This concept can be transferred to discretized optimal control problems using, e.g., the sensitivity derivatives of the boundary condition or of the discretized differential equations. The quality and applicability of both methods are illustrated by a trajectory planning problem in the context of the planar restricted problem of three bodies. In both methods the sensitivity derivatives can be used to give numerical validations of the theoretically expected convergence behaviour.
Matthias Knauer, Christof Büskens

Theory and Applications of Optimal Finite Thrust Orbital Transfers

The main author proposed a mission for the first time with a LEO to GEO orbital transfer for telecommunication application (GovSatCom). This mission will allow the use of small launchers to bring, at a lower cost, satellites of medium and large size at Geostationary orbit. This motivated the authors to develop a mathematical model in order to find the optimal thrust strategy for very long orbital transfers of satellites with electric thrusters. During the transfer, the satellite is supposed capable to steer the thrust vector in any direction. To solve the optimization problem, an averaging technique has been adopted. The authors discussed and solved this problem including the J 2 and eclipse effects. Moreover some external constraints are included in the problem in order to avoid simulations with unrealistic orbital transfers (i.e., too low perigee altitude). Referring to the papers already published by the authors, this last one is a synthetic review of the theory and the applications. After a mathematical introduction of the theoretical notions, new numerical results are presented.
L. Mazzini, M. Cerreto

Collection Planning and Scheduling for Multiple Heterogeneous Satellite Missions: Survey, Optimization Problem, and Mathematical Programming Formulation

This chapter introduces novel integrated management of multiple heterogeneous satellite missions for the purpose of intelligence collection. The focus is on optimization of acquisition planning and scheduling for various missions including single satellites and satellite constellations. The relevant optimization problem and its mathematical programming formulation that allow multiple area coverage plans for each acquisition request, as well as consideration of the quality measures of coverage plan, strip, and imaging opportunity, are presented. The chapter consists of a multi-mission planning system overview, a survey of relevant literature, a definition of the integrated acquisition scheduling optimization problem and its mathematical programming formulation.
Snezana Mitrovic-Minic, Darren Thomson, Jean Berger, Jeff Secker

Single-Stage-to-Orbit Space-Plane Trajectory Performance Analysis

The development of fully reusable launch systems has been the topic of many studies since the 1960s. Over the years, several aspects of both so-called single- and two-stage-to-orbit space planes have provided many interesting research topics. Amongst others, the constrained trajectory optimisation has proven to be a challenging subject. In this chapter, an inverse-dynamics approach is combined with trajectory optimisation and analysis, by discretising a representative (vertical-plane) ascent trajectory into a number of flight segments, and by parametrising the guidance in terms of flight-path angle as a function of altitude. When the individual guidance parameters are varied, the effect on performance indices payload mass and integrated heat load can be analysed. This can subsequently lead to a refinement of the trajectory. To do so with limited effort, design-of-experiment techniques are used. It is shown that with this relatively simple simulation scheme, combined with variance analysis and response-surface methodology, the insight in the trajectory dynamics can be increased. Alternatively, this method can be used as refinement to an otherwise (local) optimum trajectory. It is stressed, though, that the application of design of experiments to the ascent-trajectory problem cannot replace numerical optimisation. Finally, the impact of using thrust-vector control as a means to (partially) trim the vehicle shows significant fuel savings and should therefore be included in the optimisation process.
Erwin Mooij

Ascent Trajectory Optimization and Neighboring Optimal Guidance of Multistage Launch Vehicles

Multistage launch vehicles are employed to place spacecraft and satellites in their operational orbits. If the rocket aerodynamics and propulsion are modeled appropriately, optimization of their ascent trajectory consists in determining the coast duration and the thrust time history that maximize the final mass at injection. This research derives all the necessary conditions for ascent path optimization of a multistage launch vehicle. With reference to an existing rocket, the indirect heuristic method is then applied, for the numerical determination of the overall ascent trajectory. An effective approach is used with the intent of satisfying the path constraint related to the maximum dynamical pressure in the atmospheric phase. Then, the recently introduced, implicit-type variable-time-domain neighboring optimal guidance is applied to the upper stage powered arc, for the purpose of obtaining the corrective control actions in the presence of nonnominal flight conditions. The guidance approach at hand, based on the second-order analytical conditions for optimality, proves to be rather effective (in terms of propellant budget), and guarantees very accurate orbit injection in spite of perturbations.
Guido Palaia, Marco Pallone, Mauro Pontani, Paolo Teofilatto

Optimization Issues in the Problem of Small Satellite Attitude Determination and Control

The problems of synthesis of attitude determination and control system of small satellites regarding the influence of external perturbations due to their small mass and restrictions in using high-precision actuators, due to the limitations of energy budget and construction, are considered. The solutions to this problem are proposed involving the development of high-accuracy algorithms for satellite attitude determination and control, using the minimum set of sensors, various types of actuators, and optimization principles.
Zaure Rakisheva, Anna Sukhenko, Nazgul Kaliyeva

Optimized Packings in Space Engineering Applications: Part I

Packing optimization problems have a wide spectrum of real-word applications, including transportation, logistics, chemical/civil/mechanical/power/aerospace engineering, shipbuilding, robotics, additive manufacturing, materials science, mineralogy, molecular geometry, nanotechnology, electronic design automation, very large system integration, pattern recognition, biology, and medicine. In space engineering, ever more challenging packing optimization problems have to be solved, requiring dedicated cutting-edge approaches.
Two chapters in this volume investigate very demanding packing issues that require advanced solutions. The present chapter provides a bird’s eye view of challenging packing problems in the space engineering framework, offering some insight on possible approaches. The specific issue of packing a given collection of arbitrary polyhedra, with continuous rotations and minimum item-to-item admissible distance, into a convex container of minimum size, is subsequently analyzed in depth, discussing an ad hoc mathematical model and a dedicated solution algorithm. Computational results show the efficiency of the approach proposed. The following (second) chapter examines a class of packing optimization problems in space with consideration to balancing conditions.
Yuriy Stoyan, Alexandr Pankratov, Tatiana Romanova, Giorgio Fasano, János D. Pintér, Yurij E. Stoian, Andrey Chugay

Optimized Packings in Space Engineering Applications: Part II

This chapter, dedicated to a specific packing optimization scenario of considerable interest in space engineering and logistics, follows a previous one appearing in this volume [1]. Although it is presented as the second part of the whole topical discussion proposed, it can be read independently.
The layout optimization, with balancing conditions, of a given set of 3D-objects, in a container partitioned by horizontal planes into subcontainers, is considered.
We define special combinatorial configurations describing the specific structure of the problem. A mathematical model, based on the combination of the phi-function technique and the introduced configurations, is provided. The model takes into account not only the placement constraints (i.e., nonoverlapping, containment) and the mechanical characteristics of the system but also the combinatorial features relevant to the partitions of the set of objects placed inside the subcontainers. The solution strategy is proposed and the results of numerical experiments are presented.
Yu. Stoyan, I. Grebennik, T. Romanova, A. Kovalenko

A Catalogue of Parametric Time-Optimal Transfers for All-Electric GEO Satellites

In this chapter a catalogue of time-optimal low-thrust transfers from arbitrary departure orbits to the geostationary orbit is constructed. This catalogue is obtained by solving a multitude of optimal control problems with a combination of simple and multiple shooting techniques, augmented by a multi-dimensional homotopy. Modified equinoctial elements are used to describe the satellite dynamics, and state-of-the-art values for thrust and specific impulse are considered. The ultimate outcome consists of a synthetic law for transfer time, and thus cost, as function of the orbit injection parameters and engine figures. This law can be consulted in the early stages of mission design.
Francesco Topputo, Simone Ceccherini
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