Modeling and Optimization in Space Engineering

This paper extends and completes a previous investigation performed by the same authors, dedicated to the problem of minimizing the control propellant in the Next Generation Gravity Mission (NGGM). The NGGM aims to measure the temporal variations of the Earth’s gravity field by laser tracking between pairs of satellites in low Earth orbit. In the reference scenario addressed in this paper, the satellites shall operate for at least 7 years at low ≈350 km circular altitude, and active disturbance control (drag control) is required; specifically, all six components of the force/torque acting on the satellite are actively controlled by a set of electric thrusters. The previous work solved the optimization problem for a nominal (no redundancy considered) set of 8 mini-thrusters (FCT, 1-mN class) plus 1 larger thruster (DCT, 10-mN class). In this work, a more realistic mission scenario is addressed taking into account real-world requirements and constraints including functional thruster redundancy to ensure single-failure tolerance and realistic thrust levels and thrust ranges compatible with existing ion thruster types. The results show feasibility with complete single-failure tolerance using 1 + 1 DCT and 8 + 2 FCTs.

Quasi-periodic invariant tori are emerging as a powerful tool to populate the phase space and enable a better understanding of key astrodynamics problems. This chapter introduces modern numerical continuation techniques for generating two-dimensional invariant tori in the elliptical restricted three-body problem of the Earth–Moon system. As a test case and baseline trajectories for the upcoming Lunar Orbital Platform Gateway, the (2:9) and (1:4) synodic resonant near rectilinear halo orbits are hereby considered. Their dynamical substitutes are first calculated and later analyzed using two-dimensional torus maps that facilitate the creation of accurate initial guesses for real-ephemeris orbits. The transition process is demonstrated with a trajectory optimization procedure that successfully generates continuous ballistic arcs around the Moon for more than one year.

Objects travelling at hypersonic speeds typically experience significant mechanical loads, particularly during acceleration/deceleration. Excluding both technical and economic limitations, sub-orbital point-to-point travel is inevitably restricted to a group of individuals that are trained and whose health is certified prior to travel. This work seeks to explore the possibility of identifying, for a chosen route and reference vehicle, a set of parameters such that an individual could participate in hypersonic travel without health screenings or prior training.

An open-loop guidance system is used with idealised navigation and control systems. The guidance method is based on node control with the assumption of instant implementation of commanded states. After an initial design space exploration is performed with various evolutionary algorithms, the Multi-objective Evolutionary Algorithm based on Decomposition with differential evolution (MOEA/D) (DE) is selected for further use, along with a preferred set of objective functions and a decision vector length. The subsequent optimisation strategy is separated into a coupled and decoupled phase approach, where the coupled approach combines the vehicle’s ascent and descent optimisations, while the decoupled approach performs a descent phase optimisation and attempts to link an ascent phase to the optimised descent phase. Decoupling, as performed, did not allow for the identification of a linkable trajectory. An optimal trajectory was identified with the coupled approach that required a significant amount of additional propellant and dry mass, yet maximum g₀-loads approached the constraint of an increase of 1 g₀. Recommendations are given to further the study.

This study is focused on the development of mathematical models suitable for a broad study of radioisotope sails performance, in particular for materials characterized by alpha decays, both single and multiple. The models have been obtained in the framework of the Special Theory of Relativity, in order to provide mathematical tools suitable for future possible relativistic missions. The mathematical models of different relativistic spaceflight profiles were also studied and their compatibility with the radioisotope sails behavior was verified, identifying the most suitable one to represent the propulsive system under study: the profile having negative exponential proper acceleration. Finally, the theoretical results of the radioisotope sail’s possible use as a propulsion system for a mission to Pluto were calculated. The sail characteristics are such as to make this system potentially interesting, in comparison with the classic propulsion systems, as the hypothetical flight times toward Pluto are significantly lower. However, the performances are not so high as to justify, at least in this first study, their use for interstellar missions.

In the last decade, the new space economy has underlined the importance of the space sector for public and private actors. Advances in CubeSats technologies allowed widespread access to large and small businesses, with private companies and undersized players benefitting from the reduced development, manufacturing, and launch costs. As a result, the number of space assets orbiting the Earth had an exponential rise. Soon this momentum will affect outer and interplanetary space as well.

The current paradigm for deep-space missions relies on the involvement of the ground segment to perform several routinary tasks, including determining the spacecraft position, computing the reference trajectory, and uploading control maneuvers. These activities are known as Guidance, Navigation, and Control (GNC). To date, this approach has proven to be sustainable, but the increasing number of Spacecraft is showing the fragility of ground-based assets. In particular, the scarce number of deep-space ground facilities and the involvement of human operators throughout the entire mission make the approach hardly scalable to an enlarged crowd of deep-space probes.

The EXTREMA (Engineering Extremely Rare Events in Astrodynamics for Deep-Space Missions in Autonomy) project aims toward a paradigm shift on how deep-space GNC is performed by enabling CubeSats with autonomous GNC capabilities. The project has received a consolidator grant from the European Research Council (ERC), a prestigious acknowledgment that funds cutting-edge research in Europe.

EXTREMA is built upon three pillars: autonomous navigation, autonomous guidance and control, and autonomous ballistic capture. This chapter presents the motivations behind the project goals by underlining the current state of the art for each Pillar and how EXTREMA aims to overcome it. First, the methodology employed within EXTREMA is outlined by expounding the algorithms designed within each pillar. Then, their validation through tailor-made hardware-in-the-loop test benches is illustrated. The outcome of each Pillar is combined into an integrated HIL facility to test autonomous GNC systems: the EXTREMA Simulation Hub.

Eventually, the test cases to prove the feasibility of the EXTREMA vision are detailed, and the potential impact of the project is discussed. By freeing interplanetary CubeSats from human supervision, the project resolutions will be effortlessly transferrable to larger spacecraft with more powerful and diverse payloads, ultimately paving the way to systematic deep-space exploration and exploitation.

This work originates from a dedicated attitude control dispatch study (discussed in another chapter of this book) carried out in the context of the current NGGM (Next Generation Gravity Mission) ESA (European Space Agency) project. This control dispatch study investigates the optimal thruster layout (in terms of position and orientation) in order to minimize the propellant consumption relevant to the on-board attitude control. The optimization problem in question is based on a time discretization of the mission scenarios to consider. For the specific real-world problem to solve, however, this approach yields a huge set of time intervals (e.g., of the order of 100,000 elements). An ad hoc clustering approach has therefore been conceived with the objective of making the relevant optimization problem computationally treatable. It essentially involves selecting a limited number (e.g., ranging from some hundreds to some thousands) of instants as representative of the whole set of time intervals requested. The present chapter focuses on the specific clustering approach introduced. The selection of the most appropriate clustering criterion and cluster sizing for the specific application to consider is key. In the particular context of this work, the k-means and k-medoids methods are considered together with the Davies–Bouldin index (DBI) and further evaluation criteria.

We present second-order Extended Sufficiency Conditions applicable without Strict Legendre Conditions and without local controllability assumption also in the frame of saturated and bang–bang control. We provide a procedure to maximize the interval where the Sufficient Conditions can be applied using a Riccati Matrix Equation and compare this procedure with the classic conjugate point condition. Applications to Finite and Infinite Thrust Orbital Transfer cases are given at the end.

Attitude control of conventional launchers is relatively easy and straightforward and gives an adequate performance when applied to the nominal vehicle and mission. However, in the presence of environmental disturbances and vehicle design uncertainties, more robust types of controllers are required to guarantee stable attitudes. This chapter discusses the application of Simple Adaptive Control for the pitch control of a conventional flexible launcher. Because of the large number of design parameters, an optimisation procedure based on an evolutionary algorithm has been applied. With a floating-point representation for the design parameters, stochastic universal sampling selection, arithmetic crossover and non-uniform mutation, the performance of the controller is analysed, and it is identified how the developed methodology can streamline the (conceptual) design phase. Application of Pareto ranking enabled the simultaneous minimisation of the state deviation and the control effort, while the oscillation of the control has been used as an optimisation criterion. A conclusive simulation shows the controller performance for the flexible launch system.

The technological improvements in small electromechanical systems and wireless communication technologies have provided an opportunity to use wireless sensor networks (WSNs) in space applications and to explore the Moon, planets, natural moons of the planets, or asteroids. Humankind has sent humans, vehicles, satellites, and rovers to space to discover the Moon, planets, and even asteroids. WSNs can be an excellent choice for collecting and processing vital physical data including temperature, seismic, visual, infrared, light, pressure, radiation, and gas data, among others about target space objects. Therefore, this chapter analyzes applications and studies of WSN use to explore the Moon, planets, and space, and it summarizes the literature. The optimization and solution methodologies in studies are presented, and opportunities and challenges of using WSNs in space-based missions are reviewed. In brief, the environments on different planets or on other space bodies can be highly compelling for such instruments. Therefore, there are still many subjects to be studied by researchers and scientists, and as such, the capabilities and communication skills of sensors must be adapted to space applications.

In most mission scenarios, precise orbit injection represents a crucial requirement, and affects the subsequent phases of spaceflight. This research proposes a new guidance, control, and actuation architecture for upper stage orbit injection maneuvers. A novel, explicit near-optimal guidance algorithm is developed that is based on the local projection of the position and velocity variables, in conjunction with the real-time solution of the associated minimum-time problem. A new, nonlinear reduced-attitude control algorithm is introduced, which enjoys quasi-global stability properties, and is capable of driving the actual longitudinal axis toward the commanded thrust direction. Actuation is based on the joint use of modulated side jets – for the roll control action – and thrust vectoring. The overall dynamics of the upper stage, regarded as a system of two connected bodies, is modeled using Kane’s method. An upper stage with realistic propulsion parameters is selected for numerical testing. Monte Carlo simulations prove that the guidance, control, and actuation architecture at hand is effective for precise orbit injection.

A satellite system conceptual design problem is addressed in this work. A multi-objective parametric optimization problem is formulated and efficiently solved. The objectives considered are usually opposed among them, such as performance, mass, budget, and volume. By solving the optimization problem, a minimum set of different satellite configurations is obtained. Therefore, the decision-maker can select the best one, knowing that each one fulfills the requirements suite. The strategy developed in this work is based on the direct numerical simulation (DNS) of the optimization problem. The optimal Pareto front is obtained in a numerical setting. This new tool can optimize the complete system as a whole. Usually, in the standard engineering procedure, each of the interdisciplinary groups performs the optimization of their subsystem. After that, the optimized system is obtained by overlapping all of these individually optimized parts. Clearly, this standard procedure only can create a sub-optimal design. With the approach presented here the global optimal solution is guaranteed. The strategy is applied to a low orbit satellite model and a comparison with a genetic algorithm-based multi-objective optimization procedure is also presented.

In the life of a spacecraft the LEOP (Launch and Early Operational Phase) is one of the most critical, because the knowledge of the behavior of the S/C platform is still uncertain. This is, in particular, true for the propulsive maneuvers performed during the first phases of a mission, when the performance of the spacecraft and thrusters is still unknown. This chapter focuses on the determination of robust control laws to take into account dramatic thruster underperformance. The insertion of a spacecraft in a highly elliptic orbit with multiple apogee burns is considered as an example and the effect of non-nominal thrust in the first apogee maneuver is analyzed. An indirect optimization method is proposed in order to find a robust control law that guarantees “optimal” performance even in such an event.

Enhancements in evolutionary optimization techniques are rapidly growing in many aspects of engineering, specifically in astrodynamics and space trajectory optimization and design. In this chapter, the problem of optimal design of space trajectories is tackled via an enhanced optimization algorithm within the framework of Estimation of Distribution Algorithms (EDAs), incorporated with Lyapunov and Q-law feedback control methods. First, both a simple Lyapunov function and a Q-law are formulated in Classical Orbital Elements (COEs) to provide a closed-loop low-thrust trajectory profile. The weighting coefficients of these controllers are approximated with various degrees of Hermite interpolation splines. Following this model, the unknown time series of weighting coefficients are converted to unknown interpolation points. Considering the interpolation points as the decision variables, a black-box optimization problem is formed with transfer time and fuel mass as the objective functions. An enhanced EDA is proposed and utilized to find the optimal variation of weighting coefficients for minimum-time and minimum-fuel transfer trajectories. The proposed approach is applied in some trajectory optimization problems of Earth-orbiting satellites. Results show the efficiency and the effectiveness of the proposed approach in finding optimal transfer trajectories. A comparison between the Q-law and a simple Lyapunov controller is done to show the potential of the EEDA in enabling the simple Lyapunov controller to recover the finer nuances explicitly given within the analytical expressions in the Q-law.