An adaptive, in-sity, parameters estimation technique for improved spacecraft guidance and control in uncertain environments

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POLITECNICO DI MILANO

Master’s Degree Program in Space Engineering Department of Aerospace Science and Technology

AN ADAPTIVE, IN-SITU, PARAMETERS

ESTIMATION TECHNIQUE FOR IMPROVED

SPACECRAFT GUIDANCE AND CONTROL IN

UNCERTAIN ENVIRONMENTS

DAER

Dipartimento di Scienze e Tecnologie Aerospaziali del Politecnico di Milano

Supervisor: Prof. James Douglas Biggs

Master’s Degree Thesis of: Edoardo Ciccarelli, mat. 874931

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Abstract

Advances in space technology including low-thrust propulsion, such as solar sails and electric propulsion systems, are opening up new mission possibilities. Complex missions, like those about small celestial bodies, in highly perturbed environments or mission which are required to last for long operative times, are now possible. In particular, CubeSats have enabled cheaper access to space and opened space to a broader community of users. Considering the exploitation of solar sails, their use could open new frontiers in deep space exploration, both in terms of complexity and duration. However, the use of these technologies is challenging since nano-spacecraft are highly-constrained, for example, in thrust magnitude. Therefore, using such technologies for station-keeping and control in deep space requires far more efficient methods for the development of guidance and control algorithms that are able to exploit the natural perturbations of the environment. Exploiting the natural pertur-bations of the circular restricted three-body problem is a very well-known technique. However, this procedure is based on assumed values for the parameters of the model such as the spacecraft mass distribution and inertia, the main bodies sphericity, the optical properties of the spacecraft’s surface and interactions of the spacecraft’s on-board electronics and electric propulsion system with magnetic fields. To improve on these methods, the real value of these model parameters should be updated in-situ. In this thesis an approach is developed to estimate these parameters; it is based on real-time regression techniques, coupled with an extended state observer fed with the instantaneous position or angular velocity only (depending on the application). This coupling enables to build the entire system’s state, to recover information on the environment, potentially to optimise the control strategies autonomously, with no need to store data, a limited computational effort, and the need to recover the spacecraft’s position or angular velocity only. The proposed method is applied to an orbital control problem, which considers a solar sail at motion around an asteroid, with the objective of determining unknown parameters: the solar sail characteristic acceleration and the asteroid’s gravitational harmonics coefficients. It has shown to be able to recover the parameters with only 30 days of observation. In a second application, it has been demonstrated on an attitude control problem, with the aim of recovering the spacecraft residual magnetic induction. It has shown to be able to converge to the real value of the parameters in less than 15 hours of observation. The recovered pieces of information are demonstrated to be useful to preserve part of the required control action for the station-keeping, and so to preserve propellant, or to drive the operations of a set of magnetic torquer, useful to refine the attitude control of the spacecraft.

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Acknowledgements

My first thanks are for my thesis advisor, professor James Douglas Biggs. Thank you for the trust you showed in me and in this work; thank you for the serene collaboration that has been established, thank you for the enthusiasm and the tips you gave me on how to go over the problems which arised during the development, and thank you for the opportunities that are now open about this work. I have to really thank you and all the teachers from the Politecnico di Milano which have pushed me and have shown me that important results can be achieved by anyone, thanks to the hard work.

Il ringraziamento più sentito va ai miei genitori e la mia famiglia tutta. Li ringrazio per avermi permesso di fare questa bellissima esperienza. Non mi hanno fatto mai mancare il loro supporto nei momenti più o meno difficili. Non mi hanno mai messo pressione nei periodi in cui gli esami non andavano, si sono sempre fi-dati dei miei programmi, e non mi hanno mai rinfacciato nessun tipo di problema derivante dal fatto che io vivessi lontano da casa. Probabilmente è anche merito della tranquillità lasciatami e della loro fiducia in me se ho finito ingegneria con i capelli (anche se pochi).

Ringrazio Margherita. Tra mille incertezze, le difficoltà legate alla distanza, i litigi, i miei tremila impegni, ha comunque deciso di essermi affianco in questi ultimi due anni di studi. Oltre all’aver corretto il mio inglese ciceronico. Grazie per avermi sostenuto quando lo studio mi spaventava, per avermi sopportato quando avevo poco tempo, per avermi aiutato ad avere più sicurezza nei miei mezzi. Mi ha dato quel pizzico di serenità in più che mi mancava prima di conoscerla, e che solo l’avere una persona buona al tuo fianco può darti. Come ti dico sempre, sei stata e sei la scossa elettrica nella mia routin. Penso sia anche merito tuo se questa magistrale mi è piaciuta cos`ı tanto.

Grazie ai miei amici, tutti. Grazie ai miei coinquilini che favorivano sempre il casino in casa e non hanno mai creato problemi di convivenza: tornare a casa la sera, per fortuna, non è mai stato un peso, ma anzi una bella prospettiva. Grazie a tutti gli amici che il Politecnico mi ha fatto conoscere: da chi, più intimamente, condivideva le difficoltà e i problemi dovuti all’abitare lontano da casa per la prima volta, a chi animava le più disparate iniziative per puro spirito di compagnia. Grazie a tutti coloro che ci sono stati in questi ultimi sei anni a Milano, i compagni di corso, i compagni di progetto, e i compagni di mattate. E grazie a chi ha contribuito alla stesura di questa tesi con consigli, suggerimenti e risposte alle domande più sceme. Conoscere cos`ı tante diverse persone da tutte le parti d’Italia, e diverse zone del mondo, è stato curioso, divertente ed ha aperto i miei orizzonti come mai prima.

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Grazie ai miei amici di gi`u, a chi mi aspettava a Perugia ed era pronto a festeggiare e stare insieme ogni qualvolta tornassi a casa. Nonostante la distanza e le sporadiche occasioni per frequentarci, sono sempre stati presenti e disponibili. Grazie ai miei pazzi compagni di squadra per avermi inserito senza problemi nel loro gruppo. Grazie anche se non soprattutto per avermi accompagnato e riaccompagnato a casa dopo ogni partita, allenamento o incontro fuori dal campo.

Ringrazio tutti di cuore, sono sinceramente dispiaciuto per aver concluso questa esperienza, spero soltanto che i miei prossimi passi possano essere altrettanto diver-tenti e stimolanti.

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List of Acronyms

Here is reported the list of the Acronyms that can be encountered in the work: 2BP = Two Body Problem

3BP = Three Body Problem AD = Air Drag

ADRC = Active Disturbance Rejection Control AEP = Artificial Equilibrium Point

BGD = Batch Gradient Descent CMG = Control Moment Gyro

CRTBP = Circular Restricted Three-Body Problem DCM = Direct Cosine Matrix

ESA = European Space Agency ESO = Extended State Observer GD = Gradient Descent

GG = Gravity Gradient

IGRF = International Geomagnetic Reference Field LEO = Low Earth Orbit

MF = Magnetic Field ML = Machine Learning

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MO = Momentum Optimization MSE = Mean Square Error MT = Magnetic Torquer

NAG = Nesterov Accelerated Gradient ODE = Ordinary Differential Equation RHS = Right Hand Side

RW = Reaction Wheel

SGD = Stochastic Gradient Descent SRP = Solar Radiation Pressure

SSCA = Solar Sail’s Characteristic Acceleration WMSE = Weighted Mean Square Error

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Chapter 1

Introduction

1.1 General Overview

The last ten years have seen an increase in space engineering capability. Access to space is now available to a wider audience of space agencies, private companies and research teams. This advancement in technological space innovations includes the development of CubeSats, which have considerably decreased the averaged cost per launch. CubeSats have opened an incredible amount of possibilities to the scientific community. However, they present new challenges for spacecraft design due to their limited resources. This requires the need for precise optimization which in turn asks more accurate modelling and optimal control design. In this sense, the station-keeping and attitude control have always been a big challenge, because of the limited resources of this type of spacecraft.

The technology has developed some solutions for CubeSats propulsive systems like solar sails, cold gas propulsion, electric propulsion, and chemical propulsion sys-tems. Anyway, only few of them can give answers to the high energy that must be stored onboard for long-duration mission. The solutions consist in the last years’ development of low-thrust, but high total impulse systems like electric propulsive system and solar sails. Electric propulsion systems utilize onboard power to gen-erate thrust. In general, there are three types of electric propulsion systems: elec-trothermal, electromagnetic, and electrostatic. They generally generate less thrust and higher Isp than cold gas and chemical propulsion systems, but power is usually

the limiting factor for their adoption [1]. Furthermore, unlike chemical propulsion, traditional Hall thruster and ion engine systems do not scale very well into smaller

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systems. Therefore, several new concepts have been developed for use on Cube-Sats [2]. Solar sails, on the other hand, can produce thrust by exploiting the solar radiation pressure on a large reflective membrane, without relying on an onboard propulsive system [3] [4]. This unique capability allows solar sails to produce thrust over extended periods of time, building up large amounts of momentum over the mission lifetime. Solar sails are therefore particularly suited for long-duration and high-energy missions. For example, they could be useful to be employed for close-up observations of small bodies, especially to perform near-Earth asteroid rendezvous mission [5] [6].

To achieve the best guidance and control strategy, a deep knowledge of the phys-ical environment must be obtained. It is necessary to preserve fuel consumption as much as possible, such that the spacecraft operations can be undertaken as long as possible and more continuously as possible. Machine Learning (ML) could be used in this sense: it could be very useful to continuously adapt the control strategy to the changes, which are subjected to the spacecraft over the course of the mission. At the same time, ML could be useful to recover pieces of information on the space-craft’s state, to automate in-orbit processes and to optimize strategies. This thesis is focused on the analysis of the possible improvements that ML could introduce in space engineering. The objective of this work is to verify the feasibility of the ML’s algorithms adoption for some specific space applications and look at how they could perform.

In particular, ML has been used to recover information from an uncertain orbital environment such in the proximity of asteroids, where their gravitational field is not exactly known a priori. Given the known information at the beginning of the mission design, the station-keeping strategy could be extremely conservative. It could mean a waste of fuel during the first period of the mission. The control strategy could be refined exploiting the natural properties of the orbital environment, or simply, the recovered pieces of information could be useful to better understand the asteroid’s physical properties. In a second application, ML has been adopted in another appli-cation: an attitude control problem. It has been used to recover information on the spacecraft’s environment, in particular, to gain information about the spacecraft’s residual magnetic induction. This property can cause an undesired torque because of the interactions with magnetic fields; this disturbance can be neglected in real-time by compensating the perturbation with a set of magnetic torquers.

1.2 Short Description of the Work

Several nations are currently engaging in or planning for human and robotic space exploration programs, that target: the Moon Mars and near-Earth asteroids. One of the pillars of the United Nations Program on Space Applications for developing coun-tries focuses on ”basic space science, including astronomy and astrophysics, solar-terrestrial interactions, planetary and atmospheric studies and exobiology” [7]. All of those research topics are crucial to advance space exploration of the solar system. Collaboration on small low-cost missions through a worldwide CubeSat program can

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support primary exploration activities, while also enabling the participation of new space actors in a meaningful way [8].

CubeSats began as an affordable educational tool for university students in sci-ence and engineering fields, to gain hands-on experisci-ence in aerospace development programs. They represent a specific type of nanosatellite measuring 10 × 10 × 10 cm3 and weighing slightly more than 1 kg. Their small size and mass make them relatively inexpensive and simple to build; more important, it allows them to be launched as secondary payloads at much lower cost and higher frequency than traditional mono-lithic satellites. In more recent years, the wider utility of CubeSats has been increas-ingly recognized; countries now see them as cost-effective platforms for perform-ing science research, technology demonstrations, education and outreach activities. They are an ideal platform for a worldwide program that engages a wide range of space actors. Because of their standard specifications and use of mostly commercial off-the-shelf components are minimized the potential transfer of sensitive technolo-gies. Moreover, the lower costs associated with CubeSat development, deployment and operation in comparison to traditional monolithic spacecraft make it easier the space access. That is especially true for countries with more restricted budgets and limited expertise. Indeed, fast and inexpensive development from concept to launch is important not only for educational programs but also for emerging space powers and developing countries. Moreover, CubeSats have demonstrated their potential as test beds for advanced technologies as well as platforms for research in astrobiology, astronomy, Earth observation, atmospheric science and other fields [9].

There are several major organizations currently promoting them and other small-class satellites. To date, the primary method of increasing a small satellite’s capa-bility has been focused on the miniaturization of technology. The CubeSat Program embraces this approach but has also focused on developing an infrastructure to off-set unavoidable limitations caused by the constraints of small satellite missions. An important aspect of the CubeSat philosophy is to accept certain limitations to reap the benefits of standardization [10]. For example, CubeSat developers have access to regular low-cost launch opportunities, because they are willing to accept a cer-tain form factor and set of requirements. Although, limitations must be considered in all the spacecraft’s subsystems: the avionics and onboard data handling system, the attitude determination and control system, and the telecommunication system suffer from very strict design margins [11]. Furthermore, CubeSats do not typically operate at the high power or high voltage levels required to support propulsion sys-tems. Nor do they have solutions for managing the resulting high thermal loads (most CubeSats use passive thermal control techniques). In particular, if we focus on the propulsion limits, the total ∆v available over the mission lifetime range from 10 m/s [12], 11 − 40 m/s for the Strand mission [13], to < 35 m/s for CanX-2 mission [14]. While these values are compatible with Earth observations missions, as in the case of a scenario in which multiple Satellites are sharing a launch, they seem extremely reduced for a deep space mission. However, it is envisaged that CubeSats will ”piggyback” on primary orbiters travelling to the Moon and Mars to support planetary science missions. Micropropulsion has been identified as a nec-essary technology to enable these spacecraft to manoeuvre and perform formation

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flying, create constellations, and travel to interesting interplanetary destinations [15]. Orbit transfers, particularly to interplanetary destinations, are particularly challeng-ing for small spacecraft low-thrust propulsion systems because typical high-thrust manoeuvres cannot be used, and small spacecraft are extremely mass-, volume-, and power-constrained [16].

CubeSats can be used to demonstrate technologies for future space exploration too, in particular, electric propulsion and solar sail propulsion. Solar electric propul-sion has been proposed, in the last years, for a variety of interplanetary mispropul-sion architectures, including optimized trajectories to the Mars surface for scientific ex-ploration, discovery-class mission applications, and asteroid belt missions. Solar sails based CubeSats are spacecraft which utilize the momentum transfer of solar photons onto large, highly reflecting sails for passive propulsion. Therefore, no ac-tive propulsion system nor any propellant is required for primary propulsion. Using this innovative low-thrust propulsion method extended missions in our solar system which require a high ∆v of several 10km/s would become possible [17]. Especially for high-energy missions, such as a Mercury orbiter, (multiple) main belt asteroid rendezvous, small body sample return or a solar polar orbiter at about 0.5AU solar distance, inclined 90deg to the ecliptic, solar sails could be either enhancing over other means of space propulsion, or even enabling these missions. The LightSail program is the first practical exploitation example: it will investigate the viability of using solar sail propulsion in space exploration missions [18].

This is the field in which the ML could become useful: it could favour a wider participation to the space run, supporting the CubeSats need to optimize any thrust action or control strategy both for station-keeping or attitude. Because of the de-scribed limitations, an optimization or refinement of any action must be carried out. Moreover, it is believed that operations automation is key to cost reduction in fu-ture space operations, and it will be always truer as space will be accessible to more people. The gradual shift of the overall manpower cost from real-time control into day time engineering activities, such as long-term mission planning, performance evaluation, ground and onboard software development and enhancement, could be useful to counteract the natural trend to loss of expertise and motivation generated in mission control team by the long periods of low level routine activities [19]. Let’s see how ML could be helpful in this sense.

ML is the science (and art) of programming computers so they can learn from data, without being explicitly programmed. A computer program is said to learn from experience ”E” with respect to some task ”T” and some performance measure ”P”, if its performance on ”T”, as measured by ”P”, improves with experience ”E”. The first application and the most used example of ML is the email’s spam filter; it can learn to flag spam given examples of spam emails (those flagged by users) and examples of regular nonspam emails. The examples that the system uses to learn are called the training set and each training example is called a training instance or sample. In this case, the task T is to flag spam for new emails, the experience E is the training data, and the performance measure P needs to be defined; it could be the ratio of correctly classified emails. If it is required to program a spam filter using traditional programming techniques, it could be looked at what spam

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typically looks like: some words or phrases that usually come up a lot in the spam, and it could be written a detection algorithm for each of the patterns that have been noticed, so that the program would flag emails as spam if a number of these patterns are detected. Since the problem is not trivial this will likely become a long list of complex rules, pretty hard to maintain. In contrast, a spam filter based on ML techniques automatically learns which words and phrases are good predictors of spam by detecting unusually frequent patterns of words in the spam examples compared to the ham examples. The program is much shorter, easier to maintain, and most likely more accurate. ML can moreover help humans learn: algorithms can be inspected to see what they have learned. For instance. Sometimes this will reveal unsuspected correlations or new trends, and thereby lead to a better understanding of the problem. Applying ML techniques to dig into large amounts of data can help discover patterns that were not immediately apparent.

The estimation in-situ of unknown parameters, useful to improve guidance or control, is probably the first example of a working ground where could be great to leave autonomy to the spacecraft itself. As even the research field that focuses on characterising the space environment where the spacecraft is can be interested by a wider use of ML algorithms. These algorithms can be used in missions around the Solar System’s planets, or minor bodies, in order to refine or characterise, the existing physical models of the environment which are not fully known yet. The adoption of these exploring ML algorithms could be much more interesting to study those celestial bodies that have been not explored at all like asteroids. Indeed the space exploration is nowadays moving to the study of the frozen small bodies which orbit the Sun: asteroids, meteoroids and comets represent a rich source of scientific information about conditions in the early Solar System. Since they have been or-biting the Sun since the formation of the solar system, they should represent the fundamnetal bricks of the planets. Being untouched by any human activity, or in general any biological activity, they could hide information about the nebula that has originated the solar system, and the materials it was composed of. Moreover, a better understanding of their compositions could help us determine if we can rely on them as a source of useful materials, and so to possibly launch a massive run to their resources [20]. All of these observations explain why there is an always wider interest by the space community to the exploration of asteroids and comets.

What practically happens when the space exploration wants to reach these poorly known bodies, is that the overall design of the mission is oriented on granting as much as flexibility and adaptability as possible to the different problems that could arise. Only in a second moment of the mission, the control team has to adapt the spacecraft operations; a refinement of the control strategy can be achieved once more accurate information on the celestial body is obtained by the spacecraft itself. It is both a time and energy expensive activity for the mission control team that could be saved if the spacecraft would be able to recognise and adapt activities to the improved knowledge of the environment. For example, operations in the vicinity of asteroid Bennu for the OSIRIS-REx mission are affected by non-gravitational forces like so-lar radiation pressure and pressure exerted by re-emitted infrared radiation from the spacecraft and the asteroid. These forces are comparable to the force from the

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asteroid’s gravity and can create challenges due to their non-exactly predictable en-tities, during the initial phases of the mission [20]. The challenges against which the Rosetta mission’s design team faced were even more extreme: the mission had to op-erate just a few comet-nucleus radii away from the comet surface, in an environment of nucleus-emitted dust and gas. The effects of these phenomena on the spacecraft were not known in great detail during the spacecraft’s development [21] [22]. The long round-trip light times of up to 90 min, called for a highly autonomous spacecraft [19] and considering the 10 years length of the mission itself, the margins of error during the operative life were extremely restrained. It has been necessary a direct exchange of information, and software updates, between the mission control centre and the spacecraft to deal with the several surprises that the comet 67P/Churyumov-Gerasimenko had initially hidden to the mission design team. All of these attentions that missions like OSIRIS-REx or Rosetta had, cannot be applied on a large scale if deep space missions become ”more popular” and more frequent. It is an aspect that knocks with the CubeSats’ limits and their need to have more careful supervision during the mission’s operations, because of their limited possibilities.

These observations remark the need to automize the optimization of some pro-cesses, analysing the possible benefits that could arise from the ML adoption for station-keeping and attitude control problems. Another interesting aspect of the work is that the developed algorithm will analyse the perturbations recovering the interesting parameters, from the observation of the spacecraft’s state only. Other missions like GOCE [23] or BepiColombo [24], need of a specific system to analyse the perturbations and extract information. Especially, in [24], it is described the need to know, with high precision, the contribution of non-gravitational accelera-tions. They can be needful in some missions to determine the orbit at the level of accuracy desired, and to recover other pieces of information on the planet under study. An onboard accelerometer is used in BepiColombo to remove the complex non-gravitational perturbations from the list of unknowns, and then to refine the es-timation of the gravity field of Mercury. The installation of this other device implies the need to absorb the systematic noise associated inevitably with the data from the accelerometer. Then it is reported the need to elaborate somewhere else the set of data that are collected in-situ by the spacecraft. The result of this work will be instead, an algorithm that continuously adapts the spacecraft knowledge on its state or on the orbital environment where it is. It implies an advantage in terms of lower complexity of the spacecraft and saving for what concerns the mission power and mass budgets.

1.3 Structure of the Thesis

The thesis is organized in six chapters, which content is there reported in short:

Chapter 1: introduction, short description of the work and technology’s state of the art.

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Chapter 2: here is the general structure of the algorithm: the adopted tools to let it work as it is desired to do, the theoric concepts of data analysis that have been considered and the first decisions that have been taken to build-up the method.

Chapter 3: the first application’s framework, the physical and mathematical models adopted for this orbital control problem. There are reported the results to different applications, the evidence that the method works, and solutions to some possible incoming problems that can arise from the adoption of this algorithm.

Chapter 4: here are described the reference frames needed, the models adopted to simulate the dynamics of the second application’s system and the general problems that can arise from the design of an attitude control system. Then the results that come from the application of the algorithm to the attitude control problem

Chapter 5: there are reported an example of how the recovered information can be used to improve the orbital control. Then are given the results that come from the application of the algorithm to the attitude control problem, and one possible use of the incoming information to improve a real problem application.

Chapter 6: this chapter is used to remark the results of the research. It is then reserved to suggestions on how to proceed studying this method, which could be other applications and the possible future steps to be taken to inspect how machine learning can be useful for space applications.

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Chapter 2

In-situ Estimation Algorithm

2.1 Algorithm Structure

The procedure discovered during this thesis work has shown to be extremely versatile. It can be applied to any kind of dynamical problem where there is a known only part of the dynamic equations, and it is present a not perfectly known acceleration contribution. Consider a mission like the ESA’s GOCE mission [23] [25]; the aim of the research was to study the high gravitational harmonics of the Eart analysing the effects that the perturbations had on the orbit. The dynamic of the spacecraft around the planet was known, the gravitational perturbation’s model was known, but the unknown harmonics coefficients needed to be extrapolated to gain information about the Earth itself. This thesis moves on the same ground. Whenever there are some quantities that are not perfectly known whose knowledge could improve the control strategy, it could be used the method developed to determine them. Wherever there are some unknowns quantities that could be interesting for science purposes, it could be used to get them if it is present a good model on how they affect the system.

Defined a general non-linear mechanical system’s state x = [x1, x2], the set of

non-linear Ordinary Differential Equations (ODE) representing the evolution in time of the system can be written in the form:

˙ x1= x2

˙

x2= f (x1, x2)

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Inside the equations of motion, there could be some smaller accelerations, in terms of magnitude, that can be addressed as disturbances (d); the system’s dynamics can

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be affected by some kind of control action (u) designed to achieve a target objective. ˙ x1 = x2 ˙ x2 = f (x1, x2) + d(x1, x2) + u (2.2)

Usually the mathematical model for the dynamic is known, in this thesis, it is as-sumed to have a good model to describe the system’s evolution in time, but inside this model, in particular, inside the perturbations’ model some parameters are not determined in first instance. The working framework can be thought of as the fol-lowing one: ˙ x1= x2 ˙ x2= f (x1, x2)known+ d(x1, x2) + u (2.3)

The main hypothesis adopted at this level is that the interested system’s character-ising parameters appear linearly in the definition of the dynamic’s perturbations:

d = [dx, dy, dz]T d = Xθ =    g(x)1,1 . . . g(x)1,n .. . . .. ... g(x)n,1 . . . g(x)n,n       θ1 .. . θn    (2.4)

Where g(x)i,j contains the mathematical model of the j-th component of

perturba-tions alog the j-th direction, so X is a 3 × N matrix and θ is the N × 1 vector of interested unknown parameters. This formulation of the d vector enables us to per-form regression and to measure the different θ, which hide the dynamical system’s unknown information.

It has been achieved by coupling an Extended State Observers (ESO) with an online regression technique as the Stochastic Gradient Descent.

2.2 Extended Non-Linear State Observer

Since part of the problem dynamic was known, the first key part of the work was to gain information about the instantaneous perturbing accelerations that, time by time, were acting on the spacecraft; it could be done with state observers. In control theory, a state observer is a system that provides an estimate of the internal state of a given real system, from measurements of the input and output of the real system. Most estimators are made to handle slight perturbations for a modelled plant, the ESO has been designed [26] to remove the requirement of a modelled plant by rejecting un-modelled dynamics. The ESOs use a simple canonical form so the un-modelled dynamics appear at the disturbance estimation portion.

˙ˆ x1 = ˆx2+ β1(x1− ˆx1) ˙ˆ x2 = g(x) + u + ˆx3+ β2(x1− ˆx1) ˙ˆ x3 = β3(x1− ˆx1) (2.5)

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Where ˆxi is the estimation of xi, the βi are the gains of the observer that have been

tuned once we select a given observer bandwidth ω0, ˆx3 the estimate of:

h = f (x1, x2) − g(x1, x2) + d

and g(x) is a prescribed function that is defined by the control engineer depending on the control objectives, for example, it can be set to be the known nonlinear or linearized dynamics or simply set to zero.

2.2.1 Convergence Demonstration

It can be defined the unknown derivative ξ = ˙x3, and the of errors in estimations as

e = x − ˆx; the closed-loop estimation error dynamics, in this case, are: ˙ e1 = e2− 3ω0e1 ˙ e2 = e3− 3ω20e1 ˙ e3 = ξ − ω03e1 (2.6)

Then writing e2/ω0 = f2 and e2/ω02 = f3 everything can be reorganised as:

˙ e1 = f2ω0− 3ω0e1 ˙ f2 = f3− 3ω0e1 ˙ f3 = ξ/ω02− ω0e1 (2.7)

As it is reported in [27], the problem can be brought to the higher-dimensional problem here defining = [e1, f2, f3]T, and it can be rearranged as:

d dt = ω0A + B ξ ω₀2 (2.8) Where A =    −3I3X3 I3X3 03X3 −3I_3X3 03X3 I3X3 −I_3X3 03X3 03X3    B =    03X3 03X3 I3X3   

Definining i = [e1i, f2i, f3i]T, where e1 = [e11, e12, e13]T, f2 = [f21, f22, f23]T and

f3= [f31, f32, f33]T, it can be re-written the expression as:

di dt = ω0A˜ i+ ˜B ξi ω2 0 (2.9)

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Where A =    −3ω0 ω0 0 −3ω₀ 0 ω0 −ω0 0 0    B =    0 0 1   

Which has the solution:

i(t) = exp( Ãt) i(0) + Z t 0 exp[ Ã(t − τ )] ˜Bξi ω2₀ dτ (2.10) It follows that: ||i(t)|| ≤ ||exp( Ãt) i(0)|| + Z t 0 exp[ Ã(t − τ )] ˜B ξi ω₀2 dτ (2.11)

Where ||.|| is the Euclideian norm:

||i(t)|| ≤ ||exp( ˜At) i(0)|| + || ˜Bξi|| ω2 0 Z t 0 exp[ ˜A(t − τ )] dτ F (2.12)

Where ||.||F is the Frobenius norm of a matrix; than if we assume that the disturbance

is bouded such that ||ξi|| < ∂, then:

||_i(t)|| ≤ ||exp( Ãt) i(0)|| + ∂ ω₀2 − Ã−1[I − Ã(t)] F

If it is noticed that the eigenvalues of ˜A are λ1= λ2 = λ3= −ω0, then ˜A(t) → 03X3

as t → ∞, such that at the end:

||_i(t)|| ≤ ∂ ω₀2 − ˜A−1 F ≤ √ 21 ∂ ω3₀ (2.13)

This implies that x3 → d as t → ∞ within some small bounded error which can be

decreased with an increase in the gain ω0.Therefore, for a high-gain we can assume

that the error of the ESO is negligible. After a transient due to the dynamic of the ESO so, the full set of states and the perturbation will be recovered and can be used to control the orbital motion of the spacecraft around the asteroid, and to analyze the disturbs characteristics.

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2.3 Regression Techniques

In statistic, regressions are a set of possible approaches used to model the rela-tionships between a scalar response and one or more independent variables. Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications; this is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.

Figure 2.1: Example of Linear Regression

Anyway, there are multiple ways to make a multivariable Linear Regression, and to identify a set of unknown parameters; the following passages show some of them, that are used in many machine learning applications, as it is described in [28].

2.3.1 Linear Regression

Considering a function ˆy, result of a linear combination on n parameters (θ) multi-plied by n degrees of freedom (x), the function can be written in a vectorized form as:

ˆ

y = θT · x (2.14)

And the model that better fits the data is the one that fit the parameters to the set of values of θ that minimize the Root Mean Square Error (RMSE), between the measures and the fitting curve. I practice, it is simpler to minimize the Mean Square Error (MSE), and that leads to the same result.

The MSE of a Linear Regression is calculated as:

M SE(θ) = 1 m m X i=1 (θT · xi− yi)2 (2.15)

an alternative cost function that has been taken into consideration was the Weighted Mean Square Error (WMSE)[29], that includes the MSE in the case where all the weights are wi= 1. It is defined as:

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W M SE(θ, w) = 1 m m X i=1 wi(θT · xi_{− y}i₎2 _(2.16)

The adoption of the WMSE, in some applications, could be necessary to prevent problems related to the different magnitude of the different component of the mea-sured vector ˆd = [ ˆdx; ˆdy; ˆdz]. The WMSE is an interesting tool that has shown to

be useful to scale different ˆd component to the same magnitude and to contain the effect of the ESO errors on the estimation of the parameters.

To find the value of θ that minimize the MSE cost function, there is a closed-form solution that gives directly the results: the Normal Equation.

ˆ

θ = (XT · X)−1· XT · y (2.17) Where ˆθ is the value of θ that minimize the cost function, and y is the vector of target values containing y1 _{to y}m_{. Or in case it is used the WMSE, the values of ˆ}_θ

which minimize the cost function are: ˆ

θ = (XT · W · X)−1· XT · W · y (2.18)

Where W is a diagonal matrix with the different weights on the diagonal.

The main limitations to the use of the Normal Equation are the need to store a good amount of data in order to get a good fit for the model, and the need to compute the inverse of XT · X. It is an n × n matrix and requires a high amount of operations to be inverted. There are different ways to train a Linear Regression model, better suited for cases where there are a large number of measures and de-grees of freedom, methods based on the MSE function’s gradient.

2.3.2 Gradient Descent

The Gradient Descent (GD) is a very generic optimization algorithm capable of finding optimal solutions to a wide range of problems. The general idea of Gradient Descent is to tweak parameters iteratively in order to minimize a cost function. It practically measures the local gradient of the error function with regards to the parameter vector θ, and it goes in the direction of the descending gradient; once it reaches the zero, it has found the right regressions parameters, and it has reached the minimum. Simply by starting filling θ with random values, it improves gradually taking one step at a time, each step attempting to decrease the cost function until the algorithm converges to a minimum.

An important parameter in GD is the size of the steps, determined by the learning rate hyperparameter. If it is too small than the algorithm will have to go through many iterations to converge, and so long time of simulation. On the other hand, if the learning rate is too high, the algorithm could jump across the minimum even higher than how it was before, letting the estimation diverge, with larger and larger values, failing to find a good solution.

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Figure 2.2: Gradient descent method

Moreover, not all the cost function look like nice regular bowls. They can have holes, ridges, plateau, and all sorts of irregular ”terrains” making the convergence to the minimum very difficult. Fortunately, the MSE cost function for a Linear regression is a convex function: it has no local minima and just one global minimum. It is a continuous function, with a slope that never changes abruptly.

It implies that the GD is guaranteed to approach close to the global minimum if are avoided instabilities problems caused by having selected a too high learning rate.

The only problem is that the cost function can be an elongated bowl if the degrees of freedom have very different scales (if the different contributions of our d have very different magnitudes). It implies that having a constant learning rate for all the degrees of freedom, it would converge fastly to the parameter associated with the highest magnitude component of the yi_{, and will need a longer time for the other}

parameters. An example is shown in fig. 2.3. In problems like this one, the method

Figure 2.3: Example of a badly scaled problem

finds firstly a good estimation of θ2, than it spends a lot of iterations to converge

to the right θ1. One possible solution is then to adopt a vector of learning rate,

properly tuned in the different directions, in order to give a faster convergence of the algorithm. This possibility has been taken into account for the development of this work and will be described later.

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regres-sions results, let’s describe them:

2.3.3 Batch Gradient Descent

To implement the GD we need to compute the gradient of the cost function with respect to each model parameter θj. The partial derivatives of MSE with respect to

θj can be recovered as:

∂ ∂θj M SE(θ) = 2 m m X i=1 (θT · xi_{− y}i_)xi j (2.19)

The gradient vector contains all the partial derivatives of the cost function, for all the model parameters:

∇_θM SE(θ) = 2 mX

T _{· (X · θ − y)} _(2.20)

Or in the case we are using the WMSE: ∇θW M SE(θ) =

2 mX

T _{· W · (X · θ − y)} _(2.21)

Notice that this definition involves calculation over the full training set X, at each GD step. This is why the algorithm is called Batch Gradient Descent (BGD); it uses the whole batch of training data at every step. It implies that the computations are slow on a very large batch of training data, and it does not resolve the problem of the need to store a big amount of measures in the system’s memory.

Anyway, once we have the gradient vector, we just have to move the estimation of the parameter in the direction that let the MSE decrease. it means subtracting ∇θM SE(θ) from θ. This is where the learning rate vector η comes into play, giving

the size of the downhill step.

θk+1 = θk− η∇θM SE(θ) (2.22)

In this way, having properly tuned the learning rate, after a certain amount of iterations, the algorithm will converge to the right values of the parameters.

2.3.4 Stochastic Gradient Descent

The main problem with BGD is the fact that it uses the whole training set to com-pute the gradient at every step, that makes it very slow when the training set is large. At the opposite extreme, Stochastic Gradient Descent (SGD) just picks ran-dom instance in the training set at every step and computes the gradients based only on that single instance. Obviously, it makes the algorithm much faster because of the limited amount of computations that are performed and makes it possible to train on huge training sets, since only one instance needs to be in memory at each iteration. On the other hand, due to its stochastic nature, it is much less regular

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Figure 2.4: Stochastic Gradient Descent

than the BGD: the cost function could bounce up and down decreasing only on av-erage. Over time it will end up very close to the minimum, never reaching the global minimum; the final parameters will be good but not optimal. But this aspect could be useful too, in case of a badly shaped cost function, the method could even jump local minima and reach the global better than how the BGD does. Coming back to the theory of the SGD, the equation for the gradient determination is:

∇_θM SE(θ) = 2 mx

T _{· (x · θ − y)} _(2.23)

Or again in case we use the WMSE: ∇_θW M SE(θ) = 2

mx

T _{· W · (x · θ − y)} _(2.24)

Where x represent the instantaneous degrees of freedom of the regression, instead of the full set since the beginning of the simulation. Then the upgraded estimation of the parameters is performed at each time step as it happened for the BGD, and reported in eq.2.22.

In our application the randomly picked set of data were clearly be the time-by-time updated measures incoming from the ESO; after a good tuning of the learning rate, and enough simulation time, the method would converge. The SGD was the best option between the different type of gradient descent methods because it met both the requirement about the need for fast convergence and the no-need of storing data on board.

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2.3.5 Mini-Batch Gradient Descent

The last GD-based algorithm that has been considered has been the Mini-Batch Gradient Descent. At each step, instead of using the full set as in the BGD, or the last estimation as in the SGD, the Mini-Batch Gradient Descent computes the gradient on small sets of instances called mini-batches. The main advantage should be a performance boost.

The algorithm’s progress in parameter space is less erratic than with the Stochas-tic Gradient Descent, especially with fairly large mini-batches. An idea could be to use a buffer, in order to get a set of measures of disturbances, and degrees of freedom to be used in the algorithm that is contonuously updated.

The Mini-Batch will end up walking around a bit closer to the minimum than

Figure 2.5: Convergence of the different GD based methods

the Stochastic Gradient Descent, but it could be harder for it to escape from local minima.

Clearly, the wider is the time interval analyzed by the buffer, the less erratic is the descent of the gradient to the minimum, the faster is the convergence, but the bigger is the mini-batch and the wider is the memory to be installed.

2.3.6 Weighting Matrix Definition

The weighting matrix has been selected with the aim of scaling the different com-ponents of the ˆx3 vector to the same magnitude. The W matrix is diagonal by

definition, and its effect is to multiply the i-th term of ˆx3 with W (i, i) term, that’s

why it has been standardized the definition of the weighting matrix as:

W =    W11 0 0 0 W22 0 0 0 W33    (2.25) With: Wii= | ˆx3| ˆ x3(i) + (2.26)

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and where is a ”safety” factor, essential to avoid problems of singularities caused by the sinusoidal behaviour of the perturbation components.

2.3.7 Resume

The tab.2.1, shows a brief summary of the different features, the different appreci-ated aspects of the analysed GD algorithms described so far:

It has been inspected all the different algorithm’s performances: the Linear

re-Linear Regr. BGD SGD Mini-Batch

Problems due to Yes No No No

singularities

Need to store Yes Yes No Yes

data a wide dataset

Online/Offline Offline Offline Online Online

Table 2.1: Comparison between the different regression techinques

gression has shown to work well but its need to store data was disadvantageous for the selection, and at the same conclusion it has been brought the BGD. For what concerns the Mini-Batch, there were no evident improvements with respect to the SGD after 20-time stamps, so as previously explained the algorithm that has been selected for the work is the SGD.

2.4 Learning Rate Vector Definition

After a certain amount of time required by the ESO to converge to good estimations of the states and of the perturbations, the process of definition of the researched parameters can be achieved. At this point, some constants for the regression must be selected and one of these is the learning rate vector’s components in different directions. One solution to finding a good synthesis could be to gradually reduce the learning rate while the methods advance to the minimum. There are many different strategies to reduce the learning rate during descent. These strategies are called learning schedules, the most common of which are:

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Predetermined piecewise constant learning rate: for example, set the learning rate to η0 = 0.1 at first, then to η1 = 0.001 after 50 epochs. Although this

solution can work very well, it often requires fiddling around to figure out the right learning rates and when to use them.

Performance scheduling: measure the validation error every N steps and reduce the learning rate by a factor of λ when the error stops dropping.

Exponential scheduling: set the learning rate to a function of the iteration number t: η(t) = η0∗ 10−t/r. This works great, but it requires tuning η0 and

r. The learning rate will drop by a factor of 10 every r steps.

Power scheduling: set the learning rate to η(t) = η0∗ (1 + t/r)−c. The

hyper-parameter c is typically set to 1. This is similar to exponential scheduling, but the learning rate drops much more slowly

For this work, it has been decided not to let the learning rate change, but for this first analysis, it has been found an autonomous way to tune the initial value of the learning rate vector components in the different descending directions. They are defined using the information derived by the gradient of the MSE at the first useful estimation. The idea was to take a unitary step in the more ”inclined” descending direction of the MSE, and a long step in the less descending directions.

η0 = [η1, η2, . . . , ηn] (2.27) With: ηi = ∇θM SE(θ, t0)i ||∇_θM SE(θ, t0)|| (2.28)

where is a common ”safety” factor for all the different ηs, which can be set to let the longer step be shorter than unitary and to let the gradient reach the minimum of the MSE cost function without bouncing a lot around it. In this way, the conver-gence of the GD based methods would have been faster than taking the same-length step in all the regression dimensions, and are avoided problems of divergences for the gradient.

2.5 Improving the Convergence Properties

Even with the precautions on selecting different learning rates in the different direc-tions of the optimization, the speed of the GD methods could be, in term of practical use, not enough. A huge speed boost can come from the use of some faster optimizers that are described in the following subsections.

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2.5.1 Momentum Optimization

To resolve the problems related to the high difference in magnitude of the different disturb’s contributions on the measured perturbations, and the consequent difference in magnitude of the gradient’s in the different descending directions that can cause in some occasions an extremely slow convergence of the different GD-based methods, a Momentum Optimization (MO) can be introduced.

Imagine a bowling ball rolling down a gentle slope on a smooth surface: it will start out slowly, but it will quickly pick up momentum until it eventually reaches terminal velocity. This is the very simple idea behind MO, while regular GD was simply taking small regular steps down the slope, so it will take much more time to reach the bottom.

MO cares a lot what previous step’s gradients were: at each iteration, it subtracts

Figure 2.6: Momentum Optimization idea

the local gradient from the momentum vector m (multiplied by the learning rate η), and it updates the weights by simply adding this momentum vector. In other words, the gradient is used as an acceleration, not as a speed. To simulate some sort of friction mechanism and prevent the momentum from growing too large, the algorithm introduces a new hyperparameter β, simply called the friction factor, which must be set between 0 (high friction) and 1 (no friction). A typical β value that is commonly used is 0.9.

mk= βmk−1− η∇_θM SE(θ)

θk= θk−1+ mk (2.29)

if the gradient remains constant, the terminal velocity is equal to that gradient multiplied by the learning rate η, multiplied by _1−β1 ; it means that if for example β = 0.9, then the terminal velocity is equal to 10 times the gradient times the learning rate, so MO ends up going 10 times faster than GD. This allows MO to escape from plateaus much faster than GD and can also help to pass local optima.

For cases like the one that has been previously described where the cost function seems like an elongated bowl, GD goes down the steep slope quite fast, but then it takes a very long time to go down the valley. In contrast, MO will roll down the

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bottom of the valley faster and faster until it reaches the bottom, and so the global minimum.

The one drawback of MO is that it adds yet a new hyperparameter to tune. However, the momentum value of 0.9 usually works well in practice and it can be augmented (0.99, 0.999, and so on...) to have a faster and faster convergence rate, with the only consequence that the method will oscillate a bit more around the global minimum.

2.5.2 Nesterov Accelerated Gradient

There is one small variant of the MO, that has been proposed in 1983 by Yurii Nesterov, which is almost always faster than the original one. The idea of the Nesterov Accelerated Gradient (NAG) is to measure the gradient of the cost function not at the local position but slightly ahead in the direction of the momentum. So the only difference from the classical MO is that the gradient is measured at (θ + βm) rather than at θ.

mk= βmk−1− η∇_θM SE(θk−1+ βmk−1)

θk= θk−1+ mk (2.30)

This small tweak works because in general, the momentum vector will be pointing in the right direction (in the direction of the optimum), so it will be slightly more ac-curate to use the gradient measured a bit farther in that direction rather than using the gradient at the original position. After a while, these small improvements add up

Figure 2.7: Nesterov Accelerated Gradient

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As can be seen in fig. 2.7, when the momentum pushes the weights across a valley, ∇₁ continues to push further across the valley, while ∇2 pushes back toward the

bottom of the valley. This helps reduce oscillations and thus converges faster; NAG will almost always speed up training compared to regular Momentum optimization.

2.5.3 AdaGrad

Consider the problem, that has been previously described, where the cost function is the one in fig.2.3. It would be nice if the algorithm could detect this change of the slope early on and correct its direction to point a bit more toward the global optimum. The AdaGrad algorithm does this scaling down the gradient vector along the steepest dimensions.

sk= sk−1+ ∇θM SE(θk−1). ∗ ∇θM SE(θk−1)

θk= θk−1− η∇θM SE(θk−1)./

√

s + (2.31)

The first step accumulates the square of the gradients into the vector s, in practice, each si accumulates the squares of the partial derivative of the cost function with

regards to parameter θi. If the cost function is steep along the i-th dimension, then

si will get larger and larger at each iteration. The second step is almost identical to

Gradient Descent, but the gradient vector is scaled down by a factor of√s + . In short, this algorithm decays the learning rate, but it does so faster for steep di-mensions than for didi-mensions with gentler slopes. This is called an adaptive learning rate. It helps to point the resulting updates more directly toward the global opti-mum. One additional benefit is that it requires much less tuning of the learning rate hyperparameter η. AdaGrad often performs well for simple quadratic problems, but

Figure 2.8: AdaGrad optimization

unfortunately, in some applications, the learning rate can get scaled down so much that the algorithm ends up stopping entirely before reaching the global optimum.

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2.5.4 RMSProp

Although AdaGrad slows down a bit too fast and ends up never converging to the global optimum, the RMSProp algorithm fixes this by accumulating only the gradi-ents from the most recent iterations.

sk= βsk−1+ (1 − β)∇θM SE(θk−1). ∗ ∇θM SE(θk−1)

θk= θk−1− η∇_θM SE(θk−1)./psk₊ (2.32)

It does so by using exponential decay in the first step. Except on very simple prob-lems, this optimizer almost always performs much better than AdaGrad.

2.5.5 Adam Optimization

Adam which stands for ”adaptive moment estimation”, combines the ideas of M0 and RMSProp: just like MO it keeps track of an exponentially decaying average of past gradients, and just like RMSProp it keeps track of an exponentially decaying average of past squared gradients.

mk= β1mk−1− (1 − β1)∇θM SE(θk−1) sk= β2sk−1+ (1 − β2)∇θM SE(θk−1). ∗ ∇θM SE(θk−1) mk= m k 1 − β₁i sk= s k 1 − βi 2 θk= θk−1− ηmk./psk₊ (2.33)

With ”i” that is the iteration number, and start at 1. It is evident, in the first steps, the similarity to both MO and RMSProp. The only difference is that step 1 computes an exponentially decaying average rather than an exponentially decaying sum, but these are actually equivalent except for a constant factor. Steps 3 and 4 are included because since m and s are initialized at 0, they will be biased toward 0 at the beginning of training, so these two steps will help boost m and s at the beginning of training.

2.5.6 Resume

All the optimization techniques discussed so far only rely on the Jacobian of the cost function. The optimization literature contains algorithms based on the Hessians too but these algorithms are very hard to apply to deep neural networks because there are n2 Hessians per output (where n is the number of parameters), as opposed to just n Jacobians per output. The tab.2.1, shows a brief summary of the different Pro&Cons of the optimizations algorithms described so far:

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MO NAG AdaGrad RMSProp Adam

Parameters 1 1 1 2 3

To Tune

Speeding Good Good Very Bad Very

Capability bad good

Table 2.2: Comparison between the different optimizers

Since the target of this study was to find a method that is as automatic as pos-sible, and as versatile as possible to be used to different problems, the choice on the used optimizers has gone on the MO and the NAG.

2.6 Resulting Algorithm

In eq.2.1 is reported the final structure of the algorithm, whose steps are there described:

1 - The real dynamics are integrated and the system’s instantaneous state is obtained.

2 - The measure of the state (or part of it) reaches the ESO, which builds back information about the instantaneous state and the perturbations acting on the spacecraft.

3 - On the basis of the spacecraft state, since the perturbations’ model are known, are built the features for the regression on the perturbation measure, and the weighting matrix.

4 - It is performed the regression based on the real-time measurement of the disturbances that the ESO gives to the SGD. The resulting performances of the SGD joined to MO or to NAG are compared.

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˙ x1= x2 ˙ x2= f (x1, x2) + d(x1, x2) + u ↓ x1 ↓ ˙ˆ x1= ˆx2+ β1(x1− ˆx1) ˙ˆ x2= g(x) + u + ˆx3+ β2(x1− ˆx1) ˙ˆ x3= β3(x1− ˆx1) ↓ ˆ x1 xˆ2 xˆ3 u ↓ X =    g( ˆx)1,1 . . . g( ˆx)1,n .. . . .. ... g( ˆx)n,1 . . . g( ˆx)n,n    W =     | ˆx3| ˆ x3(1)+ 0 0 0 | ˆx3| ˆ x3(2)+ 0 0 0 | ˆx3| ˆ x3(3)+     ↓ ∇θW M SE(θk−1) = 2 mX T _{· W · (X · θ}k−1_{− ˆ}_x 3) η0= ∇_θM SE(θ, t0) ||∇_θM SE(θ, t0)|| mk= βmk−1− η0∇θW M SE(θk−1) θk= θk−1+ mk

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Chapter 3

Parameters Estimation in

Orbital Dynamic: Application

to Solar Sails in Deep Space

As the first example of application, it has been chosen the orbital control in Perturbed Hill’s Problem. In such an environment, orbit design can be achieved by computing periodic orbits or equilibrium points. Solar sail spacecraft can enable a vast port-folio of vantage points for the long-term, observation of planetary bodies. However, without being overly conservative these approaches cannot easily or efficiently adapt to changes in the environment, parameter drift or faults, that occur over the life-time of the mission. That’s why the in-situ measurement of this parameter can be useful to react to these changes and to improve the knowledge of the dynamic system.

3.1 Reference frames

To properly begin working with this application, it is necessary to fix some useful reference frames. It is necessary to begin by describing the fundamental Sun-centered inertial reference frame Rin(X, Y, Z); it is characterised by having the X-axis aligned

with the γ point, the Z-axis perpendicular to the orbital plane and the Y-axis that completes the reference frame.

Then, to easily describe the relative motion of the three different bodies, it is useful to set up a rotating reference frame. If the two main celestial bodies are the

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Sun and one of the Solar System planets it can be introduced the hypothesis of the Circular Restricted Three-Body Problem (CRTPB)[30], that implies that the orbit of the Planet around the Sun has a null eccentricity. In these cases, the reference frame angular speed is constant and is a characteristic of the system’s couple of main bodies:

||Ω|| = dθ

dt = cost

The resulting reference frame is the RCR3BP(x, y, z), it is Sun-centered (since the

µ = M2/(M1+ M2) parameter shown in fig.3.1 is close to zero), is characterised by

the x-axis aligned with the Sun-Planet direction, again the the z-axis perpendicular to the orbital plane and the y-axis that completes the reference frame.

Figure 3.1: The RCR3BP reference frame

The same hypothesis of CRTBP can be applied to many asteroids orbit around the Sun with almost null eccentricity, and so it can be identified as the most use-ful reference frame for the development of this work. By applying a simple rigid translation of the reference frame’s origin, it can be described the one that is mainly involved in the work: it is a rotating frame R1(ˆx, ˆy, ˆz), centered in the main asteroid,

characterised by having the ˆx axis along the Sun-asteroid line, the ˆz axis perpen-dicular to the asteroid’s heliocentric orbital plane, and the ˆy axis completing the right-handed reference frame as in fig.3.2.

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Figure 3.2: The R1 reference frame

The motion around asteroid can be suitably described in this reference frame rather than in RCR3BP, due to the usual proximity of the spacecraft to the asteroid

itself.

3.2 Dynamical Models

For the development of the work, it has been decided to operate in the framework of Hill’s problem which consists of a special case of the restricted three-body problem.

3.2.1 The Restricted Three-Body Problem

The Three-Body Problem (3BP) [30] is one of the most adopted dynamical models to describe the relative orbital motion of three different objects in space. It is used in many space engineering applications, in particular in its ”Restricted” formulation, to describe the behaviour of a spacecraft under the gravitational influence of a couple of celestial bodies, such stars, planets, moons and asteroids. It is mostly adopted to describe the dynamics of the Earth-Moon system or the Sun-Earth system for example, and derives from the definition of the gravitational force acting on the spacecraft itself: F = msca = − GM1msc |r1|3 r1− GM2msc |r2|3 r2 (3.1)

Where M1 is the first celestial body mass, M2 is the second celestial body mass

and msc is the spacecraft mass. r1 and r2 are the distance vector between the

spacecraft and the respective body, and G is the universal gravitational constant. That acceleration, which is described in this way if it is adopted the inertial Rin

reference frame, must be derived in the rotating RCR3BP reference frame.

The position of the spacecraft in RCR3BP can be described as:

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And the absolute velocity can be derived as: v = dr dt = ˙xri + xr di dt + ˙yrj + yr dj dt + ˙zrk = ˙xri + ˙yrj + ˙zrk + xr di dt + yr dj dt = ∂r ∂t + (Ω × r) (3.3)

Where Ω is the previously described reference frame’s angular speed. The absolute acceleration is again: a = dv dt = ∂v ∂t + (Ω × v) = ∂ ∂t ∂r ∂t + (Ω × r) + Ω × ∂r ∂t + (Ω × r) = ∂ 2_r ∂t2 + ( ˙Ω × r) + Ω × ∂r ∂t + Ω × ∂r ∂t + Ω × (Ω × r) = ∂ 2_r ∂t2 + Ω × (Ω × r) + 2Ω × ∂r ∂t (3.4)

Because in the CRTBP the angular speed Ω is constant and its derivative with respect to time is consequently zero. If it is recalled Eq. 3.1, it can be expressed the equation of motion of the spacecraft in the rotating reference frame:

∂2_r ∂t2 = −G M1 |r1|3 r1+ M2 |r2|3 r2 − Ω × (Ω × r) − 2Ω ×∂r ∂t (3.5) The first two terms of the right-hand side can be shown to be the gradient of a scalar function, that is the three-body potential energy:

U = −G M1 |r₁|+ M2 |r₂| −1 2Ω 2_(x2_{+ y}2₎ _(3.6)

And so the equation of motion can be resumed as: ∂2r

∂t2 = −∇U − 2Ω ×

∂r

∂t (3.7)

3.2.2 The Hill’s Problem

While the restricted 3BP is a special case of the generic 3BP, Hill’s problem is itself a special case of the restricted 3BP. It was first studied by George W. Hill at the end of the nineteenth century in his efforts to produce a reliable theory that could describe the perturbations of the orbit of the Moon [31]. The approximation introduced in Hill’s theory is to consider trajectories very close to the smaller primary object whose mass, moreover, is much smaller than that of the other primary.

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It can be stated the general equations of motion in the R1 reference frame; they

are reported in [32] and are given by: ¨

r + 2Ω × ˙r = ∇U (3.8)

That once explicit is the following set of scalar equations: ¨ x = 2Ωy + Ux ¨ y = −2Ωx + Uy ¨ z = Uz (3.9)

In Eq.3.8, r is the spacecraft position vector in the R1(ˆx, ˆy, ˆz); Ω = Ωˆz, with Ω

the asteroid’s orbit angular velocity around the Sun, in other words the R1 angular

velocity; U is the Hill potential. The Ui nomenclature reported in the equations of

motions represents the partial derivative of the potential U with respect to the i-th coordinate.

The derivation of the Hill potential is the result of a series of hypothesis and simplifications reported in, and recalled in the following lines: beginning with the normalized potential of the restricted 3-body problem:

U = 1 2(x 2_{+ y}2_{) +}(1 − µ) r1 + µ r2 (3.10)

Where the unit of length has been fixed as the distance between the two principal bodies, the time unit has been defined as G = 1, and as a consequence the system angular velocity Ω = 1, and where:

µ = Gm2 m1+ m2 r1 = p (x + µ)2_{+ y}2_{+ z}2 r2 = p (x − 1 + µ)2_{+ y}2_{+ z}2

The approximation introduced in Hill’s theory is to consider trajectories very close to the smaller primary (r2 r1) whose mass is much smaller than that of the

other primary (µ 1). This approximation is required, if we want to describe in a faithful way, the motion in such a dynamic environment, due to the very small mass of the asteroid with respect to the sun and the expected close proximity of the spacecraft with respect to the asteroid [33].

Starting with the potential of the restricted 3-body problem, it can be moved the origin to coincide with the smaller primary x → x + (1 − µ); in this way, the potential becomes: U = 1 2(x 2_{+ y}2_{) + (1 − µ)x +} 1 − µ p(x + 1)2_{+ y}2_{+ z}2 + µ p x2_{+ y}2_{+ z}2 (3.11)

(45)

It can be applied the first approximation, expanding the third term on the right in Eq.3.11 and discarding all terms with powers greater than two.

1 − µ

p(x + 1)2_{+ y}2_{+ z}2 ' (1 − µ)[1 − x + x 2₋1

2(y

2_{+ z}2_)]

The potential becomes:

U = 1 2(x 2_{+ y}2_{) + (1 − µ)x + (1 − µ)[1 − x + x}2₋1 2(y 2_{+ z}2_{)] +} µ p x2_{+ y}2_{+ z}2

After once again, dropping constant terms. Because both the components (x,y,z) and the mass µ are small in the current units, it can be neglected those terms where they are multiplied. It is obtained:

U = µ r − 1 2(z 2_{− 3x}2₎ _(3.12) With r =px2_{+ y}2_{+ z}2_.

Once it get back the non-normalized form of the Hill potential, recalling its definition from [32], it is reached: U = µ˜ r − 1 2Ω 2_(z2_{− 3x}2₎ _(3.13)

Where, this time:

˜

µ = G masteroid

Ω =r µSun aOrbit

It can be resumed the orbital dynamic in state space form, assuming as state vector the following set of free coordinates: x = [r, ˙r] = [x, y, z, vx, vy, vz], and the following

set of ODE: ˙x = f (x) ˙ x = vx ˙ y = vy ˙ z = vz ˙ vx= 2Ωvy+ 3Ω2x − µ r3x ˙ vy = −2Ωvx− µ r3y ˙ vz = −Ωz − µ r3z (3.14)

An adaptive, in-sity, parameters estimation technique for improved spacecraft guidance and control in uncertain environments

POLITECNICO DI MILANO

AN ADAPTIVE, IN-SITU, PARAMETERS

ESTIMATION TECHNIQUE FOR IMPROVED

SPACECRAFT GUIDANCE AND CONTROL IN

UNCERTAIN ENVIRONMENTS

Abstract

Acknowledgements

Contents

List of Acronyms

Chapter 1

Introduction

1.1

General Overview

1.2

Short Description of the Work

1.3

Structure of the Thesis

Chapter 2

In-situ Estimation Algorithm

2.1

Algorithm Structure

2.2

Extended Non-Linear State Observer

2.3

Regression Techniques

2.4

Learning Rate Vector Definition

2.5

Improving the Convergence Properties

2.6

Resulting Algorithm

Chapter 3

Parameters Estimation in

Orbital Dynamic: Application

to Solar Sails in Deep Space

3.1

Reference frames

3.2

Dynamical Models