Development and testing of model predictive control strategies for spacecraft formation flying

Satellite Formation Flying (SFF) is a key technology for several future missions, since, with respect to a single spacecraft, it allows better performances, new capabilities, more flexibility and robustness to failure and cost reduction. Despite these benefits, however, this new concept poses several signicant design challenges and requires new technologies. The Guidance, Navigation and Control (GNC) system is a key element in the SFF concept since it must be reliable in coordinating all the satellites fying in formation during each mission phase, guaranteeing formation integrity and preventing from formation evaporation, and, at the same time, efficient in using the limited on board resources. Model Predictive Control (MPC), also referred to as Receding Horizon Control, is a modern optimal control technique that seems to be suitable for these purposes because of its three main features: model-based control scheme, constraints handling ability and replanning nature. The final aim of my Ph.D. activities was to develop and test MPC strategies for SFF applications. This task was accomplished by means of both computer simulations and experimental tests conducted on both the MIT Synchronized Position Hold Engage & Reorient Experimental Satellites (SPHERES) testbed and the SFF Hardware Simulator under development at the Center of Studies and Activities for Space "Giuseppe Colombo" (CISAS), University of Padova. MPC capabilities were first tested in computer simulations in carrying out a formation acquisition maneuver for two space vehicles, taking into account two scenarios: a Leader-Follower (LF) formation and Projected Circular Orbit (PCO) formation. The performances of the MPC-based controller were compared with those of a Linear Quadratic Regulator (LQR) based controller in the presence of active constraints on the maximum control acceleration, evaluating also the effects of the gravitational harmonics J2 and J3 and atmospheric drag perturbations on the proposed maneuvers. Simulation results of both scenarios showed that, with similar performances in tracking the same reference state trajectory in terms of settling time, the MPC controller is more efficient (less delta-v requirement) than the LQR controller also in the perturbed cases, allowing a delta-v requirement reduction by 40% in the LF formation scenario and by 30% in the PCO formation scenario. The next activity concerned the development of some guidance and control strategies for a Collision-Avoidance scenario in which a free-flying chief spacecraft follows temporary off-nominal conditions and a controlled deputy spacecraft performs a collision avoidance maneuver. The proposed strategy consists on a first Separation Guidance that, using a computationally simple, deterministic and closed-form algorithm, takes charge of avoiding a predicted collision. When some safe conditions on the relative state vector (position and velocity) are met, a subsequent Nominal Guidance takes over. Genetic Algorithms are used to compute a pair of reference state trajectories in order to place the deputy spacecraft in a bounded safe or "parking" trajectory, while minimizing the propellant consumption and avoiding the formation evaporation. The performances of a LQR and a MPC in tracking these reference trajectories were compared, showing how a MPC controller can reduces the total delta-v requirement by 5 - 10% with respect to a LQR controller. MPC capabilities were then evaluated on the MIT SPHERES testbed in simulating the close-proximity phase of the rendez-vous and capture maneuver for the Mars Orbital Sample Return (MOSR) scenario. Better performances of MPC with respect to PD in executing this maneuver were conrmed both in a Matlab simulator and in the MIT SPHERES software simulator, with a total delta-v requirement reduction by 10-15 %. The proposed MPC control strategy was then tested using the SPHERES Flat Floor facility at the MIT Space System Laboratory. The last part of my research activities was devoted to the SFF Hardware Simulator of the University of Padova. My contributions to this project dealt with: (a) conclusion of the designing, building and testing of the five main subsystems of the hardware simulator; (b) software development for the hardware simulator and its Matlab software simulator; (c) preparatory experimental activities aimed at characterizing the thrust force performed by the on board thrusters and estimating the hardware simulator inertia properties; and (d) test of attitude control maneuvers with the use of predictive controllers. In particular, three main tests were carried out with the hardware simulator moving at one degree of freedom about the yaw axis. The first one aimed at tuning a Kalman Filter to properly estimate the yaw axis angular velocity using a double-integrator as dynamic model and angular position measurements provided by the yaw quadrature encoder. With the use of a simple Kalman Filter, the yaw angular position and velocity could be estimated with an error less than 0.1 ° and 0.1°/s, respectively. In the second test, an explicit MPC was used to perform a 170° slew maneuver of the hardware simulator attitude module about the yaw axis. The final target angular position was reached with an error less than 0.5° in 20 s. In the third test, a 3 degrees of freedom attitude reference trajectory was first computed using pseudospectral optimization methods for a repointing maneuver with active constraints on the attitude trajectory. The state trajectory was then projected along the satellite z-Body axis and tracked in the hardware simulator using an explicit MPC. Experimental results showed that with an explicit MPC the reference trajectories can be tracked with an error less that 1.5° for the angular position and less than 1°/s for the angular velocity, both in dynamic conditions. The final target state was reached with an error less than the estimation accuracy. The SFF Hardware Simulator is a ground-based testbed for the development and verification of GNC algorithms that in the present configuration allows the development and testing of advanced controls for attitude motion and in its final form will enable the derivation of control strategies for Formation Flight and Automated Rendezvous and Docking

Metrological characterization of a vision-based system for relative pose measurements with fiducial marker mapping for spacecrafts

An improved approach for the measurement of the relative pose between a target and a chaser spacecraft is presented. The selected method is based on a single camera, which can be mounted on the chaser, and a plurality of fiducial markers, which can be mounted on the external surface of the target. The measurement procedure comprises of a closed-form solution of the Perspective from n Points (PnP) problem, a RANdom SAmple Consensus (RANSAC) procedure, a non-linear local optimization and a global Bundle Adjustment refinement of the marker map and relative poses. A metrological characterization of the measurement system is performed using an experimental set-up that can impose rotations combined with a linear translation and can measure them. The rotation and position measurement errors are calculated with reference instrumentations and their uncertainties are evaluated by the Monte Carlo method. The experimental laboratory tests highlight the significant improvements provided by the Bundle Adjustment refinement. Moreover, a set of possible influencing physical parameters are defined and their correlations with the rotation and position errors and uncertainties are analyzed. Using both numerical quantitative correlation coefficients and qualitative graphical representations, the most significant parameters for the final measurement errors and uncertainties are determined. The obtained results give clear indications and advice for the design of future measurement systems and for the selection of the marker positioning on a satellite surface

Space tethers: parameters reconstructions and tests

In the last several years, the need for an alternative to chemical propulsive systems for low-orbit satellite deorbiting has become increasingly evident; a Tethered System can provide adequate thrust or drag without the complications of combustions and with a minimal impact on the environment. In this context, the authors are part of a team that is studying various tether applications and building a prototype of an electrodynamic tether system. The goal of this paper is to characterize tether materials in order to find valid solutions for future space tether missions. Mission requirements (e.g., the survivability to hypervelocity impacts and the capability to damp oscillations in electrodynamic tethers) influence the choice of tether parameters such as cross section geometry (round wires or tapes), materials, length, and cross section sizes. The determination of the elastic characteristics and damping coefficients is carried out through a campaign of experiments conducted with both direct stress/strain measurements and the laboratory facility SPAcecRraft Testbed for Autonomous proximity operatioNs experimentS (SPARTANS) on a low friction table at the University of Padova. In the latter case, the stiffness and damping of a flexible line were verified by applying different tensile load profiles and then measuring the tether-line dynamic response in terms of tension spike amplitude, oscillation decay, and estimation of the damping coefficient

EXPERIMENTAL VALIDATION OF A DEPLOYMENT MECHANISM FOR TAPE-TETHERED SATELLITES

The number of space debris orbiting our Earth has been continuously increasing since the beginning of the space era. The space community is converging on responsible conducts and self-regulations to address this serious problem that is degrading the near-Earth environment. In this context, green deorbiting technologies and strategies alternative to the traditional chemical propulsion are under investigation, including Electrodynamic Tethers (EDTs) because they are a promising option. To increase EDT technology maturity level, some critical points shall be addressed and experimentally evaluated, including the deployment of tape tethers, to demonstrate their reliability. This paper presents results of an experimental validation of the Deployment Mechanism (DM) proposed for the H2020 FET OPEN Project E.T.PACK \u2013 Electrodynamic Tether Technology for Passive Consumable-less Deorbit Kit. We developed a mockup that hosts the DM and other elements that are on board the tip mass of a tethered system, using off-the-shelf components. The DM is tested for the first part of the tether deployment maneuver employing the SPARTANS facility of the University of Padova. This facility includes a Testing Table where the mock-up can move with almost no friction and a Motion Capture system that provides an accurate estimation of the mock-up motion during this first part of the tether deployment maneuver

Deployment requirements for deorbiting electrodynamic tether technology

In the last decades, green deorbiting technologies have begun to be investigated and have raised a great interest in the space community. Among the others, electrodynamic tethers appear to be a promising option. By interacting with the surrounding ionosphere, electrodynamic tethers generate a drag Lorentz force to decrease the orbit altitude of the satellite, causing its re-entry in the atmosphere without using propellant. In this work, the requirements that drive the design of the deployment mechanism proposed for the H2020 Project E.T.PACK—Electrodynamic Tether Technology for Passive Consumable-less Deorbit Kit—are presented and discussed. Additionally, this work presents the synthesis of the reference profiles used by the motor of the deployer to make the tethered system reach the desired final conditions. The result is a strategy for deploying electrodynamic tape-shaped tethers used for deorbiting satellites at the end of their operational life.Open Access funding provided by Università degli Studi di Padova. This work was supported by European Union’s H2020 Research and Innovation Programme under Grant Agreement No. 828902 (E.T.PACK Project). Gonzalo Sánchez-Arriaga's work is supported by the Ministerio de Ciencia, Innovación y Universidades of Spain under the Grant RYC-2014-15357

Development and testing of model predictive control strategies for spacecraft formation flying

Satellite Formation Flying (SFF) is a key technology for several future missions, since, with respect to a single spacecraft, it allows better performances, new capabilities, more flexibility and robustness to failure and cost reduction. Despite these benefits, however, this new concept poses several signicant design challenges and requires new technologies. The Guidance, Navigation and Control (GNC) system is a key element in the SFF concept since it must be reliable in coordinating all the satellites fying in formation during each mission phase, guaranteeing formation integrity and preventing from formation evaporation, and, at the same time, efficient in using the limited on board resources. Model Predictive Control (MPC), also referred to as Receding Horizon Control, is a modern optimal control technique that seems to be suitable for these purposes because of its three main features: model-based control scheme, constraints handling ability and replanning nature. The final aim of my Ph.D. activities was to develop and test MPC strategies for SFF applications. This task was accomplished by means of both computer simulations and experimental tests conducted on both the MIT Synchronized Position Hold Engage & Reorient Experimental Satellites (SPHERES) testbed and the SFF Hardware Simulator under development at the Center of Studies and Activities for Space "Giuseppe Colombo" (CISAS), University of Padova. MPC capabilities were first tested in computer simulations in carrying out a formation acquisition maneuver for two space vehicles, taking into account two scenarios: a Leader-Follower (LF) formation and Projected Circular Orbit (PCO) formation. The performances of the MPC-based controller were compared with those of a Linear Quadratic Regulator (LQR) based controller in the presence of active constraints on the maximum control acceleration, evaluating also the effects of the gravitational harmonics J2 and J3 and atmospheric drag perturbations on the proposed maneuvers. Simulation results of both scenarios showed that, with similar performances in tracking the same reference state trajectory in terms of settling time, the MPC controller is more efficient (less delta-v requirement) than the LQR controller also in the perturbed cases, allowing a delta-v requirement reduction by 40% in the LF formation scenario and by 30% in the PCO formation scenario. The next activity concerned the development of some guidance and control strategies for a Collision-Avoidance scenario in which a free-flying chief spacecraft follows temporary off-nominal conditions and a controlled deputy spacecraft performs a collision avoidance maneuver. The proposed strategy consists on a first Separation Guidance that, using a computationally simple, deterministic and closed-form algorithm, takes charge of avoiding a predicted collision. When some safe conditions on the relative state vector (position and velocity) are met, a subsequent Nominal Guidance takes over. Genetic Algorithms are used to compute a pair of reference state trajectories in order to place the deputy spacecraft in a bounded safe or "parking" trajectory, while minimizing the propellant consumption and avoiding the formation evaporation. The performances of a LQR and a MPC in tracking these reference trajectories were compared, showing how a MPC controller can reduces the total delta-v requirement by 5 - 10% with respect to a LQR controller. MPC capabilities were then evaluated on the MIT SPHERES testbed in simulating the close-proximity phase of the rendez-vous and capture maneuver for the Mars Orbital Sample Return (MOSR) scenario. Better performances of MPC with respect to PD in executing this maneuver were conrmed both in a Matlab simulator and in the MIT SPHERES software simulator, with a total delta-v requirement reduction by 10-15 %. The proposed MPC control strategy was then tested using the SPHERES Flat Floor facility at the MIT Space System Laboratory. The last part of my research activities was devoted to the SFF Hardware Simulator of the University of Padova. My contributions to this project dealt with: (a) conclusion of the designing, building and testing of the five main subsystems of the hardware simulator; (b) software development for the hardware simulator and its Matlab software simulator; (c) preparatory experimental activities aimed at characterizing the thrust force performed by the on board thrusters and estimating the hardware simulator inertia properties; and (d) test of attitude control maneuvers with the use of predictive controllers. In particular, three main tests were carried out with the hardware simulator moving at one degree of freedom about the yaw axis. The first one aimed at tuning a Kalman Filter to properly estimate the yaw axis angular velocity using a double-integrator as dynamic model and angular position measurements provided by the yaw quadrature encoder. With the use of a simple Kalman Filter, the yaw angular position and velocity could be estimated with an error less than 0.1 ° and 0.1°/s, respectively. In the second test, an explicit MPC was used to perform a 170° slew maneuver of the hardware simulator attitude module about the yaw axis. The final target angular position was reached with an error less than 0.5° in 20 s. In the third test, a 3 degrees of freedom attitude reference trajectory was first computed using pseudospectral optimization methods for a repointing maneuver with active constraints on the attitude trajectory. The state trajectory was then projected along the satellite z-Body axis and tracked in the hardware simulator using an explicit MPC. Experimental results showed that with an explicit MPC the reference trajectories can be tracked with an error less that 1.5° for the angular position and less than 1°/s for the angular velocity, both in dynamic conditions. The final target state was reached with an error less than the estimation accuracy. The SFF Hardware Simulator is a ground-based testbed for the development and verification of GNC algorithms that in the present configuration allows the development and testing of advanced controls for attitude motion and in its final form will enable the derivation of control strategies for Formation Flight and Automated Rendezvous and Docking.Il volo in formazione tra satelliti è una tecnologia fondamentale per molte missioni future, poiché, rispetto ad un satellite singolo, permette migliori prestazioni, nuove capacità, maggiore flessibilità e robustezza alle avarie e riduzione dei costi. Nonostante questi benefici, tuttavia, questo nuovo concetto pone svariate sfide progettuali e richiede nuove tecnologie. Il sistema di Guida, Navigazione e Controllo (GNC) è un elemento chiave per il volo in formazione, poiché deve essere affidabile nel coordinare tutti i satelliti che volano in formazione durante ciascuna fase della missione, garantendo l'integrità della formazione e prevenendo l'evaporazione della stessa, e, allo stesso tempo, efficiente nell'utilizzo delle limitate risorse di bordo. Il Model Predictive Control (MPC), chiamato anche Receding Horizon Control, è una moderna tecnica di controllo ottimo che sembra essere adeguata a queste finalità per le sue tre principali caratteristiche: schema di controllo basato su modello, abilità nel gestire i vincoli e ripianificazione. L'obbiettivo finale delle mie attività di dottorato è stato quello di sviluppare e testare strategie di controllo MPC per applicazioni di volo in formazione. Questo obiettivo è stato raggiunto sia mediante simulazioni al computer sia attraverso test sperimentali condotti e sul sistema Synchronized Position Hold Engage & Reorient Experimental Satellites (SPHERES) del MIT e sul simulatore hardware per volo in formazione che è in fase di sviluppo al Centro di Ateneo di Studi ed Attività Spaziali "Giuseppe Colombo" (CISAS) dell'Università di Padova. Le capacità del controllo MPC sono state dapprima testate mediante simulazioni al computer nell'eseguire una manovra di acquisizione di formazione per due veicoli spaziali, prendendo in considerazione due scenari: una formazione Leader-Follower (LF) e una formazione Projected Circular Orbit (PCO). Le prestazioni del controllore MPC sono state confrontate con quelle di un controllore LQR in presenza di vincoli attivi sulla massima accelerazione di controllo, valutando inoltre gli effetti perturbativi delle armoniche gravitazionali J2 e J3 e dell'attrito atmosferico sulle manovre proposte. I risultati delle simulazioni per entrambi gli scenari hanno mostrato che, per simili prestazioni nel seguire la stessa traiettoria di stato di riferimento in termini di tempo di assestamento, il controllore MPC è più efficiente (minor requisito di delta-v) rispetto al controllore LQR anche nei casi con perturbazioni, permettendo una riduzione del requisito di delta-v totale del 40% nello scenario LF e del 30% in quello PCO. L'attività successiva ha riguardato lo sviluppo di alcune strategie di guida e controllo per uno scenario di Collision-Avoidance in cui un satellite chief non controllato segue temporaneamente condizioni non nominali e un satellite controllato deputy esegue una manovra di anti-collisione. La strategia proposta consiste in una prima Separation Guidance che, utilizzando un algoritmo semplice, deterministico e in forma chiusa, ha lo scopo di evitare una collisione prevista. Quando vengono soddisfatte alcune condizioni di sicurezza sullo stato relativo (posizione e velocità), subentra una successiva Nominal Guidance. Gli Algoritmi Genetici sono usati per calcolare una coppia di traiettorie di stato di riferimento al fine di collocare il satellite deputy in una traiettoria chiusa "di parcheggio", minimizzando il consumo di carburante ed evitando l'evaporazione della formazione. Le prestazioni di un controllo LQR e di uno MPC nel seguire queste traiettorie di riferimento sono state messe a confronto, dimostrando come un controllo MPC può ridurre il requisito totale di delta-v del 5 - 10% rispetto ad un controllo LQR. Le capacità del controllo MPC sono state valutate anche nel sistema SPHERES del MIT nel simulare la fase di prossimità della manovra di rendez-vous and capture per lo scenario Mars Orbital Sample Return (MOSR). Migliori prestazioni del controllo MPC rispetto al controllo PD nell'eseguire questa manovra sono state confermate sia in un simulatore Matlab che nel simulatore software di SPHERES del MIT, con una riduzione del requisito totale di delta-v del 10 - 15%. La strategia di controllo MPC proposta è stata poi testata nella SPHERES Flat Floor facility presso lo Space System Laboratory del MIT. L'ultima parte dell'attività di ricerca si è concentrata sul simulatore hardware per il volo in formazione dell'Università di Padova. Il mio contributo a questo progetto ha riguardato: (a) la conclusione delle fasi di progettazione, costruzione e test dei cinque principali sottosistemi del simulatore hardware; (b) lo sviluppo di software per il simulatore hardware e del suo simulatore software in Matlab; (c) alcune attività sperimentali preparatorie finalizzate a caratterizzare la spinta prodotta dai razzetti di bordo e stimare le proprietà d'inerzia del simulatore hardware; e (d) il test di manovre di controllo d'assetto con l'utilizzo del controllo predittivo. In particolare, sono stati eseguiti tre principali test con il simulatore hardware in moto ad un grado di libertà attorno all'asse di yaw. Il primo test è stato finalizzato al tuning di un Filtro di Kalman per stimare in modo opportuno la velocità angolare di yaw usando un doppio integratore come modello dinamico e misure della posizione angolare fornite dall'encoder di yaw. Utilizzando un semplice Filtro di Kalman, è stato possibile stimare la posizione e la velocità angolare con un errore inferiore a 0.1° e 0.1°/s, rispettivamente. Nel secondo test, è stato utilizzato un controllo MPC esplicito per eseguire una manovra di ri-orientazione di 170° del modulo d'assetto del simulatore hardware attorno all'asse di yaw. La posizione angolare obiettivo è stata raggiunta con un errore inferiore a 0.5° in 20 s. Nel terzo test, una traiettoria d'asseto di riferimento è state dapprima calcolata utilizzando metodi di ottimizzazione pseudospectral per una manovra di ripuntamento con vincoli attivi sulla traiettoria di stato. La traiettoria di stato è stata poi proiettata lungo l'asse z-Body del satellite ed inseguita nel simulatore hardware utilizzando un controllo MPC esplicito. I risultati sperimentali hanno dimostrato che con un controllo predittivo esplicito le traiettorie di riferimento possono essere inseguite con un errore inferiore a 1.5° per la posizione angolare e inferiore a 1°/s per la velocità angolare, entrambi in condizioni dinamiche. Lo stato finale obiettivo è stato raggiunto con un errore inferiore all'accuratezza di stima. Il Simulatore Meccanico per il volo in formazione costituisce un banco di prova per lo sviluppo e la verifica in laboratorio di algoritmi di GNC; nella configurazione attuale il simulatore permette lo sviluppo ed il test di controlli avanzati per il moto d'assetto, mentre nella sua configurazione finale consentirà di sviluppare strategie di controllo per Formation Flight e Automated Rendezvous and Docking

Regional L 2m gain analysis for linear saturating systems

International audienceSufficient conditions are presented for regional stability and nonlinear L2m gain analysis of linear systems subject to saturation, based on piecewise polynomial Lyapunov functions. The proposed conditions are formulated in terms of convex optimization problems and improve existing results both for the quadratic (L2 gain) and the polynomial (L2m gain) cases

Global stability and finite L2m-gain of saturated uncertain systems via piecewise polynomial Lyapunov functions

In this technical note, we provide sufficient conditions for global asymptotic stability and global L2m gain estimation of linear systems subject to saturations and/or deadzones with structured parametric uncertainties, based on piecewise polynomial Lyapunov functions. By using sum-of-squares relaxations, these conditions are formulated in terms of linear matrix inequalities. Example studies are used to comparatively illustrate the proposed techniques