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Universidad Politécnica de MadridEscuela Técnica Superior de Ingenieros Industriales

École Polytechnique Fédérale de Lausanne

ANGULAR VELOCITY ESTIMATION

OF A REACTION SPHERE ACTUATOR

FOR ATTITUDE SATELLITE CONTROL

Trabajo Fin de Máster

Máster en Ingeniería Industrial

Autor: GUZMÁN BORQUE GALLEGODirector: MANUEL FERRE PÉREZCodirector: ALIREZA KARIMI

Madrid, 12th September 2016

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Acknowledgements

The research work developed for this Master's thesis was developed at the Swiss Center ofElectronics and Microtechnology CSEM in Neuchâtel, Switzerland, in collaboration with theAutomatic Laboratory (LA) of the Swiss Federal Institute of Technology (EPFL), Switzerland.

I would like to thank the valuable help and courage given by Prof. Alireza Karimi LA,providing the guidance required for developing the work and overcoming all the diculties foundduring these months. I also owe my deepest gratitude to Leopoldo Rossini of CSEM for super-vising all my work and development at the company thoroughly during my internship. The workwould not have been possible without all the help provided by Max Boegli and Olivier Chételat.

Finally, I have to sincerely thank my family, for their unconditional support during my wholelife and studies, and the new family that I have surprisingly found in Switzerland, for welcomingme so warmly and making me feel like home.

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Abstract

In dierent types of spacecraft, such as stabilised satellites, the Attitude Determination andControl System (ADCS) is responsible for stabilisation and achieving the desired rotationalmovement of the spacecraft. Depending on its application, many satellites may require a three-axis stabilisation system, thus a device or set of devices capable of applying three-axis rotationalmotion is needed. These torques are commonly applied by a set of minimum three ReactionWheels (RW), but commonly four of them are used for redundancy and optimization purposes.An alternative device to this set of RWs was proposed more than 60 years ago: Reaction Spheres(RSs), but the performance of these designs was not comparable to the one obtained with RWs.The main idea behind Reaction Spheres is substituting the three or four rotating masses by asingle mass in the shape of a hollow sphere that can be accelerated, and thus apply torque, aboutany given axis.

More recently, a novel concept of Reaction Sphere was proposed and manufactured, inwhich the spherical actuator is magnetically levitated and can be torqued about any desiredaxis electronically. The design consists of an eight-pole Permanent Magnet spherical rotor anda twenty-pole stator with electromagnets. Magnetic ux model, required for optimization of thedesign and for deriving force and torque models, was developed and validated by using a hybridanalytical-FEM approach, and taking advantage of Spherical Harmonic decomposition, whichallowed to model and control the system by means of magnetic information, called magneticstate or magnetic orientation, without explicitly knowing the physical orientation of the rotor.

This design, developed at CSEM SA, is able to magnetically levitate and acquire angularvelocities up to 300 rpm about any given axis, but it faced some limitations. In order to managethe stored angular momentum, and for closed-loop control purposes, measurement of the angularvelocity of the rotor inside the stator is required. As there is no direct measure of this mag-nitude, a technique based on determining the back-EMF voltage induced in the stator coils wasdeveloped and experimentally validated. Nevertheless, this direct method is specially susceptibleto magnetic ux density distortions of the rotor. Alternatively, some work on implementing anExtended Kalman Filter by using rotor orientation instead of the magnetic state has been done,but the algorithm was too heavy to be implemented in real-time.

In this thesis, a novel approach based on a Linear Parameter-Varying Kalman Filter observerhas been proposed. This method tries to combine the main advantages of the previous approachesfor angular velocity estimation, such as fast computations and execution time obtained by makinguse of the aforementioned Spherical Harmonic decomposition and back-EMF voltage estimation,

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and the noise ltering and optimality of estimations, under specic circumstances and conditions,obtained with Kalman Filter.

A state-space model of the available system is derived, by making use of the magnetic stateand sensor measurements as parameters and input respectively, yielding a Linear Parameter-Varying model, in which the proposed Kalman Filter observer is based, and obtaining this waythe novel LPV KF estimator.

This estimator is validated and analysed both in simulation and experimentally with thereal prototype, obtaining promising results when used in the angular velocity closed-loop controlsystem, specially for high angular velocities, in which the oscillations around the desired valueare reduced by a factor of two or three in amplitude, and these noisy oscillations are substitutedby sinusoidal deviations. However, it is believed that the obtained performance could be furtherimproved by improving the control technique used in the closed-loop system to be better adaptedto the proposed observer, as the whole system (sphere, estimator and controller) should be takeninto account.

We believe that the work developed in this thesis continues with the baseline marked atthe Centre Suisse d'Electronique et de Microtechnique SA for the available Reaction Sphereprototype, improving and proposing a promising alternative to one of the critical problems:angular velocity estimation. Nonetheless, the system is still not fully prepared, requiring tostudy some of the issues shown in this thesis.

Key Words

Satellite attitude control, reaction sphere, spherical actuator, spherical harmonics, magneticstate, angular velocity estimation, linear parameter-varying system, Kalman Filter.

UNESCO Codes

33 Technology sciences:

3304 Computer technology : 3304.12 Control devices, 3304.17 Real-time systems3306 Electrical technology and engineering : 3306.03 Electric motors3311 Instrumentation Technology : 3311.02 Control engineering3324 Space Technology : 3324.01 Articial satellites, 3324.07 Vehicle control

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Resumen

En distintos tipos de naves espaciales o astronaves, como pueden ser los satélites articiales,el sistema de determinación y control de actitud (SDCA) es el responsable de la estabilizacióny obtención del movimiento de rotación deseado para la nave. Dependiendo de su aplicación,muchos satélites requieren el uso de sistemas capaces de actuar sobre tres ejes (tres grados delibertad) para poder controlar la orientación completa del satélite. El par necesario para modicarla orientación del satélite es generado generalmente por un sistema, como mínimo, de tres ruedasde reacción, pero comúnmente un total de cuatro son empleadas por motivos de optimizacióny redundancia. Un dispositivo alternativo al conjunto de ruedas de reacción fue propuesto hacemás de 60 años: la esfera de reacción, pero el rendimiento obtenido con estos dispositivos noes comparable al obtenido por el conjunto de ruedas de reacción. La idea principal reside en lasustitución del conjunto de tres o cuatro masas rotativas por una única esfera hueca capaz deser acelerada y ejercer un par alrededor de cualquier eje.

Recientemente, un nuevo concepto de esfera de reacción fue propuesto y fabricado, en el cualel actuador esférico es levitado magnéticamente y puede ser acelerado electrónicamente alrede-dor de cualquier eje. El diseño consiste en un rotor esférico con ocho polos creados con imanespermanentes y un estátor de veinte polos con electroimanes. El modelo de ujo magnético, nece-sario para la optimización del diseño y para obtener los modelos de fuerza y par que relacionanlas corrientes aplicadas a las bobinas del estátor con la fuerza y el par aplicado al rotor, fuedesarrollado y validado haciendo uso de técnicas híbridas de modelado analítico y por elementosnitos. Para estos modelos se empleó la denominada descomposición en armónicos esféricos, lacual permitió realizar el modelado y controlar el sistema utilizando únicamente información re-lacionada con el campo magnético generado por el dispositivo, denominada estado magnético uorientación magnética, sin calcular explícitamente la orientación del rotor.

Este diseño, desarrollado en la empresa CSEM SA en Suiza, es capaz de levitar por acciónmagnética y rotar hasta un máximo de 300 rpm alrededor de cualquier eje, pero no sin elloafrontar distintas dicultades y limitaciones. Con el objetivo de controlar el momento angularalmacenado en el rotor, y de cerrar el bucle de control, es imprescindible el conocimiento de lavelocidad angular del rotor. Debido a que no existe medida directa de dicha magnitud, un métodode estimación a base del cálculo de la fuerza contra-electromotriz inducida en las bobinas delestátor ha sido desarrollado y validado. Sin embargo, esta técnica directa es especialmente sensiblea distorsiones en la densidad de ujo magnético del rotor. Por otro lado, un estudio preliminaracerca de la implantación de un estimador basado en un Filtro de Kalman Extendido que haceuso de la estimación de la orientación del rotor. Este último método es descartado por su elevadocoste computacional, que lo hace imposible de ser implementado en tiempo real.

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En el presente trabajo n de máster, un nuevo estimador de velocidad angular es propuestoy analizado. Dicho estimador está basado en un Filtro de Kalman para sistemas lineales conparámetros variables. Este nuevo método trata de combinar las ventajas mostradas por las dosalternativas anteriormente estudiadas, como son la rapidez de cálculo obtenida gracias al usodel estado magnético, a la descomposición en armónicos esféricos y a las ecuaciones linealesdesarrolladas para la estimación de la fuerza contra-electromotriz inducida en las bobinas, y lacancelación o reducción del ruido y optimalidad de las estimaciones realizadas mediante un ltrode Kalman, bajo ciertas condiciones.

Un modelo de espacio de estado es desarrollado para el sistema estudiado, haciendo uso delestado magnético del rotor y de la información de la densidad de ujo magnético medida porlos sensores disponibles, obteniéndose un sistema lineal con parámetros variables sobre el cual seaplicará un Filtro de Kalman para la estimación del estado del sistema: la velocidad angular delrotor.

Este estimador es validado y analizado haciendo uso tanto de modelos de simulación desa-rrollados para la esfera durante la tesis, como del propio prototipo fabricado en la empresa CSEMSA de Suiza, obteniéndose resultados prometedores al ser empleado en el bucle de control deldispositivo. Para relativamente elevadas velocidades angulares, la amplitud de las oscilacionesobtenidas mediante el uso del estimador propuesto son reducidas por un factor de dos o tres conrespecto al uso del método basado en el cálculo de la fuerza contra-electromotriz. Además, coneste último estimador las oscilaciones obtenidas poseen una gran componente de ruido, mientrasque con el uso del ltro de Kalman son principalmente sinusoidales, y por tanto podrían serreducidas mediante un mejor ajuste o mejora del algoritmo de control empleado.

Se considera que el trabajo desarrollado para la obtención del presente documento sigue lalínea marcada por las necesidades de la empresa Centre Suisse d'Electronique et de Microtech-nique SA en Suiza para el desarrollo y mejora del prototipo ya disponible de esfera de reacción.Los prometedores resultados obtenidos con este trabajo deben ser utilizados como base para con-tinuar con el progreso y solución de los principales problemas encontrados durante el desarrollodel presente concepto de esfera de reacción.

Palabras Clave

Control de actitud de satélites, esfera de reacción, actuador esférico, armónicos esféricos, estadomagnético, estimación de velocidad angular, sistema lineal con parámetros variables, Filtro deKalman.

Códigos UNESCO

33 Ciencias tecnológicas:

3304 Tecnología de ordenadores: 3304.12 Dispositivos de control, 3304.17 Sistemas en tiemporeal3306 Ingeniería y tecnología eléctricas: 3306.03 Motores eléctricos3311 Tecnología de la instrumentación: 3311.02 Ingeniería de control3324 Tecnología del espacio: 3324.01 Satélites articiales, 3324.07 Control de vehículos

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Introducción

Este proyecto ha sido desarrollado durante 6 meses de periodo de prácticas en el Centre Suissed'Electronique et de Microtechnique, CSEM SA (Centro suizo de electrónica y microtecnologíaen francés), una empresa de investigación y desarrollo situada en Neauchâtel, Suiza.

El estimador de velocidad angular propuesto continua con el trabajo desarrollado anterior-mente por Leopoldo Rossini durante su tesis doctoral, habiendo obtenido ayuda por parte deempleados de CSEM como Emmanuel Onillon u Olivier Chételat, de otras empresas como Ma-xon Motor y organizaciones como la Agencia Espacial Europea (ESA por sus iniciales en inglésEuropean Space Agency), entre otras, relacionada con el desarrollo de un actuador esférico levi-tado por acción magnética para el control de actitud de satélites.

El actuador esférico estudiado, o Esfera de Reacción ha sido desarrollado bajo el trabajo dedistintos proyectos a lo largo de los últimos años. Un primer prototipo fue diseñado y fabricadobajo un proyecto de la Agencia Espacial Europea denominado SPHERE [1]. Posteriormente, unnuevo rotor esférico fue diseñado y ensamblado, el cual fue concebido para mejorar y facilitar suproceso de fabricación [2].

En [3], una nueva electrónica de control y potencia (Elegant BreadBoard en inglés) fuedesarrollada bajo el abrigo de un proyecto europeo FP-7 bajo el nombre de Actuador EsféricoLevitado Europeo (del inglés European Levitated Spherical Actuator, ELSA), realizado conjunta-mente por el Centro suizo de electrónica y microtecnología (CSEM), Maxon Motor, la Sociedadanónima belga de construcciones aeronáuticas (del francés Societé Anonyme Belge de Construc-tions Aéronautiques, SABCA), el Centrum Badan Kosmicznych (CBK), Sener y Redshift. Losrequerimientos empleados en el proyecto fueron escogidos de una misión espacial especíca yrealista (Proba-3). Los resultados del conjunto de proyectos se encuentra incluido en la tesisdoctoral realizada por Leopoldo Rossini [4].

Tras el desarrollo de los proyectos ya mencionados, distintas limitaciones del sistema fueronidenticadas, algunas de ellas relacionadas con la estimación y control de la velocidad angulardel rotor del actuador esférico, lo que llevó a la proposición del presente trabajo n de máster enla empresa CSEM y en colaboración con la Escuela politécnica federal de Lausana (del francésÉcole Polytechnique Fédérale de Lausanne, EPFL), el cual corresponde con el desarrollo de unestimador de velocidad angular alternativo para la ya mencionada esfera de reacción para controlde actitud de satélites.

Objetivos

Ante el comienzo del presente trabajo n de máster, se jaron una serie de objetivos a cumplirdurante el desarrollo del mismo, los cuales son mencionados a continuación:

• Objetivo 1 : Desarrollar un algoritmo implementable en tiempo real, capaz de mejorar lasestimaciones de velocidad angular del rotor de una esfera de reacción empleando los sensoresy dispositivos disponibles en el prototipo actual y mejorando así el rendimiento del sistemaactual sin incrementar su complejidad en hardware y coste.

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• Objetivo 2 : Mejorar y validar el modelo de simulación disponible de la esfera de reacciónen la plataforma Simulink, el cual es requerido para la vericación del comportamiento delestimador propuesto.

• Objetivo 3 : Vericar el comportamiento del algoritmo propuesto empleando el modelo desimulación y datos experimentales obtenidos con el dispositivo real antes de su implemen-tación en el sistema de control en tiempo real.

• Objetivo 4 : Implementar el algoritmo desarrollado en el sistema real y comparar su com-portamiento y rendimiento con el sistema actualmente empleado para la estimación develocidad angular del rotor.

Una vez expuestos los objetivos marcados para el desarrollo del presente trabajo, en laspróximas secciones se presentará, en primer lugar, una breve explicación acerca de distintos con-ceptos introducidos en el trabajo previo realizado por Leopoldo Rossini [4] así como la descripcióndel sistema en el que se empleará el estimador.

A continuación se introducirán los métodos previamente estudiados para la estimación develocidad angular, con el n de comprender su funcionamiento y poder comparar virtudes ydefectos de los mismos. Posteriormente se desarrollará teóricamente el modelo del sistema yel estimador propuesto de velocidad angular, para nalizar con una muestra de los distintosexperimentos y ensayos realizados para analizar el comportamiento del mismo.

Por último, se expondrán las conclusiones extraídas tras el análisis anteriormente mencio-nado, así como las líneas de trabajo futuro que se consideran necesarias para la continuación ymejora del sistema de la esfera de reacción disponible.

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Esfera de reacción

A lo largo de esta sección se explicarán las principales características del diseño de la esfera dereacción realizada en [4], así como distintos conceptos necesarios para la comprensión del algorit-mo de estimación propuesto. Se comenzará por la descripción del diseño general de la esfera dereacción. Posteriormente se mostrarán los resultados del modelo de densidad de ujo magnético,los modelos de fuerza y par, así como la estimación del estado magnético del rotor necesariospara el control en bucle cerrado de la esfera. Se terminará la presente sección con la descripcióndel prototipo desarrollado y de la electrónica de control empleada para su funcionamiento.

Diseño general de la esfera de reacción

El modelo de la Esfera de Reacción estudiada en el presente texto se basa en el diseño de unmotor eléctrico síncrono de 3 grados de libertad con imanes permanentes, en el cual el rotor eslevitado por acción magnética y puede ser acelerado alrededor de cualquier eje. Para ello, el rotoresférico dispone de 8 polos magnéticos creados por imanes permanentes situados en los vérticesde un octaedro, y el estátor, también esférico, presenta 20 electroimanes constituidos por bobinascon núcleo de aire, situados en los vértices de un dodecaedro. Un dibujo esquemático del diseñopuede apreciarse en la Figura 1.

Figura 1: Esfera de reacción con un rotor de 8 polos y un estátor de 20. Rotor completo y medio estátor(izquierda) y medio estátor (derecha).

El último prototipo fabricado consiste en 8 bloques polos de imanes permanentes, magnetiza-dos paralelamente y que poseen forma de esfera truncada, los cuales son jados a una estructurade hierro con forma de octaedro truncado. Anteriormente los polos habían sido discretizadosmediante el uso de 728 imanes cilíndricos para aproximarse el armónico fundamental del campomagnético generado. En el prototipo nal la densidad de ujo magnético creada se encuentracon mayor distorsión, pero se consiguió mayor facilidad de fabricación y estabilidad mecánica.

Para su control, se consideró la inclusión y uso de los siguientes actuadores y sensores:

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• 20 electroimanes con núcleo de aire capaces de aplicar la fuerza y par necesarios al rotor dela esfera de reacción mediante interacción electromagnética con el mismo. Para conseguir lalevitación magnética y velocidad angular requerida se controlarán las corrientes de entradaa dichas bobinas.

• 3 sensores de desplazamiento láser para medir la posición del rotor en el interior del estátor.

• 15 sensores Hall de campo magnético, capaces de medir la densidad de ujo magnético, cuyadistribución permitirá conocer el estado magnético de la esfera de reacción o la orientacióndel rotor dentro del estátor.

Una vez sea explicado en mayor detalle el sistema de control empleado para la esfera dereacción se proporcionará más información acerca del prototipo desarrollado en CSEM.

Modelo de densidad de ujo magnético

El modelo de densidad de ujo magnético desarrollado es necesario para conocer la relaciónexistente entre la rotación u orientación del rotor con el ujo magnético generado por el mismo.A continuación se detallarán los resultados obtenidos y el modelo nal, por lo que si mayorinformación es necesaria,se recomienda al lector la consulta de [4].

Para la obtención del modelo se ha hecho uso de un método basado en la descomposición enarmónicos esféricos de la distribución espacial de la densidad de ujo magnético generada porel rotor esférico de imanes permanentes. Los armónicos esféricos son un conjunto completo defunciones ortogonales denidas en la supercie de una esfera, las cuales son empleadas común-mente para la representación de ciertas funciones como la suma de diferentes armónicos, de unamanera similar al uso de las series de Fourier para expresar funciones periódicas.

El modelo de densidad de ujo magnético es necesario tanto para el desarrollo y formulaciónde los modelos que serán posteriormente mostrados , como para la optimización del diseño delactuador esférico. El presente modelo es obtenido haciendo uso de una técnica híbrida, en la cualse combina el uso de ecuaciones analíticas, que proporcionan información acerca de la distribucióndel ujo magnético en el entrehierro, y simulaciones de elementos nitos junto con medidas realespara la jación de las condiciones de contorno.

El modelo de densidad de ujo magnético para la esfera de reacción se ha obtenido partiendode la solución analítica a las ecuaciones de Laplace y Poisson. Gracias a la elección de un rotormagnetizado según un octupolo cúbico, la distribución de densidad de ujo magnético puedeser expresada como la combinación lineal de un número nito de funciones armónicas esféricas.Inicialmente, estas funciones son expresadas bajo un sistema de referencia ligado al rotor, paradespués realizar el cambio de coordenadas a un sistema ligado al estátor mediante el uso de laspropiedades de los armónicos esféricos bajo rotación.

La densidad de ujo magnéticoB resultante en el punto (rs, θs, φs) del entrehierro del actua-dor, expresado en coordenadas esféricas, puede ser formulado siguiendo la estructura siguiente:

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B(rs, θs, φs) =

Nh∑n=3

n∑m=−n

cmn (α, β, γ)Bmn (1)

donde cmn (α, β, γ) corresponden a los coecientes de descomposición en armónicos esféricosde grado n y orden m, dependientes de la orientación del rotor expresada mediante los ángulos deEuler ZYZ α, β y γ, Bm

n a las contribuciones a la densidad de ujo magnético de cada armónicoesférico de grado n y orden m, y Nh al máximo grado de armónicos esféricos considerado en elmodelo.

Por un lado, la contribución a la densidad de ujo magnético Bmn puede ser calculada ana-

líticamente a priori, puesto que dichos valores son constantes, tal y como es mostrado en [4].Por otro lado, los coecientes de descomposición en armónicos esféricos cmn (α, β, γ) pueden serobtenidos, explotando las propiedades de los armónicos esféricos en rotación, como una combi-nación lineal de dichos coecientes expresados bajo el sistema de referencia ligado al rotor (rotorinmóvil) cln,imm(α, β, γ) del mismo grado n:

cmn (α, β, γ) =∑l

Dnm,l(α, β, γ)cln,imm (2)

donde Dnm,l(α, β, γ) representan matrices de rotación unitarias. Dichos coecientes para l

rotor inmóvil cln,imm(α, β, γ) pueden ser calculados ya sea mediante simulaciones de elementosnitos o mediante datos experimentales del rotor fabricado.

En las próximas secciones se presentará un método para estimar dichos coecientes de des-composición en armónicos esféricos directamente mediante la utilización de las medidas de densi-dad de ujo magnético y el estado magnético, lo cual permitirá su obtención sin el conocimientoexplícito de la orientación y el aligeramiento del algoritmo de control.

Modelos de fuerza y par

Los modelos de fuerza y par que van a ser presentados relacionan la fuerza y el par aplicadoal rotor del actuador esférico como función de las corrientes inyectadas en las 20 bobinas delestátor. Dicha relación dependerá también de la orientación del rotor, la cual será proporcionadaindirectamente mediante el estado magnético empleando un enfoque similar al realizado para elmodelo de densidad de ujo magnético.

Pueden diferenciarse dos modelos de fuerza y par: modelo directo, que proporciona la fuerzay el par aplicado al rotor en función de las corrientes de entrada al actuador, y el modelo inverso,que devuelve el vector de corrientes óptimo para aplicar un determinado par y fuerza.

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Modelos directos

Para la obtención de los modelos, se hará uso de la expresión de la fuerza de Lorentz, y aplicandoel principio de superposición, la fuerza F ∈ <3×1 y el par T ∈ <3×1 aplicado al rotor puede serexpresado mediante la relación:

F = KF (α, β, γ)i

T = KT (α, β, γ)i(3)

donde i ∈ <20×1 representa el vector de corrientes de entrada a las bobinas del estátor,KF (α, β, γ) ∈ <3×20 y KT (α, β, γ) ∈ <3×20 son las matrices características de fuerza y parrespectivamente, del actuador estudiado.

Estas matrices, al igual que con el modelo de densidad de ujo magnético, dependen de laorientación del rotor, y pueden ser caracterizadas haciendo uso de los coecientes de descompo-sición en armónicos esféricos, o estado magnético cmn (α, β, γ), siguiendo las expresiones:

KF (α, β, γ) =

Nh∑n=3

n∑m=−n

cmn (α, β, γ)KmF,n

KT (α, β, γ) =

Nh∑n=3

n∑m=−n

cmn (α, β, γ)KmT,n

(4)

donde KmF,n y K

mT,n representan la contribución de cada armónico esférico de grado n y orden

m a las matrices características de fuerza y par, y cmn (α, β, γ) los valores de estado magnéticodel actuador. De esta manera, las matrices características de fuerza y par son calculadas comocombinación lineal de un conjunto de matrices constantes y calculadas a priori Km

F,n y KmT,n.

Modelos inversos

Para control en bucle cerrado, es necesario conocer qué conjunto de corrientes es necesario aplicara los electroimanes del estátor para aplicar un determinado par y fuerza al rotor. Partiendodel modelo directo anteriormente explicado, la expresión (3) puede ser reformulada en formamatricial:

[FT

]=

[KF (α, β, γ)KT (α, β, γ)

]i = KF,T i (5)

Para poder calcular el conjunto de corrientes de control, el sistema de ecuaciones anterior-mente mostrado debe ser resuelto. Nótese en este caso, que el sistema estudiado es indeterminado,

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ya que el número de ecuaciones, 6, es inferior al número de incógnitas, 20, y por tanto, la soluciónde mínimos cuadrados será empleada:

iLS = K>F,T(KF,TK

>F,T

)−1 [FT

]=[K>F K>T

]([KF

KT

] [K>F K>T

])−1 [FT

]=[K>F K>T

] [KFK>F KFK

>T

KTK>F KTK

>T

]−1 [FT

] (6)

Debido a la simetría en el diseño del rotor y estátor, y al patrón seguido para la magnetizacióndel rotor, las matrices características de fuerza y par son mutuamente ortogonales, cumpliéndosepor tanto KFK

>T = 0, dando lugar a:

i = MFF +MTT (7)

donde las matricesMF yMT son denidas comoMF = K>F (KFK>F )−1, yMT = K>T (KTK

>T )−1

respectivamente. Las fuerzas y pares aplicados en el rotor por las corrientes de las bobinas delestátor denen siempre un espacio tridimensional, y por tanto el rango de las matrices KF y KT

es igual a 3, lo que garantiza siempre la existencia de las matrices MF y MT . La expresión (7)muestra que si el rotor se encuentra perfectamente centrado, no existe interacción entre la fuerzay el par aplicado, lo que permite el control independiente, por un lado, de la posición o levitacióndel rotor, y por otro, de la velocidad angular del mismo.

Estimación de estado magnético

El estado magnético mencionado con anterioridad, es una magnitud ligada con la descomposiciónen armónicos esféricos de funciones como la distribución de densidad de ujo magnético o la fuerzay par aplicados al rotor, la cual contiene la información relacionada con la orientación del rotor.

Durante la explicación del modelo de densidad de ujo magnético, se mencionó un métodopara el cálculo de los coecientes de descomposición en armónicos esféricos mediante el uso delas propiedades de los mismos en rotación, pero con la desventaja de necesitar el conocimientoexplícito de la orientación exacta del rotor con respecto al estátor.

A causa del número de sensores disponibles para la medición del componente radial de ladensidad de ujo magnético, el grado máximo de armónicos esféricos que puede considerarsees Nh = 3. El estado magnético del correspondiente armónico esférico de grado 3 se trata delconjunto de valores complejos denidos como:

c3 = [c−33 , c−23 , c−13 , c03, c13, c

23, c

33]>

xv

Teniendo en cuenta únicamente la componente radial de la densidad de ujo magnético,denido en la posición del sensor de efecto Hall número k, (Rsens, θk, φk) en coordenadas esféricas,la ecuación (1) puede ser simplicada como se muestra a continuación:

B⊥k =3∑

m=−3cm3 (α, β, γ)B⊥3,m,k (8)

en donde se ha empleado una notación simplicada: Brs(Rsens, θk, φk) = B⊥k para la com-ponente radial de la densidad de ujo magnético en la posición del sensor k, y B⊥3,m,k =Bmrs,3(Rsens, θk, φk) para la contribución del armónico de grado 3 y orden m sobre la densidad

de ujo magnético. Mediante la descomposición en parte real e imaginaria del estado magnético,cm3 = am3 + ibm3 , |m| ≤ 3, y aplicando las propiedades de los armónicos esféricos, la funciónestudiada puede expresarse como combinación lineal de coecientes reales de la forma:

B⊥k = a03R03(θk, φk) + 2

3∑m=1

am3 Rm3 (θk, φk) + 2

3∑m=1

Bm3 I

m3 (θk, φk) (9)

siendo Rm3 (θk, φk) = Re[B⊥3,m,k(θk, φk)] y Im3 (θk, φk) = −Im[B⊥3,m,k(θk, φk)]. Considerando

ahora todo el conjunto de sensores de efecto Hall disponible, B⊥ ∈ <Nm×1, y reformulando laecuación anterior en forma matricial se obtiene:

B⊥ = A(Γ)c∗3 (10)

considerando la matriz A(Γ), de dimensiones Nm × 7 como:

A(Γ)> =

R03(ς1) R0

3(ς2) . . . R03(ςNm)

2R13(ς1) 2R1

3(ς2) . . . 2R13(ςNm)

2R23(ς1) 2R2

3(ς2) . . . 2R23(ςNm)

2R33(ς1) 2R3

3(ς2) . . . 2R33(ςNm)

2I13 (ς1) 2I13 (ς2) . . . 2I13 (ςNm)2I23 (ς1) 2I23 (ς2) . . . 2I23 (ςNm)2I33 (ς1) 2I33 (ς2) . . . 2I33 (ςNm)

donde Γ representa el conjunto de coordenadas esféricas del conjunto de sensores Γ =

ς1, ς2, . . . , ςNm, siendo ςk = (θk, φk) para k = 1, 2, . . . , Nm, c∗3 el estado magnético descompues-

to en partes real e imaginaria c∗3 =[a03, a

13, a

23, a

33, b

13, b

23, b

33

]> ∈ <7×1. Nótese que los elementosde la matriz A(Γ) dependen únicamente de la posición de los sensores de efecto Hall, y por tantopueden ser calculados a priori analíticamente.

Resolviendo la ecuación (10) mediante mínimos cuadrados, los valores de c∗3 son calculadosde la forma:

xvi

c∗3 = (A>A)−1A>B⊥ = HprojB⊥ (11)

La matriz A>A es invertible siempre y cuando los sensores no sean mutuamente colineales,y los factores (A>A)−1A> son constantes y pueden ser calculados a priori.

Una vez que se dispone de los valores del estado magnético estimados mediante el uso dela expresión (11), debido a que el formato de los coecientes empleados ha sido modicado, esnecesario reformular los modelos de fuerza y par para hacer uso directamente de los coecientesreales c∗3. Procediendo de una forma similar a la descomposición en partes real e imaginaria de ladensidad de ujo magnético, las matrices características de fuerza y par deben ser modicadasde manera que:

KF = a03K0F,R + 2

3∑m=1

am3 KmF,R + 2

3∑m=1

Bm3 K

mF,I

KT = a03K0T,R + 2

3∑m=1

am3 KmT,R + 2

3∑m=1

Bm3 K

mT,I

(12)

donde KmF,R = Re[Km

F,3], KmF,I = −Im[Km

F,3], KmT,R = Re[Km

T,3] y KmT,R = −Im[Km

T,3] sonvalores constantes y calculados a priori. Esta aproximación empleada para el cálculo de lasmatrices, mediante el uso de armónicos esféricos hasta grado 3, presentará su impacto importanteen la estimación de velocidad angular, debido a que el par y fuerza aplicado al rotor no puedeser conocido con exactitud, por lo que para estimación y control es necesario considerar dichasdiscrepancias.

Una vez explicada la base teórica del modelado de la esfera de reacción, se explicará acontinuación tanto el algoritmo de control empleado como el prototipo y su electrónica de control.

Sistema de control

El bucle de control de la esfera de reacción, tanto para levitación como para velocidad angular,diseñado en [4], es mostrado en la Figura 2. Tal y como se ha mencionado con anterioridad, laesfera de reacción es siempre considerada centrada en el interior del estátor, lo que permite ladescomposición del control de posicionamiento del rotor (levitación) y del control de velocidadangular del mismo, y por tanto emplear bucles de control independientes.

En la presente sección, únicamente el bucle de control de velocidad angular será explicado enmayor detalle, debido a que es el único directamente relacionado con la estimación de velocidadangular. Si se necesita mayor información acerca del mismo o del bucle de control de levitaciónmagnética consúltese [3], [4], [5], o [6].

En la Figura 2, por un lado, el bucle de control mostrado en la parte superior es el encargadode posicionar el rotor dentro del estátor en el punto pref , siendo este valor siempre el origen

xvii

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cción.Fuente:

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xviii

o punto central de la esfera de reacción, y p la estimación de la posición obtenida mediantetriangulación de las medidas de los sensores láser de desplazamiento. La señal de salida delcontrolador de posición es la componente de fuerza del vector de corrientes para los actuadoresde la esfera de reacción iF . Por otro lado, el bucle mostrado en la parte inferior se encarga delcontrol de la velocidad angular del rotor bajo la señal de referencia ωref , y haciendo uso de laestimación de estado magnético obtenido mediante el método mostrado en la expresión (11).La señal de salida del controlador de velocidad angular es la componente de par del vector decorrientes para los actuadores de la esfera de reacción iT .

Para ambos bucles de control, las matrices características de fuerza y par son calculadasmediante la expresión (12), la cual hace uso de los coecientes reales c∗3 para cada periodo demuestreo. Finalmente la salida de ambos controladores es combinada empleando superposicióniF,t = iF + iT .

El controlador implementado en el sistema es simplemente un control proporcional, dondela constante de proporcionalidad Kp = 4 es jada para obtener un ancho de banda del sistemaen bucle cerrado de 6 Hz.

Electrónica de control y sistema de desarrollo

El hardware de control empleado en el prototipo, así como la estructura general de los dispositivosde control requiere una especial mención. En la Figura 3 se muestra el esquema general delhardware de control y las conexiones existentes entre los distintos componentes. Esencialmente,es sistema consta de los siguientes elementos principales:

• Sistema de control central (CCS del inglés Central Control System Rack): elemento centralde control del sistema completo. Recibe las medidas de los sensores, registra los valoresnecesarios según la conguración realizada y ja el tiempo de ejecución del algoritmo decontrol enviando las interrupciones necesarias para su ejecución.

• Control de la electrónica de potencia: actualiza la señal a aplicar al bucle de control decorriente de las bobinas del estátor mediante la ejecución del algoritmo de control a cadainterrupción generada por el CSS empleando las medidas de los sensores proporcionadaspor el mismo.

• Esfera de reacción: prototipo físico de la esfera de reacción, con 20 bobinas (actuadores), 15sensores de efecto Hall y 3 sensores de desplazamiento. También se dispone de 20 sensoresde temperatura para cada electroimán.

• Amplicadores de potencia: proporcionan la corriente y el voltaje necesarios para sensoresy actuadores de la esfera de reacción.

En términos generales, el proceso principal es ejecutado para cada instante de muestreo comose explica a continuación: el sistema de control central recibe las medidas de todos los sensores auna frecuencia de 5 kHz, y una de cada dos, a una frecuencia de 2.5 kHz, una interrupción por

xix

Figura 3: Esquema general de la electrónica de control. Fuente: [7]

software es enviada al dispositivo de control de la electrónica de potencia, así como las medidasrecibidas. Antes de que una nueva interrupción sea recibida, la señal de control de las corrientes esactualizada ejecutando el algoritmo de control mostrado en la Figura 2. Para mayor información,los dispositivos electrónicos, así como las pruebas de calibración, realizados para el desarrollo delsistema de control son explicados en mayor detalle en [7].

La conguración del sistema de control, así como el diseño del mismo, ha sido realizadohaciendo uso de la tecnología proporcionada por dSPACE GmbH. Dicha empresa proporcionatanto hardware como software para el desarrollo, prueba y calibración de unidades de controlelectrónicas (ECUs del inglés Electronic Control Units) para el sector de las ingenierías de auto-moción, aeroespacial y médica. Mayor información acerca de la empresa puede obtenerse en supágina web ocial [8].

El código a ejecutar es desarrollado haciendo uso de la interfaz en tiempo real (RTI delinglés Real-Time Interface) para MATLAB R© y Simulink, los cuales permiten la implementacióndirecta en la plataforma dSPACE.

El prototipo nal de la esfera de reacción empleado se encuentra en las instalaciones dela compañía CSEM SA, el cual es mostrado en la Figura 4a. Además, el set-up del sistema,incluyendo el prototipo (esfera de reacción con actuadores y sensores), electrónica de control yde potencia, así como el controlador empleado ds1005 de dSPACE, es mostrado en la Figura 4b.

xx

(a) Prototipo de la esfera de reacción. (b) Set-up completo de la esfera de reacción.

Figura 4: Prototipo y set-up en laboratorio de la esfera de reacción desarrollada en CSEM SA.

Estimación de velocidad angular

En la presente sección se mostrarán en primer lugar los métodos anteriormente estudiados parala estimación de velocidad angular de la esfera de reacción, y se nalizará con la presentacióndel estimador propuesto en el presente texto. Los métodos estudiados serán: estimador basadoen fuerza contra-electromotriz, estimador basado en Filtro de Kalman Extendido, y un observa-dor de estado basado en un Filtro de Kalman para sistemas lineales con parámetros variables(algoritmo propuesto). Se completará la sección mediante un análisis teórico del bucle cerrado ydel controlador empleado.

Estimadores anteriores

A continuación se mostrarán únicamente las expresiones resultantes de cada método, por loque si se necesita mayor información acerca del desarrollo de las mismas, consúltese la seccióncorrespondiente a la estimación de velocidad angular en inglés del presente trabajo n de másteren 4, o el desarrollo original en [4].

• Estimador basado en fuerza contra-electromotriz: Este estimador es el que ha sidoempleado hasta ahora para el control en bucle cerrado de la esfera de reacción. Presentacomo ventajas su rapidez y simplicidad de cálculo, y como inconvenientes su excesivasensibilidad a las distorsiones existentes en la distribución de densidad de ujo magnéticogeneradas por el rotor.

Este método se basa en la estimación de la fuerza contra-electromotriz inducida en lasbobinas del estátor a través de la ley de Faraday de inducción electromagnética, y mediante

xxi

el principio de conservación de la energía en las mismas, la obtención de la estimación develocidad angular.

Haciendo uso de la ley de Faraday, puede modelarse la fuerza contra-electromotriz inducidaen el conjunto de bobinas uemf , a partir de la variación del ujo magnético que las atraviesaΨav y el número de vueltas de las mismas Nt. Haciendo uso, al igual que en casos anteriores,de la descomposición en armónicos esféricos, dicho ujo magnético puede parametrizarsemediante la utilización del estado magnético c3, dando lugar a la siguiente expresión:

uemf = −Nt

3∑m=−3

[Ψmav,1 Ψm

av,2 · · · Ψmav,20

]T ( d

dtcm3

)= Φ

d

dtc3 (13)

donde Ψmav,k representa la contribución sobre el ujo magnético medio que atraviesa la bobi-

na k del armónico esférico de grado 3 y orden m, y ddtc

m3 la derivada temporal del componen-

te de orden m del estado magnético. Para simplicar la notación, en forma vectorial, estoscomponentes pueden expresarse como uemf = [uemf,1 uemf,2 · · · uemf,20]

T ∈ <20 para

el vector de voltajes inducidos en el conjunto de bobinas, c3 =[c−33 c−23 · · · c33

]T ∈ <7

los coecientes de los armónicos esféricos de grado 3, y Φ ∈ <20×7 la matriz que contieneel conjunto de ujos magnéticos para cada bobina y armónico considerado.

Una vez obtenida la estimación de la fuerza contra-electromotriz inducida en cada bo-bina, mediante conservación de la energía, pueden relacionarse dichas magnitudes con lavelocidad angular del rotor de la forma:

uemf = KTT ω =⇒ ω =

(K>T)+

uemf =(K+T

)>uemf = M>T uemf (14)

donde MT representa la matriz pseudoinversa de la matriz de par caracterítica KT , yaempleada para el control de las corrientes a aplicar a las bobinas, por lo que no es necesarioningún cálculo adicional para su obtención.

Combinando ambos pasos se puede obtener una expresión que relacione directamente lavariación temporal del estado magnético con la velocidad angular del rotor:

ω = M>T Φd

dtc3 (15)

lo que muestra de nuevo la relación existente entre el estado magnético y la orientación delrotor, ya que sus derivadas temporales se encuentran relacionadas mediante la expresiónanterior.

• Estimador basado en ltro de Kalman extendido: este método fue estudiado paraintentar solventar la elevada sensibilidad de la técnica anterior y las limitaciones existentes.Las ventajas que presenta se encuentran relacionadas con el uso de un ltro de Kalman,como puede ser la optimalidad en las estimaciones, así como la reducida sensibilidad alruido existente en las medidas de los sensores. En cambio como principal desventaja es laimposibilidad de ser implementado en tiempo real, debido al elevado tiempo de ejecucióndel algoritmo.

El estimador se basa en el desarrollo de un modelo de espacio de estado del sistema,empleando como vector de estado x ∈ <7 la orientación (expresada en cuaternios) q ∈ <4 y

xxii

velocidad angular del rotor ω ∈ <3. La estimación de las medidas de los sensores B⊥ ∈ <15

(vector de salida del sistema y ∈ <15) se realiza a través de la rotación de armónicosesféricos de grado n y orden m mediante el uso de la orientación del rotor, mostrado enlas expresiones (1) y (2), y representado en la función h3. La entrada del sistema es el parque desea aplicarse al rotor Tcmd ∈ <3. Las ecuaciones del modelo de espacio de estado delsistema continuo son:

x(t) = f(x(t),u(t),w(t)) ⇒ω = J−1Tcmd + w1

q = 12Ωq + w2 = 1

2GTω + w2

y(t) = h(x(t),u(t),v(t)) ⇒ B⊥ = h3(q) + v

(16)

siendo J el tensor de inercia del rotor, el cual por diseño es diagonal e igual para los tresejes principales, y las matrices Ω y GT relacionan la evolución de la orientación q con lavelocidad angular. En cuanto al modelado del ruido, el elemento condicionante es el uso,como matriz de covarianzas del ruido en las medidas de los sensores Rk, es la consideraciónde armónicos de grado superior (hasta grado 11 h11) de forma que:

Rk = diag(abs(h11(qk)− h3(qk)

))(17)

El cálculo de ambas funciones de rotación de armónicos para máximo grado 11 y 3 represen-tan el elemento más pesado, computacionalmente hablando, que impide la implementacióndel presente método en tiempo real. Para mayor información puede consultarse la seccióncorrespondiente en inglés del presente texto o el documento original [9].

Estimador propuesto

Debido a las limitaciones existentes en los métodos expuestos anteriormente para la estimación develocidad angular de la esfera de reacción, una nueva técnica de estimación ha sido desarrolladay será propuesta a continuación. Se expondrá una versión resumida de dicho desarrollo, por loque para su completa comprensión se pide al lector la consulta de la Sección 4.2 del texto originalen inglés.

Estimador basado en ltro de Kalman para sistemas lineales con parámetros varia-bles

Para el método propuesto se intentará combinar las ventajas obtenidas con ambas alternativas,lo que corresponde, por un lado, al uso de la descomposición es armónicos esféricos para laobtención de ecuaciones lineales y simplicadas del modelo gracias al estado magnético, y porotro lado, al uso de un ltro de Kalman para reducir la sensibilidad existente con el métodoempleado actualmente.

Para la implementación de un ltro de Kalman es necesaria la descripción del sistema me-diante un modelo de espacio de estado, y en este caso se busca hacer uso del estado magnético

xxiii

para beneciarse de la rapidez de cálculo obtenida mediante su uso. En este caso, el vector deestado x lo compone simplemente la velocidad angular del rotor ω ∈ <3, la salida del sistemay será la derivada temporal del estado magnético en coecientes reales c∗ ∈ <7 empleado enla expresión (10), y la entrada u el par que se desea aplicar al rotor T cmd ∈ <3. El modelo deespacio de estado para el sistema continuo estudiado, tras las simplicaciones pertinentes, es:

x(t) = Bu(t) + w(t)

y(t) = C(t)x(t) + v(t)

=⇒

ω = J−1Tcmd + w(t)

c∗ = Φ+KTT (c∗)ω + v(t)

(18)

donde se han combinado las expresiones (11) y (16) para su obtención. La versión discreta,necesaria para la implementación en un controlador digital, desarrollada en detalle en la Sección4.2, resulta:

xk+1 = Fxk +Guk + wk

yk = Hkxk + vk

=⇒

ωk+1 = ωk + TsJ−1Tcmd,k + wk

c∗k = Φ+KTT (c∗k)ωk + vk

(19)

siendo F = I3×3,G = TsB = TsJ−1 yHk = Φ+KT

T (c∗k) las matrices del modelo discretizadodel sistema. Nótese la dependencia de la matriz Hk con respecto al estado magnético c∗, el cualrepresenta el conjunto de parámetros variables en el tiempo de los que dependen las ecuacionesdel sistema.

Una vez desarrollado el modelo del sistema, se presentará la implementación del ltro deKalman sobre dicho modelo. Para ello se hará uso de la misma notación empleada en [10],denotando con superíndice − aquellas variables estimadas a priori (antes de recibir las medidas delos sensores), y con + aquellas efectuadas a posteriori (una vez recibidas las medidas). Entre cadainstante de muestreo el ltro de Kalman realiza dos etapas: predicción de estado y actualizacióncon medidas. La primera hace uso del modelo de estado existente para predecir el valor delestado en el instante siguiente, y en la segunda, una vez recibidas las medidas se corrigen lasestimaciones anteriormente realizadas. Además del estado, también se predicen y actualizan lasmatrices de covarianzas del estado, que muestra el grado de conanza en cada medida.

El algoritmo resultante puede expresarse en la ejecución de los siguientes puntos:

1. Inicialización del ltro de Kalman, mediante el uso del estado inicial esperado x+0 y de la

matriz de covarianzas P+0 , para el instante k = 0:

x+0 = E[x0]

P+0 = E

[(x0 − x+

0 )(x0 − x+0 )T

] (20)

2. Para los instantes k = 1, 2, 3 . . . es preciso calcular:

xxiv

(a) Estimación del estado magnético mediante proyección de mínimos cuadrados de lasmedidas de los sensores:

c∗k = HprojB⊥k

c∗k =c∗k − c∗k−1

Ts

KT,k =7∑

m=1

KmT c∗k,m

(21)

siendo c∗k,m el m-ésimo componente del estado magnético en el instante tk.

(b) Estimación a priori del estado del sistema x−k y de la matriz de covarianzas P−k :

x−k = x+k−1 +Guk−1

P−k = P+k−1 +Qk−1

(22)

donde Qk = E[(xk − E[xk])(xk − E[xk])

T]

= E[wkw

Tk

]es la matriz de covarianzas

de la propagación del estado.

(c) Cálculo de la matriz de ganancias del observador Lk, y estimación a posteriori delestado x+

k y matriz de covarianzas P+k :

Lk = P−k HTk (HkP

−k H

Tk +R)−1

x+k = x−k + Lk(yk −Hkx

−k )

P+k = (I − LkHk)P

−k (I − LkHk)

T + LkRLTk

(23)

siendo R = E[(yk − E[yk])(yk − E[yk])

T]

= E[vkv

Tk

]la matriz de covarianzas del

ruido de las medidas de los sensores de efecto Hall.

El modelado del ruido no se ha incluido en el presente resumen para garantizar la continuidaddel desarrollo efectuado y facilitar la comprensión del algoritmo propuesto, pero puede consultarseen la sección 4.3.

Análisis del controlador y sistema en bucle cerrado

Debido a la complejidad y naturaleza de la esfera de reacción, ésta debe funcionar siempreen bucle cerrado. Esta situación implica que el comportamiento del estimador se encuentracondicionada e inuida por la del controlador y viceversa, así como por el resto de componentesdel sistema, como sensores y actuadores, lo que complica el análisis del estimador en sí.

Además, a causa de la naturaleza del sistema estudiado, siendo éste lineal de parámetrosvariables, la estabilidad del sistema es difícil de garantizar analíticamente, ya que no podríanemplearse técnicas lineales para su análisis, requiriendo estudios basados en la estabilidad segúnLyapunov para sistemas no lineales. Debido a la complejidad de realizar dicho análisis teniendoen cuenta elementos como un ltro de Kalman, y a los resultados obtenidos en la práctica, en los

xxv

Compensator

Controller Plant

Observer

r e u y

x

-

δu δy

Figura 5: Bucle de control de velocidad angular.

cuales mediante una elección adecuada de ganancia del controlador, el sistema resulta estable,se ha decidido implementar dicho algoritmo sin la garantía analítica de estabilidad del mismo.

Para el análisis en bucle cerrado del sistema se han seguido las pautas establecidas en [11]para el diseño en espacio de estado. En este caso, el sistema considerado para el control de lavelocidad angular del rotor dispone de: la planta, o esfera de reacción, un observador de estado,ya que no existe medida directa de la velocidad angular, y un controlador. El sistema puederepresentarse en diagrama de bloques tal y como se muestra en la Figura 5.

Tal y como se estipula en [11], cuatro pasos son necesarios para el análisis en bucle cerradode un sistema en espacio de estado, partiendo del sistema estudiado de la forma:

x = Ax +Bu

y = Cx(24)

1. Diseño de la ley de control: un control proporcional es considerado, siendo la señalgenerada por el controlador up,k = Kp(xref,k − xk) = Kpek. Considerando una señal dereferencia nula xref = 0, y un conocimiento perfecto del estado x = x, la dinámica delsistema viene denida por:

x = Ax +Bup = Ax−BKpx (25)

lo que resulta en una ecuación característica del bucle cerrado denida por el polinomio:

det [sI − (A−BKp)] = αc(s) = 0 (26)

2. Análisis de la dinámica del estimador: considerando el error en la estimación, denidocomo x = x− x, y analizando su evolución temporal, resulta:

˙x = x− ˙x

= [Ax +Bup]− [Ax +Bup + L(y − Cx)]

= (A− LC)x

(27)

xxvi

dando lugar a un polinomio característico del sistema en bucle cerrado, considerando úni-camente el estimador, denido como:

det [sI − (A− LC)] = αe(s) = 0 (28)

En este caso la matriz de ganancias del observador L es calculada siguiendo las ecuacionesdeterminadas por el ltro de Kalman. Al no ser constante en el tiempo, la posición de lospolos del observador variarán tanto a causa de la matriz L como de C(t), por lo que elanálisis lineal de estabilidad sólo puede ser considerado como primera aproximación.

3. Combinación del controlador y estimador: combinando el estudio del estado y delerror en la estimación, puede estudiarse el comportamiento de ambos elementos en conjunto:

[x˙x

]=

[A−BKp BKp

0 A− LC

] [xx

](29)

por lo que la ecuación característica del sistema completo viene denida como la combina-ción de los polinomios característicos de la ley de control y del observador, y por tanto eldiseño del controlador y del estimador puede realizarse por separado:

det [sI − (A−BKp)] det [sI − (A− LC)] = αc(s)αe(s) = 0 (30)

4. Inclusión de la señal de referencia: como se ha mencionado con anterioridad, al tratarsede un controlador proporcional, la señal de referencia es introducida como:

up,k = Kp(xref,k − xk) = Kpek (31)

En el sistema estudiado, la única matriz que depende del conjunto de parámetros, el estadomagnético, es la matriz de salida C para sistemas continuos, o H para sistemas discretos, queafecta únicamente a la dinámica del observador. Por otro lado, al emplear un ltro de Kalmancomo estimador, la matriz de ganancias L también varía en el tiempo, pero se ha demostradoexperimentalmente, que para el sistema estudiado, debido a que el estado magnético se encuentraúnicamente denido en el interior del hipercubo [−0,5, 0,5]7 ∈ <7, la variabilidad de la matriz Co H es limitada, y los polos del observador se encuentran también limitados a un cierto intervalonito de valores.

La ganancia del controlador será escogida de tal manera que los polos del controlador seanmás lentos (más cercanos al círculo unidad para sistemas discretos o al eje imaginario parasistemas continuos) que los medidos del observador. Este paso puede ser considerado únicamentecomo primera aproximación, y debe ser ajustado posteriormente mediante simulación primero,y en el prototipo real después.

Tal y como se mostrará en la próxima sección, el uso del estimador propuesto generará unoset en la velocidad angular del rotor, pro lo que debería considerarse, para trabajos futuros,la incorporación de una acción integradora en el controlador para cancelar dicha desviación.

xxvii

Resultados

En la presente sección se presentarán los resultados obtenidos tras la realización de diversosensayos y experimentos llevados a cabo tanto en simulación como en el sistema real. Con el nde guiar al lector a la extracción de las conclusiones más signicativas tras la implementación delestimador propuesto, no se mostrará la totalidad de resultados obtenidos, puesto que extenderíaexcesivamente el contenido del presente resumen, y mediante la elección de especícos ensayosno se perderá información requerida para su análisis.

Tal y como se ha mencionado, el análisis del estimador propuesto se ha realizado tanto ensimulación como en el sistema real, lo cual sigue la tendencia reciente para facilitar y acelerar eldiseño de sistemas. Para ello, ha sido necesario desarrollar y completar un modelo de la esferade reacción, para el cual se ha hecho uso de MATLAB R© y Simulink. Los detalles acerca de losmodelos desarrollados pueden consultarse en la Sección 5.2 del texto original en inglés.

Se han desarrollado dos modelos de simulación con niveles de abstracción distintos: unmodeloreducido en el que se considera como entrada directamente el par aplicado al rotor (descompuestoen par comandado, y par error), y un modelo completo en el que como entrada se tiene el conjuntode corrientes en las bobinas, y considera así, por tanto, la interacción electromagnética entre rotory estátor.

Para ambos casos (simulación y experimental), con el n de realizar una implementaciónprogresiva del estimador propuesto, garantizando la estabilidad del sistema (especialmente elprototipo real), se ha dividido el conjunto de ensayos en dos grupos principales:

• Ensayos en bucle abierto: el estimador propuesto es ejecutado en paralelo con el buclecerrado de control, por lo que las estimaciones del mismo no son consideradas por el al-goritmo de control. Estos ensayos permiten el análisis y corrección de posibles fallos en laimplementación del estimador de forma independiente, lo que garantiza el correcto funcio-namiento del sistema. Para el bucle de control se ha hecho uso del método basado en lafuerza contra-electromotriz para la estimación de velocidad angular.

• Ensayos en bucle cerrado: El estimador propuesto es incluido en el bucle de control ensustitución del método anteriormente empleado para la estimación de velocidad angular.

Para cada grupo de ensayos se mostrará el experimento más representativo dentro de losmismos. Para mayor información y detalle, consúltese el capítulo 5 del texto principal.

xxviii

Modelo de simulación completo

El modelo de simulación sobre el cual se mostrarán los resultados tras la implementación del esti-mador de velocidad angular es mostrado en la Figura 6, donde pueden apreciarse tres elementosprincipales:

• Esfera de reacción: modelada mediante un bloque electromagnético, que simula la interac-ción electromagnética entre estátor y rotor, por lo que computa la fuerza y par aplicadosal rotor a partir de la orientación del rotor, y las medidas de los sensores en función dela misma. El bloque dinámico simula la evolución temporal de la orientación y velocidadangular del rotor a partir del par aplicado. Para la simulación de la esfera se hará usode armónicos esféricos hasta grado 11, con el n de obtener resultados más cercanos a larealidad.

• Estimador de velocidad angular : ejecuta el algoritmo propuesto para la estimación de ve-locidad angular del rotor. Dispone de un estimador de estado magnético, como se muestraen la expresión (11), una derivación discreta y el algoritmo de ltro de Kalman explicadocon anterioridad.

• Generador de corriente de control : el cual computa la corriente necesaria para aplicar laseñal de par. Para dicho cálculo, se hace uso del estado magnético, y por tanto se consideraúnicamente el armónico de grado 3, por lo que este cálculo no es exacto, lo que repercutiráen una discrepancia entre el par realmente aplicado y la señal de control.

Reaction Sphere

Alngular Velocity Estimator

Control

Current

Estimation

Electromagnetic

Model

Kinematic & Dynamic

Model

Kalman

Filterz−1z

cEstimation

T cmd icmd T ω

q

B⊥

B⊥ c

˙c ω

Figura 6: Setup del modelo de simulación completo de la esfera de reacción.

xxix

Ensayos en bucle abierto

• Ensayos en simulación: Ensayo no 5. Se aplica un escalón en la señal de control de parde amplitud 0.1 Nm durante 0.3 segundos en la dirección [1, 1, 1]. La frecuencia de muestreoempleada es de 2.5 kHz que corresponde con la frecuencia del sistema real, y la desviacióntípica del ruido de los sensores es denida en 0.1 mT.

Este experimento muestra que la velocidad angular estimada sigue la velocidad angularsimulada por el modelo, pero se encuentra distorsionada por la diferencia existente entre laseñal de control del par (empleada como entrada al estimador propuesto) y el par realmenteaplicado al rotor (calculada a partir del conjunto de corrientes generado por el controlador).A pesar de dichas diferencias, el estimador es capaz de seguir la velocidad angular real dela esfera de reacción simulada.

Esta situación combina la inuencia de armónicos esféricos de grado superior (superioresa 3), no modelados para el estimador, pero sí para el cálculo de la fuerza y par realesaplicados al rotor, así como para la generación de las medidas de los sensores de efectoHall.

Puede observarse que el error máximo obtenido en este experimento se encuentra alrededorde los 5 rpm para todas las componentes (solamente se ha mostrado la componente x, perolos resultados son equivalentes para y y z).

• Ensayos experimentales: Ensayo no 7. Se aplica una señal escalonada de referenciade velocidad angular al bucle de control que emplea el estimador basado en la fuerzacontra-electromotriz, mientras se ejecuta el algoritmo propuesto en paralelo haciendo usode las medidas obtenidas por los sensores. La velocidad máxima alcanzada es de 300 rpmen dirección [1, 1, 1]. La frecuencia de muestreo empleada es de 2.5 kHz tanto para elcontrolador como para el estimador.

El principal objetivo de este ensayo es estudiar el posible rendimiento del estimador pro-puesto, y compararlo con el obtenido mediante el método anteriormente empleado basadoen la fuerza contra-electromotriz (back-EMF en inglés). Los resultados obtenidos se mues-tran en la Figura 8. Sólo se ha incluido la componente x en las grácas para facilitar sulectura e interpretación, pero los resultados son equivalentes para los otros dos ejes.

En la parte superior de la gura puede observarse que las estimaciones arrojadas por elobservador propuesto siguen la señal de referencia introducida al controlador, y en la parteinferior puede compararse el comportamiento de ambos estimadores. A bajas velocidadeslas oscilaciones obtenidas son similares para ambos, pero en cuanto la señal de referenciacomienza a aumentar, puede apreciarse una mejora considerable con el ltro de Kalmanpropuesto. Esta situación hace esperar que las oscilaciones sean reducidas aún más con eluso del estimador propuesto en el bucle cerrado.

Debido a que el método basado en la fuerza contra-electromotriz es empleado para elcontrol, las oscilaciones existentes en la velocidad angular no son fruto únicamente deerrores en la estimación, sino de desviaciones reales de la velocidad angular del rotor conrespecto a la señal de referencia.

xxx

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

0

5

10

·10−2

Time (s)

Torque(N

m)

Command and Applied Torque along X-axis

Command Tx

Applied Tx

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

−140

−120

−100

−80

−60

Time (s)

AngularVelocity

(rpm)

Angular Velocity along X-axis

Real ωx

Estimated ωx

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7−10

−5

0

5

10

Time (s)

AngularVelocity

(rpm)

Angular Velocity Error along X-axis (Real-Estimation)

Figura 7: Resultados del ensayo no 5. Señal de control de par y par aplicado al rotor (superior). Com-ponente x de la velocidad angular real y estimada (centro). Componente x del error de estimación en lavelocidad angular: error = real − est (inferior).

xxxi

Figura 8: Resultados del ensayo no 7. Componente x de la señal de referencia y la estimación de lavelocidad angular mediante ltro de Kalman (superior). Componente x de la estimación de la velocidadangular mediante ltro de Kalman y mediante fuerza contra-electromotriz (inferior).

xxxii

Ensayos en bucle cerrado

• Ensayos en simulación: Ensayo no 10. Se pretende simular una situación similar a lamostrada en el ensayo 7. Para ello se incrementa de forma escalonada la señal de referenciade velocidad angular hasta alcanzar un valor de 300 rpm en dirección [1, 1, 1]. La frecuenciade muestreo establecida es también de 2.5 kHz.

Al tratarse del primer ensayo en el que se incluye el estimador propuesto en el bucle decontrol es preciso realizar los ajustes pertinentes en el controlador y observador para obtenerel comportamiento deseado. Para ello se han congurado y ajustado ambos siguiendo lasdirectrices marcadas durante el análisis en bucle cerrado del sistema. Haciendo el estimadormás inmune al ruido y oscilaciones puede conseguirse una reducción considerable del mismo,pero a costa de una menor velocidad de convergencia hacia el estado real del sistema, por loque un compromiso debe ser alcanzado. Una vez jada la rapidez del estimador, la gananciadel controlador se ja para que los polos del mismo sean ligeramente más lentos, con eln de evitar saturaciones en la señal de control debido a que el controlador quiera corregircon demasiada celeridad un estado que todavía no ha convergido al valor real debido a ladinámica introducida por el observador.

Los resultados obtenidos en el presente ensayo pueden verse en la Figura 9, en donde semuestra en la parte superior una comparación entre la señal de referencia y la estimaciónproporcionada por el estimador, mientras que en la gura inferior se representa el errorobtenido (referencia - estimación). Puede observarse que el error en régimen transitorioalcanza valores de hasta 10 rpm para cada componente de velocidad angular, mientras queen régimen permanente se mantiene inferior a 5 rpm.

• Ensayos experimentales: Ensayo no 17. Se pretende simular, en este caso también,una situación similar a la mostrada en el ensayo 7, generando una señal de referenciade velocidad angular escalonada hasta alcanzar unos 300 rpm en dirección [1, 1, 1]. Lafrecuencia de muestreo del sistema real es de 2.5 kHz.

En el presente ensayo con el prototipo real se ana el ajuste de los parámetros del con-trolador y observador realizado en el ensayo anterior hasta obtener un comportamientoaceptable. Se muestra en la Figura 10 el resultado de la ejecución del algoritmo de controlhaciendo uso del estimador propuesto. Se ha intentado vericar la mayor velocidad angularque se puede alcanzar con el sistema actual, habiéndose, al menos, igualado la obtenidacon el método anterior.

Puede observarse que, a diferencia de como ocurría en los anteriores casos, un oset enla velocidad angular ha aparecido, lo que empuja a considerar la implementación de unaacción integradora en el controlador para compensar dicha desviación. Por otro lado, uncomportamiento deteriorado es obtenido para velocidades angulares obtenidas, pero escorregido para velocidad angulares superiores, para las cuales las oscilaciones son menores.Además del oset anteriormente mencionado, las oscilaciones existentes pasan de ser denaturaleza ruidosa (ruido blanco) a presentar forma sinusoidal, por lo que un mejor ajustedel controlador podría solventarlas o reducirlas.

En denitiva, para velocidades reducidas, un mejor comportamiento es obtenido si se con-sidera un estimador y controlador más rápidos, tal y como se muestra en el experimentono 12 (no incluido en el presente resumen), mientras que con la disposición del ensayo no

17, un mejor comportamiento es obtenido para altas velocidades de rotación, en donde las

xxxiii

Figura 9: Resultados del ensayo no 10. Componente x de la señal de referencia y estimación de velocidadangular de la esfera de reacción (superior). Componente x del error en la estimación de velocidad angular:ωerr = ωref − ωest (inferior).

xxxiv

Figura 10: Resultados del ensayo no 17. Señal de referencia y estimación de velocidad angular (superior).Error en la estimación de velocidad angular: ωerr = ωref − ωest (inferior).

oscilaciones son mucho menos agresivas que con el método anteriormente empleado y demenor amplitud.

Comparación de estimadores

En el presente apartado se realizará una comparación más en detalle de los dos métodos deestimación de velocidad angular implementables en el prototipo real: el estimador basado en lafuerza contra-electromotriz empleado anteriormente, y el método basado en un ltro de Kalmanpropuesto en el presente texto. El tercer método existente basado en un ltro de Kalman exten-dido no se ha considerado en la comparación debido a su imposibilidad a ser implementado entiempo real.

Para llevar a cabo dicha comparación, se han escogido ensayos en los que se disponga de unasituación equivalente de la esfera de reacción mediante el uso de ambos estimadores en el sistema

xxxv

Rango de Velocidad Back-EMF ErrorLPV-KF Error

(con oset)

LPV-KF Error

(sin oset)

Baja velocidad (≤ 50 rpm) 6.63 rpm 14.78 rpm 14.78 rpm

Alta velocidad (> 50 rpm) 29 rpm 23.6 rpm 11.05 rpm

Cuadro 1: Comparación de estimadores de velocidad angular. Máximo error absoluto para cada estimador:ωAE = |ωref − ωest|.

real. Por ello se ha decidido comparar los ensayos no 7 y no 17, cuyos resultados son mostradosen las guras 8 y 10 respectivamente.

Puesto que con anterioridad únicamente se han mostrado los resultados del ensayo 7 haciendouso del estimador propuesto y no del originalmente empleado para control, en la Figura 11 seincluyen las estimaciones realizadas mediante el método basado en la fuerza contra-electromotriz.Esta gura debe compararse con los resultados mostrados en la Figura 10. Para resumir lasprincipales características de los resultados de cada estimador se ha incluido a continuación laTabla 1.

En dicha tabla, se ha clasicado la comparación en dos secciones diferenciadas: baja veloci-dad, para velocidades inferiores o iguales a 50 rpm para cada componente, y alta velocidad paravalores superiores. Por otro lado, debido a la existencia de un oset en la velocidad angular resul-tante haciendo uso del estimador propuesto, se han comparado 3 posibles situaciones: Back-EMFpara el estimador basado en fuerza electromotriz, LPV-KF con oset para el estimador basadoen ltro de Kalman para sistemas lineales con parámetros variable (LPV-KF de sus siglas eninglés), y LPV-KF sin oset como el anterior, pero eliminando el oset, con el n de medir lasoscilaciones máximas obtenidas y dar una primera impresión acerca de unos posibles resultadostras la implementación de una acción integral en el controlador.

Puede observarse que para bajas velocidades, el primer estimador presenta mejores resulta-dos, con aproximadamente la mitad de error absoluto máximo, mientras que para altas velocida-des la situación se invierte. Por otro lado, el error máximo obtenido con el estimador propuestopuede reducirse, como ya se ha mencionado con anterioridad, mediante la modicación de losparámetros de observador y controlador, obteniéndose errores similares comparados con el mé-todo anterior (Back-EMF), tal y como se muestra en la gura 5.19 no incluida en el presenteresumen.

Un desarrollo más extenso acerca de las conclusiones extraídas en los distintos experimentosrealizados y en la presente comparación ha sido recogido en la próxima sección.

xxxvi

Figura 11: Experiment 17 Results with Back-EMF Estimator. Stepped reference angular velocity andBack-EMF's estimation (top). Angular velocity error: ωerr = ωref − ωest (bottom).

xxxvii

Conclusiones

Una vez mostrados los resultados obtenidos tras la implementación en la estera de reacción,del estimador de velocidad angular propuesto, basado en un ltro de Kalman para sistemaslineales con parámetros variables, se presentarán a continuación las conclusiones extraídas de lacomparación entre este método con la técnica anteriormente empleada basada en la estimaciónde la fuerza contra-electromotriz.

El principal objetivo jado para el presente trabajo n de máster fue el desarrollo de un nuevométodo de estimación de velocidad angular para el prototipo disponible de esfera de reacciónpara control de actitud de satélites. Teniendo esto en mente se ha llevado a cabo el análisis ycomparación del método propuesto, habiéndose extraído las siguientes conclusiones:

• Un nuevo estimador de velocidad angular basado en un ltro de Kalman para sistemaslineales con parámetros variables ha sido presentado y desarrollado teóricamente, y validadoy analizado en la práctica tanto en simulación como en el sistema real de la esfera dereacción.

• El observador de estado propuesto ha sido implementado en tiempo real el el hardware decontrol empleado en el prototipo, afectando en muy poca medida en el tiempo de ejecucióncon respecto al método anteriormente empleado.

• Haciendo uso de la descomposición en armónicos esféricos los cálculos a realizar en el buclede control son simplicados, lo que permite la implementación de un ltro de Kalman entiempo real, lo cual no es posible mediante el uso de la orientación en cuaternios comoestudiado en [9].

• Mediante el uso del estimador propuesto, para bajas velocidades de rotación del rotor(inferiores a 50 rpm para cada componente) similar o peor comportamiento es obtenido encomparación con el estimador anterior.

• Sin embargo, para velocidades de rotación superiores (mayores que 50 rpm para cada com-ponente), las oscilaciones son reducidas considerablemente mediante el uso del observadorpropuesto.El máximo error obtenido es reducido en un factor de dos o tres para una veloci-dad angular de módulo 300 rpm, y las vibraciones anteriormente existentes son canceladas.

• El mejor comportamiento para bajas y altas velocidades del ltro de Kalman es obtenidocon parámetros distintos del mismo, por lo que podría considerarse el uso de un observadoradaptativo que modicase sus parámetros en función de la referencia introducida para lavelocidad angular.

• Si se hace uso del observador propuesto, un oset es obtenido en la velocidad angular, yel error existente es sinusoidal, el cual podría reducirse mediante la incorporación de unaacción integral en el controlador y mediante un mejor ajuste de sus parámetros.

• La estabilidad del bucle cerrado del sistema con la inclusión del ltro de Kalman propuestono ha sido garantizada analíticamente, pero los experimentos realizados muestran una con-vergencia de las estimaciones al valor real de velocidad angular y que ésta se mantiene convalores nitos en todo momento empleando los parámetros adecuados para el controlador.

xxxviii

ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

De las conclusiones extraídas y enumeradas anteriormente, puede apreciarse el cumplimientode todos los objetivos marcados al inicio del comienzo del presente trabajo n de máster, y elcomportamiento obtenido con el estimador propuesto dan lugar a una situación prometedorapara la continuación del desarrollo del sistema existente de la esfera de reacción.

El sistema existente, a pesar del progreso realizado en el presente texto, se encuentra aúnlejos de la perfección requerida para su uso en misiones espaciales debido a la existencia aún deuna serie de limitaciones, las cuales se muestran a continuación:

1. Existencia de oscilaciones sinusoidales en la velocidad angular del rotor al hacer uso delestimador propuesto y un controlador proporcional. El mejor ajuste de los parámetros delcontrolador debería reducir estas oscilaciones.

2. Aparición de un oset en la velocidad angular mediante la utilización del algoritmo estu-diado en bucle cerrado. La inclusión de una actuación integradora en el controlador podríaeliminar dicha desviación.

3. Mejor comportamiento para bajas y altas velocidades de rotación es obtenida con distintosvalores de los parámetros del ltro de Kalman, por lo que una variación adaptativa segúnla referencia de velocidad angular puede ser estudiada para mayor mejora del sistema.

4. Existencia de perturbaciones repetidas periódicamente debidas, por ejemplo, a armónicosesféricos de grado superior no considerados en el algoritmo de control. El estudio de laimplementación de un ltro notch generalizado, como los empleados para el control de ro-damientos magnéticos presentados en [12], ayudará a la reducción de dichas perturbaciones.

5. Durante todo el estudio del bucle de control de velocidad angular no se ha consideradointeracción alguna con el bucle de control de posicionamiento del rotor, lo cual únicamentees válido si éste se encuentra perfectamente centrado en todo momento, lo cual no es estric-tamente cierto. Una consideración de la interacción entre ambos bucles debe ser estudiadapara la mejora del comportamiento de ambos.

Mediante el estudio de las limitaciones mencionadas, una mejora en el comportamientodel sistema completo puede obtenerse, lo que permitiría un aumento de la velocidad máximade rotación adquirible por la esfera de reacción, así como una reducción en las vibraciones delsistema, mejorando la durabilidad del mismo.

Guzman Borque Gallego xxxix

CAPÍTULO 0. RESUMEN

xl ETSII-UPM & EPFL

ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

Contents

Acknowledgements iii

Abstract v

Resumen vii

List of Figures xlv

List of Tables xlix

Acronyms li

1 Introduction 1

1.1 Project Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Work Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 ACS and Spherical Actuators Overview 5

2.1 Satellite Attitude Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Attitude Determination and Control System (ADCS) . . . . . . . . . . . . 6

2.1.2 Attitude Control Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Attitude Control Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Reaction Sphere Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 3-DoF Spherical Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Inductive Electromagnetic Spherical Actuators . . . . . . . . . . . . . . . 12

2.3.2 Variable Reluctance Electromagnetic Spherical Actuators . . . . . . . . . 14

2.3.3 Permanent Magnet Synchronous Electromagnetic Spherical Actuators . . 14

Guzman Borque Gallego xli

CONTENTS

3 Reaction Sphere 19

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Reaction Sphere Concept Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Magnetic Flux Density Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Force and Torque Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.1 Force and Torque Forward Models . . . . . . . . . . . . . . . . . . . . . . 23

3.4.2 Force and Torque Inverse Models . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Magnetic State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Control Scheme and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6.1 Angular Velocity Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6.2 Prototype Electronics and Development Environment . . . . . . . . . . . . 29

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Angular Velocity Estimation 33

4.1 Previous Work on Angular Velocity Estimation . . . . . . . . . . . . . . . . . . . 34

4.1.1 Back-EMF-based Angular Velocity Estimator . . . . . . . . . . . . . . . . 34

4.1.2 Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Linear Parameter-Varying Kalman Filter Development . . . . . . . . . . . . . . . 38

4.2.1 State-Space Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.2 Linear Parameter-Varying Kalman Filter . . . . . . . . . . . . . . . . . . . 40

4.2.3 Final Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Controller and Closed-loop Design . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.1 Full State Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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5 Simulation and Experimental Results 51

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 Simulation Models Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2.1 Reduced Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2.2 Complete Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Open-loop Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4 Closed-loop Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.5 Back-EMF and LPV Kalman Filter Comparison . . . . . . . . . . . . . . . . . . . 83

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6 Conclusion and Future Work 87

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

A Time Planning and Budget 91

B Simulation Model Validation 99

Bibliography 103

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CONTENTS

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List of Figures

1 Modelo esquemático de la esfera de reacción estudiada . . . . . . . . . . . . . . . xi

2 Diagrama de bloques del bucle de control de la esfera de reacción. . . . . . . . . . xviii

3 Esquema general de la electrónica de control. . . . . . . . . . . . . . . . . . . . . xx

4 Prototipo y set-up en laboratorio de la esfera de reacción desarrollada en CSEMSA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

5 Bucle de control de velocidad angular. . . . . . . . . . . . . . . . . . . . . . . . . xxvi

6 Setup del modelo de simulación completo de la esfera de reacción. . . . . . . . . . xxix

7 Resultados del ensayo no 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi

8 Resultados del ensayo no 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxii

9 Resultados del ensayo no 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv

10 Resultados del ensayo no 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv

11 Experiment 7 Results with Back-EMF Estimator . . . . . . . . . . . . . . . . . . xxxvii

2.1 Reaction Sphere proposed by Ormsby. . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Schematic of the Reaction Sphere proposed by Downer et al. . . . . . . . . . . . . 10

2.3 Schematic of the Reaction Sphere proposed by Chételat. . . . . . . . . . . . . . . 11

2.4 Induction Spherical motor proposed by Williams et al. . . . . . . . . . . . . . . . 12

2.5 2-DoF Induction Spherical Motor proposed by Dehez et al. . . . . . . . . . . . . . 13

2.6 3-DoF Induction Spherical Motor proposed by Kumagai and Hollis . . . . . . . . 13

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LIST OF FIGURES

2.7 3-DoF Variable Reluctance Spherical Motor proposed by Lee et al. . . . . . . . . 14

2.8 3-DoF Spherical Wheel Motor proposed by Lee et al. . . . . . . . . . . . . . . . . 15

2.9 3-DoF PM Spherical Motor proposed by Yan et al. . . . . . . . . . . . . . . . . . 15

2.10 3-DoF Spherical Actuator with cylindrical PM proposed by Yan et al. . . . . . . 16

2.11 CAD model of 6-DoF PM Spherical Actuator for haptic applications proposed byBai et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.12 3-DoF PM Stepper Motor proposed by Chirikjian and Stein . . . . . . . . . . . . 17

2.13 3-DoF PM Spherical Actuator proposed by Chen et al. . . . . . . . . . . . . . . . 17

2.14 Multi-pole synchronous spherical motor proposed by Yano et al. . . . . . . . . . . 18

2.15 Polyhedron-based spherical stepper motor proposed by Yano et al. . . . . . . . . 18

3.1 Reaction Sphere's Model Schematics . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Main Control Scheme for magnetic bearing and angular velocity control of theReaction Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Main Control Electronics block schematic. . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Reaction Sphere's prototype and laboratory set-up of the system. . . . . . . . . . 31

3.5 ECUs manufactured by dSPACE used in the System . . . . . . . . . . . . . . . . 31

4.1 Graphical Representation of Angular Velocity Estimator. . . . . . . . . . . . . . . 43

4.2 Angular Velocity Control Loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.1 Reaction Sphere's Reduced Model for simulation. . . . . . . . . . . . . . . . . . . 53

5.2 Reaction Sphere's Complete Model for simulation. . . . . . . . . . . . . . . . . . 54

5.3 Simulation Set-Up for Reduced Model of the Reaction Sphere. . . . . . . . . . . . 56

5.4 Experiment 1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.5 Experiment 2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.6 Experiment 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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5.7 Simulation Set-Up for Complete Model of the Reaction Sphere. . . . . . . . . . . 60

5.8 Experiment 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.9 Experiment 5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.10 Experiment 6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.11 Experiment 7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.12 Experiment 8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.13 Experiment 9 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.14 Simulation Set-Up for Closed-loop analysis of the Reaction Sphere control system. 70

5.15 Experiment 10 Results (Nmax = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.16 Experiment 10 Results (Nmax = 11) . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.17 Experiment 11 Results (Nmax = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.18 Experiment 11 Results (Nmax = 11) . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.19 Experiment 12 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.20 Experiment 13 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.21 Experiment 14 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.22 Experiment 15 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.23 Experiment 16 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.24 Experiment 17 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.25 Experiment 7 Results with Back-EMF Estimator . . . . . . . . . . . . . . . . . . 84

B.1 Reaction Sphere's set-up for model validation. . . . . . . . . . . . . . . . . . . . . 100

B.2 Model Validation. Experimental-Simulated Comparison . . . . . . . . . . . . . . 101

B.3 Model Validation. Experimental-Simulated Results . . . . . . . . . . . . . . . . . 102

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LIST OF FIGURES

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List of Tables

1 Comparación de estimadores de velocidad angular. . . . . . . . . . . . . . . . . . xxxvi

5.1 Simulation Experiments for Reduced Model. . . . . . . . . . . . . . . . . . . . . . 56

5.2 Simulation Experiments for Complete Model. . . . . . . . . . . . . . . . . . . . . 61

5.3 Experiments in Real Prototype with open-loop estimator. . . . . . . . . . . . . . 65

5.4 Closed-Loop experiments in Simulation. . . . . . . . . . . . . . . . . . . . . . . . 70

5.5 Closed-Loop experiments in the real prototype. . . . . . . . . . . . . . . . . . . . 75

5.6 Angular Velocity Estimators comparison. . . . . . . . . . . . . . . . . . . . . . . . 83

A.1 Cost for developing the current Master's Thesis. . . . . . . . . . . . . . . . . . . . 98

B.1 Reaction Sphere Model Validation Comparison for Nmax = 3, 7, 9, 11. . . . . . . 100

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LIST OF TABLES

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Acronyms

ADCS Attitude Determination and Control System. v, 6

CBK Centrum Badan Kosmicznych. ix, 2

CMG Control Moment Gyroscope. 79

CSEM Centre Suisse d'Electronique et de Microtechnique. iii, vix, xii, xx, 2, 19, 85

DoF Degrees of Freedom. 11, 12, 14, 15, 17

EBB Elegant BreadBoard. ix, 2

EKF Extended Kalman Filter. v, 34, 36, 38, 83

ELSA European Levitated Spherical Actuator. ix, 2

EPFL École Polytechnique Fédérale de Lausanne. ix, 2

ESA European Space Agency. ix, 2

KF Kalman Filter. v, vi, 33, 37, 38, 4044, 46, 48, 49, 51, 55, 62, 65, 68, 69, 8385, 8790

LPV Linear Parameter-Varying. v, vi, 33, 39, 40, 42, 43, 46, 48, 49, 51, 8385, 8790

PM Permanent Magnet. v, 11, 1418, 20

RS Reaction Sphere. v, vi, xlvi, 2, 5, 6, 8, 9, 11, 1931, 33, 34, 46, 48, 5155, 57, 60, 62, 80, 83,85, 8790

RW Reaction Wheel. v, 79

SABCA Societé Anonyme Belge de Constructions Aéronautiques. ix, 2

SH Spherical Harmonic. v

SWM Spherical Wheel Motor. 14

VR Variable Reluctance. 14

VRSM Variable Reluctance Spherical Motor. 14

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Chapter 1

Introduction

The present document focus on the development and implementation of an angular velocityestimator of a reaction sphere used for satellite attitude control. This chapter will introduce thework developed during this project, including its background, motivation and objectives xed forthis thesis. Finally, the chapter will conclude with a brief overview of the document's structureand the content that will be developed in each section.

The chapter will position the reader in the context and general background of the projectunder which the described Reaction Sphere has been developed.

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CHAPTER 1. INTRODUCTION

1.1 Project Background and Motivation

This project has been developed during an Internship and Master's Thesis at the Swiss Centre ofElectronics and Microtechnology, CSEM SA (Centre Suisse d'Electronique et de Microtechniquein French), a Research and Development company located at Neuchâtel, Switzerland.

The proposed angular velocity estimator continues with the work previously done at CSEMmainly by Leopoldo Rossini during the development of his PhD, also with the help from Em-manuel Onillon and Olivier Chételat of CSEM, Yves Perriard of École Polytechnique Fédérale deLausanne (EPFL), or other companies and organisations as Maxon Motor or the European SpaceAgency (ESA), among others, related to the development of a magnetically-levitated sphericalactuator for attitude control for satellites.

This spherical actuator, or Reaction Sphere, has been developed within dierent projectsin the last years. A rst prototype was designed and manufactured under a European SpaceAgency (ESA) project named SPHERE [1]. Afterwards, a new spherical rotor was was designand assembled, optimised to improve the manufacturability, as shown in [2].

In [3], a novel Elegant BreadBoard (EBB) of the new RS design is presented. The EBBwas developed under the frame of a European FP-7 project named European Levitated SphericalActuator (ELSA), which was carried out by the Centre Suisse d'Electronique et de Microtechnique(CSEM), Maxon Motor, the Societé Anonyme Belge de Constructions Aéronautiques (SABCA),the Centrum Badan Kosmicznych (CBK), Sener, and Redshift. This breadboard requirementswere based on the ones of a realistic and specic space mission (Proba-3). The results of theoverall progress are included in [4].

After the development of these projects, dierent limitations, specially regarding the angularvelocity control and estimation were identied, which led to the proposal of the developmentof the current internship and Master's Thesis at CSEM and EPFL, which corresponds to thedevelopment of an Angular Velocity Estimator of the aforementioned Reaction Sphere intendedto be applied for Satellite Attitude Control.

1.2 Objectives

For any type of human work, the establishment of concrete objectives is essential for the correctand ecient development of the aforementioned work before its beginning, and in this sectionthe objectives of the present document are going to be mentioned.

These goals were xed in order to improve some of the main problems found during thedevelopment of the reaction sphere prototype that has been used for the angular velocity estim-ation. As it is going to be explained during this text, the control of the angular velocity of therotor is one of the key elements of the system, as its variation will generate the torque requiredfor the rotation of the satellite.

Then, the main objectives xed for this thesis are explained as follows:

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• Objective 1 : Develop an algorithm, implementable in Real-Time, capable of improvingthe estimations of the angular velocity of a reaction sphere's rotor by using the availablesensors and controller, and thus, improving the performance of the current system withoutincreasing its complexity and cost.

• Objective 2 : Improve and validate the Simulated Reaction Sphere model already imple-mented using Simulink, which will be required for algorithm tests.

• Objective 3 : Test the algorithm using simulation and available experimental data in orderto analyse its behaviour before its implementation.

• Objective 4 : Implement, if possible, the developed algorithm in the real system and compareits behaviour and performance with the current estimations.

1.3 Work Overview

The work developed during this Master's Thesis has been structured in the following chapters,and its content is explained below:

• Chapter 2: Attitude Control Systems and Spherical Actuators Overview. Anbrief introduction to dierent concepts related with the developed project, such as attitudecontrol systems for satellites is performed, as well as a general vision of dierent sphericalactuators, and their control, in which the reaction sphere studied in this document can beclassied.

• Chapter 3: Reaction Sphere. During this chapter the modelling and nal prototypeof the reaction sphere is going to be explained. This will include the magnetic ux densitymodel, the force and torque models, the magnetic state estimation and an overview of thereal prototype and its control architecture.

• Chapter 4: Angular Velocity Estimation. The chapter will start with a brief in-troduction of previous approaches used for estimating the angular velocity. Afterwards aKalman Filter estimator is going to be derived, including system and noise modelling.

• Chapter 5: Simulation and Experimental Results. During this chapter the resultsobtained both in simulation and real-time experiments are going to be shown and studiedin order to analyse the behaviour of the developed angular velocity estimator.

• Chapter 6: Conclusion and Future Work. The main body of this document will nishwith the conclusions that can be extracted from the work done during this Master's Thesisand the future development required for solving the current limitations of the reactionsphere implementation.

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Chapter 2

Attitude Control Systems and Spherical

Actuators Overview

In this chapter, the state of the art of satellite attitude control systems and spherical actuatorswill be developed. It will start by explaining the dierent satellite attitude actuators that areavailable in real systems and a classication of them will be shown. Afterwards a historicalbackground of spherical actuators, in which the reaction sphere can be classied, is included.

This chapter will serve as an introduction and as an historical overview of the dierentapproaches proposed until now, related to the application in which the studied reaction spherewill be used. In the following chapters, the actuator by itself will be explained and described.

A good and comprehensive overview of these topics and its main aspects has already beendeveloped in [4] as introductory work to the development of the Reaction Sphere initially studiedin the Thesis, and now in this document. For convenience and in order to help the reader tobetter understand the background of these topics, the same structure has been preserved and asummarised content has been included in this chapter.

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CHAPTER 2. ACS AND SPHERICAL ACTUATORS OVERVIEW

2.1 Satellite Attitude Control

The current prototype of Reaction Sphere was developed with a specic application in mind:Satellite Attitude Control, therefore several concepts regarding this topic are going to be explainedin this section.

Any spacecraft, independently from its intended application, is composed of two main sys-tems: the payload and the bus, as stated in [13]. The former corresponds to the hardware andsoftware that is designed to interact with the subject, the element in the outer world that thespacecraft is intended to point at or interact with, in order to accomplish the spacecraft's mis-sion. The latter provides the required resources for the payload to carry out its work. Thisrequirements in the bus will lead to the breakdown of the spacecraft into dierent subsystemswith a specic function, such as the mechanical structure, the propulsion, the thermal control,power supply, telecommunication and Attitude Determination and Control System among others.This last mentioned subsystem will be explained more in detail hereunder.

2.1.1 Attitude Determination and Control System (ADCS)

The movement of any given rigid body can be described by using four magnitudes: position,linear velocity, attitude (or orientation) and attitude motion (or angular velocity). The rst twodescribe the translational motion of the center of mass of the rigid body, the spacecraft in thiscase, while the latter two specify the rotational motion of the satellite with respect to its centerof mass.

The Attitude Determination and Control System (ADCS) is the system that is involvedin stabilisation and achieving the desired rotational movement of the spacecraft. In general,satellites are sent to space in order to orient an instrument to a specic point in space, e.g. acommunications satellite should transmit and receive information from the Earth, and thus, pointits antennas towards it, or for environmental monitoring, a camera and other sensors towardsthe Earth for recording information such as the structure of the atmosphere, snow levels or themovement and formation of clouds.

Generally, every ADCS consists of two subsystems: Attitude Determination correspondsto the measurement and identication of the orientation of the spacecraft, and the forecast ofthe future required attitude of the body, for which sensors such as magnetometers, gyroscopes,Inertial Measurement Units (IMUs), sun or earth sensors are used, and Attitude Control whichstabilises and modies the attitude of the satellite according to the requirements of the situationand application. The latter makes use of dierent actuators that will be exposed in detailafterwards.

Every body in the outer space whose orientation is not controlled, tends to rotate aroundall axes as a consequence of existent external and natural forces, such as gravity gradients, solarpressure and magnetic torques. This arbitrary rotation is not acceptable for a spacecraft, as thepayload should be able to accomplish its mission by pointing it towards the target.

A spinning body tends to hold one axis, the rotation axis, which can be enough for manymissions, requiring only one-axis control. For more complex missions, a three-axis control isrequired in order to stabilise and control the full orientation of the spacecraft.

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2.1.2 Attitude Control Methods

In order to control the attitude of the spacecraft, dierent approaches have been proposedthroughout history. They can be mainly classied in passive and active. The former grouptakes advantage of the spacecraft's natural dynamics to guarantee its stability at a given atti-tude. This can only achieve coarse levels of precision. The latter group makes use of dierenttypes of actuators to modify and stabilise the spacecraft's attitude at a desired orientation. Ingeneral, both approaches are commonly combined for attitude control.

The main attitude control systems used in spacecraft are:

• Spin-stabilised Systems. The spacecraft rotates around the axis with highest moment ofinertia.

• Dual-spin Systems. A spacecraft with a spinning segment and an inertially xed section.

• Gravity-gradient Systems. Takes advantage of the natural tendency of aligning the longaxis of the spacecraft with the gravity gradient.

• Three-axis-stabilised Systems. The complete attitude of the spacecraft of all the three axisis actively controlled.

• Momentum bias Systems. This system makes use of a rotating will to increase the stinessin two axes, and the wheel speed to control the third and remaining axis.

As the studied device can be categorised as a Three-axis-stabilised System, this group mustbe explained in more detail. These systems utilise dierent type of actuators for maintainingthe orientation of the spacecraft aligned to a given reference frame. Some of the actuators thatcan be used for stabilisation are Reaction Wheel (RW), thrusters, magnetic torquers or ControlMoment Gyroscopes (CMGs).

A common situation in which these actuators can be used can be: a disturbance slightlyrotates the spacecraft, creating an error in the payload direction. The spacecraft's control systemaccelerates a Reaction Wheel, which creates a torque that brings the spacecraft back to thedesired orientation. If the disturbances are cyclic, in the next deviations the Reaction Wheel'sspeed will be reduced, staying far from saturation limits. If this does not happen, an externaltorque must be applied by using a thruster or a magnetic torquer, to force the wheel speed backto zero in a process called desaturation, momentum unloading or momentum dumping.

On the one hand, as main advantages, three-axis-stabilised systems have an unlimited point-ing capability and good pointing accuracy, only limited by sensor accuracy. On the other hand,these devices are complex, which implies a bigger source of failure; heavy; have a high powerconsumption and they are expensive.

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CHAPTER 2. ACS AND SPHERICAL ACTUATORS OVERVIEW

2.1.3 Attitude Control Actuators

The main alternatives for actuating over the orientation of a spacecraft are:

• Magnetic Torquers: composed of a magnetic core and a coil, which when energised, thecore gets magnetised and generates a magnetic moment. The magnitude of the torquesthat can be applied with magnetic torquers is generally low.

• Thrusters: make use of the reaction force generated by the expulsion of gas or ion particles.They can be used directly to control the orientation of the spacecraft or as momentumdesaturation actuator for RWs. Even if the amplitude of the pulse provided by thrustersis almost unbounded, no smooth control can be achieved, as the control is non-linear, inthe sense that the amplitude is constant and variable duration.

• Momentum-Exchange Devices: based on rotating masses inside the spacecraft, which allowsangular momentum exchange between the bodies, without modifying the overall inertialangular momentum. The main actuators of this type are RWs, CMGs, momentum wheels.

The rst two techniques are inertial controllers, which change the overall angular momentumof the spacecraft, while the last type preserves it. The latter technique will be explained in moredetail, as it corresponds to the working principle of a Reaction Sphere.

Momentum-Exchange Devices

As stated before, momentum-exchange devices are based on rotating masses inside the spacecraft,allowing angular momentum exchange between the device and the spacecraft without aectingthe overall inertial angular momentum, and thus, actuating over the attitude of the spacecraft.

The main momentum-exchange devices used in space applications are: Reaction Wheels, mo-mentum wheels, and Control Moment Gyroscopes (CMGs). CMGs are mainly used for mannedspacecraft, as it can achieve torques up to 200 Nm, but they are heavy and rarely used fornormal-sized satellites. Momentum wheels are used for stabilising the attitude of the satellitearound the rotating axis of the wheel's axis of rotation, providing stiness in the other two. Butfor very accurate attitude control, RW are preferred. They allow continuous manoeuvres withlow torque disturbances. The achievable torques are located between 0.05 and 2 Nm.

RW are mainly small ywheels powered by an electrical motor. The change of angular speedof the rotor generates a torque, and the reaction torque on the satellite rotates it around therotor axis by conservation of angular momentum in the spacecraft. In the presence of externaldisturbances, the overall angular momentum of the satellite can be modied, which could guidetowards RW's saturation. To prevent this desaturation is required by using thrusters or magnetictorquers.

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

With one Reaction Wheel, only one axis of rotation can be controlled, thus, for three-axiscontrol, a minimum of three RWs are required, while in general a fourth one is included in orderto increase redundancy and reliability of the system.

Several approaches have been tried throughout history to combine a multi-axis actuationin one only device, a compact alternative to a four-wheel system previously explained. Thecapability of these devices to work in a real mission in space has yet to be proven.

An active magnetic bearing wheel with ve axes actively controlled has been proposed in [14].This wheel can be tilted by 1o, which allows it to be used both as a RW and a CMG. Otherproposed approaches are Reaction Spheres (RSs), a spherical actuator that can produce a torquearound any given axis. These devices and the dierent approaches that have been tried will bepresented in the next section.

2.2 Reaction Sphere Concepts

In order to solve some of the problems existing on using Reaction Wheels, such as bearing andgyroscopic cross coupling between axes, the rst concepts of Reaction Spheres were proposed in1960. The main idea behind this concept, was to substitute the three or four rotating massesof the RWs by a hollow sphere that could be accelerated about any given direction in order toprovide three-axis attitude control with a single device.

As expressed by Haeussermann in [15], the author shows the advantages of using a sphereto avoid coupling between three RW and to reduce the number of actuators. For suspendingthe sphere, air and magnetic bearings are proposed, being the latter the most promising onefor being operated in vacuum, in which a set of four or more coils allow proper centring of thesphere, made of ferromagnetic material. For applying a torque, air nozzles or electric means areproposed, being the latter the most promising alternative, by two or more phase AC torquers.

In [16], an inertial sphere made of a conducting shell positioned by a high-frequency magneticeld and torqued by a low-frequency magnetic eld is considered. A broader study of the systemis done in [17] for attitude control of satellites. The bearing force is generated by the interactionof the magnetic eld and the eddy currents induced in the surface of the rotor, and the torqueis applied by a set of mutually perpendicular coils.

An electrically-suspended reaction sphere was rst proposed by Ormsby in [18]. The sus-pension of the metallic rotor is done by an electric eld generated by mutually perpendicularelectrodes. In order to generate the required torque, three pairs of induction windings situatedin orthogonal planes. An illustration of the design is shown in Figure 2.1 in which only twowindings are shown. A similar design is also studied by Ormsby in [19], in which importantcharacteristics of the electrically levitated reaction sphere are presented. The proposed statorhas eight zones to provide four pairs of electrodes which can be paired to obtain four independentsuspension forces. The major limitation of electric eld levitation was the maximum positioningforce that could be applied to the sphere.

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CHAPTER 2. ACS AND SPHERICAL ACTUATORS OVERVIEW

Figure 2.1: Reaction Sphere proposed by Ormsby [18].

Figure 2.2: Schematic of the Reaction Sphere proposed by Downer et al. [22]. Rotor with bearing rings(left) and rotor with torquing coils (right).

A similar conguration is developed in [20]. The reaction sphere is made of a thin-walledspherical rotor inside a spherical cavity bounded by 8 electrodes and three orthogonal stators.The arrangement of electrodes divides the surface into 8 equal parts, and each stator consists ofa two-phase induction motor that creates a rotating magnetic eld that applies a specic torqueto the rotor about any given axis.

Two decades after, the rst magnetically-levitated reaction sphere was patented in 1986 [21].The designed is composed of a rotor with at least an outer conductive and magnetizable layer, andstator with three pair of mutually orthogonal sectors carrying a centring (bearing) winding andtorquing winding. The applied torque results from the interaction of the eddy currents, inducedin the surface of the rotor by the magnetic ux of the centring winding, and the magnetic uxgenerated by the torquing winding.

A magnetic bearing suspension system with a rotor that can be rotated about 360 waspatented in 1990 in [22]. The rotor consists of a ywheel, machined to a spherical shape, attachedto an annular magnet. The stator is made of three orthogonal pairs of ring inductive motors usedfor bearing control, and other three orthogonal pairs of cylindrical coils that apply the requiredtorque to the sphere. The schematic design of the proposed reaction sphere is shown in Figure2.2.

In 1995, a spherical actuator which can be used for satellite attitude control was patented byNakanishi et al. [23]. The general design consists of a hollow spherical rotor with several magnets

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

Figure 2.3: Schematic of the Reaction Sphere proposed by Chételat [24].

arranged in alternate polarity, and a stator with a plurality of magnets in the inner surface. Eitherstator or rotor magnets could be electromagnets and several bearings are proposed: hydrostaticbearing, magnetic bearing, sliding bearing, and roller bearing.

In 2006, a Reaction Sphere, in which the design studied in this Thesis is based on, waspatented by Chételat [24]. The RS consists of a concentric assembly of a spherical rotor andspherical actuator. On the one hand, a spherical rotor with 8 poles arranged symmetricallyin such a way that when projecting radially on a concentric octahedron, the same pattern isobtained in the faces of the polyhedron, and having opposite polarity to the adjacent poles. Onthe other hand, a spherical stator with at least 20 poles magnetised by electromagnets arrangedin the vertices of a icosahedron. Due to the symmetry of the system, and the constant source ofmagnetic eld generated by the rotor, bearing and driving problems can be decoupled. One ofthe possible embodiments is shown in Figure 2.3.

2.3 3-DoF Spherical Actuators

A spherical actuator is a device capable of performing 3-DoF rotational movements in a singlejoint. Thus, the studied Reaction Sphere can be classied as a spherical actuator. In this sectionan historical overview of the dierent alternatives and dierent approaches that have been triedthroughout history.

These actuators are presented as a more compact and high performance alternative to acombination of multi-DoF actuators, specially for robotic applications, avoiding the dierentdrawbacks that a combination of actuators can have, such as backlash, friction or kinematicsingularities.

Spherical actuators have been subject of study during the last decades, and generally, theyall consist of a spherical rotor inside a stator, and in order to export the desired torque tothe external world, a shaft emerges from the rotor. Even though piezoelectric and ultrasonicdevices have been proposed, the most common studied approach has been the development ofelectromagnetic devices, which are the ones that are going to be explained in this section.

These electromagnetic devices can be classied, according to its main working principle,in three main types: inductive, variable reluctance, and Permanent Magnet (PM) synchronouselectromagnetic spherical actuators.

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CHAPTER 2. ACS AND SPHERICAL ACTUATORS OVERVIEW

Figure 2.4: Induction Spherical motor proposed by Williams et al. [25]. Rotor and stator block (left) andstator structure (right).

2.3.1 Inductive Electromagnetic Spherical Actuators

• Williams et al. designed the rst inductive shperical actuator in 1959 [25]. This 2-DoFspherical motor schematic design is shown in Figure 2.4. The design consists of a sphericalbarrel-shaped rotor, with a stator with two winding blocks. The current is induced in thesurface of the rotor, in which conducting rings, joint at every point forming a grid, are cre-ated by performing slots in the rotor, allowing current circulation in any direction. Rotor'sspeed is modied by changing the direction of the moving magnetic eld, by changing theorientation of the stator around an axis perpendicular to the one of the rotor.

• Vachtsevanos et al. proposed in 1987 a robotic manipulator with a spherical actuatorcapable of 3-DoF [26]. The designed is composed of a spherical conducting rotor lled witha high permeability material and a stator with three sets of windings which are excited tocreate the induced current on the rotor. Due to the mechanical complexity of the design,no prototype has never been manufactured.

• More recently, in 2006, Dehez et al. proposed a 2-DoF induction spherical actuator [27].The design consists of a tow-layer spherical rotor with teeth held in position by aerostaticbearings, and a stator made of two sets of three-phase windings placed orthogonally onefrom the other. The proposed prototype is shown in Figure 2.5.

• In 2013, Kumagai and Hollis proposed a 3-DoF spherical induction motor which is the rstinduction prototype that has allegedly been operated in closed-loop [28]. The designed iscomposed of a two-layer copper-over-iron spherical rotor and a stator with four independentinductors that generate the required forces at the surface of the rotor. The design alsocontains four optical mouse sensors that measure the velocity of the surface, which is usedfor computing the angular velocity of the rotor for closed-loop control. The actuator iscapable of rotating at 300 rpm around any given axis with 4 Nm torque. The manufacturedprototype is shown in Figure 2.6.

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

Figure 2.5: 2-DoF Induction Spherical Motor proposed by Dehez et al. [27].

Figure 2.6: 3-DoF Induction Spherical Motor proposed by Kumagai and Hollis [28].

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CHAPTER 2. ACS AND SPHERICAL ACTUATORS OVERVIEW

Figure 2.7: 3-DoF Variable Reluctance Spherical Motor proposed by Lee et al. [3135]. Schematic design(left) and real prototype (right).

2.3.2 Variable Reluctance Electromagnetic Spherical Actuators

• A spherical stepper motor has been proposed by Lee et al. [29,30] capable of 3-DoF motionin a single joint, which has been based on the principle of a hybrid permanent magnet anda conventional Variable Reluctance stepper motor. The motor consists of a hemisphericalstator that houses the stator coils and a rotor that contains a pair of permanent magnets.The rotor is supported freely by means of air bearings or gimbals.

• Lee et al. have developed and proposed a Variable Reluctance Spherical Motor (VRSM)[3135]. In general terms, the design is composed of a hemispherical stator with electro-magnets and a rotor with a number of poles made of ferromagnetic materials or permanentmagnets. The motor is shown in Figure 2.7. By controlling the current applied to the coilsof the stator, a magnetic energy function of the relative position of the rotor and stator iscreated in the airgap. The movement is generated as the rotor positions itself to minimisethis energy.

• A particular form of Variable Reluctance Spherical Motor has been proposed by Lee et al.which has been referred as Spherical Wheel Motor (SWM) [36,37]. The main focus of thisdesign was to control in open-loop the orientation of the rotating shaft in a spherical joint.The SWM consists of a stator with two layers of 10 aluminium-core coils (total of 20 coils)arranged in the shape of a regular decagon, and a rotor with two layers of 8 cylindricalPermanent Magnet (total of 16 PMs) arranged in the shape of a octahedron. The proposeddesign is shown in Figure 2.8.

2.3.3 Permanent Magnet Synchronous Electromagnetic Spherical Actuators

• The progressive design of a prototype of a 3-DoF Permanent Magnet spherical actuator hasbeen proposed by Yan et al. [3842]. Initially, as shown in [38], the design is composed of arotor with 8 Permanent Magnets with the shape of a dihedral cone, placed with alternate

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

Figure 2.8: 3-DoF Spherical Wheel Motor proposed by Lee et al. [36, 37]. Motor Assembly (left) androtor assembly (right).

Figure 2.9: 3-DoF PM Spherical Motor proposed by Yan et al. [38].

polarities around the equator and a stator of two layers of 12 air-core coils. This initialdesign is shown in Figure 2.9. In [39], the magnetic ux density is formulated by solving, inspherical coordinates, the Laplace equation, yielding a torque model, which linearly relatesthe set of currents applied to the coils with a torque vector.

Alternative designs were studied by Yan et al. On the one hand, in [40], the inuence ofusing an iron stator on the magnetic eld and output torque. By using an iron stator,the radial component of the magnetic ux density can increased, and similarly, the outputtorque of the actuator. On the other hand, in [41,42], due to the high manufacturing costsand system complexity, a substitution of the dihedral-shaped Permanent Magnet with astacked distribution of cylindrical-shaped magnets. This prototype can be seen in Figure2.10.

• A 6-DoF Permanent Magnet Spherical Actuator has been proposed by Bai et al. [43].This actuator is intended for haptic applications and can be operated in displacement androtational modes. The actuator consists of a stator with 24 cylindrical electromagnets anda rotor with 24 cylindrical Permanent Magnet. The actuator allows to manipulate objectsin a virtual environment while providing force and torque feedback. The design is shownin Figure 2.11.

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CHAPTER 2. ACS AND SPHERICAL ACTUATORS OVERVIEW

Figure 2.10: 3-DoF Spherical Actuator with cylindrical PM proposed by Yan et al. [41, 42].

Figure 2.11: CAD model of 6-DoF PM Spherical Actuator for haptic applications proposed by Bai etal. [43].

The magnetization axes of stator electromagnets and rotor PM pass radially through thespherical center of the actuator. Articial neural networks are used for directly mapping themagnetic torque and the required force and torque feedback without explicitly measuringthe orientation of the rotor. In [44], a direct eld-feedback control of the proposed actuatoris derived. In this case, magnetic ux density measurements are used for feedback insteadof using rotor orientation, which allows parallel computations of the control law and torquemodel through articial neural networks.

• A Spherical Stepper Motor is proposed by Chirikjian and Stein [45]. The Design consists ofa plastic spherical rotor with 89 PMs that is partially surrounded by a stator composed of16 iron-core electromagnets arranged on a hemisphere. The developed prototype is shownin Figure 2.12. By approximating the interaction of one stator pole with one rotor pole topotential functions the commutation problem is addressed, and these potential functionsare expressed using spherical harmonics, allowing simplied computations. Experimentalresults are shown in [46].

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

Figure 2.12: 3-DoF PM Stepper Motor proposed by Chirikjian and Stein [45].

Figure 2.13: 3-DoF PM Spherical Actuator proposed by Chen et al. [50].

• A 3-DoF spherical actuator was proposed by Wang et al. [47]. This actuator is composedof a 4-pole PM rotor made of two pairs of parallel-magnetized quarter spheres and a statorwith 4 sets of windings arranged in such a way that 3 independent torque components canbe controlled. An analytical model of the magnetic eld for force and torque models isdeveloped based on the Laplace equation. Also a rotor dynamics model and rotor positionsensing and control algorithms are introduced in [48]. An application of the actuator ispresented in [49] as a high-delity force and torque feedback joystick.

• Chen at al. proposed a 3-DoF Permanent Magnet spherical actuator [50]. The design iscomposed of a 8-PM-pole rotor and a two-layer stator with a total of 24 air-core coils.The PM poles are arranged with alternate polarization in a aluminium rotor shell. Theorientation of the rotor is measured by a rotary encoder and two-axis tilt sensors includedin the spherical joint. The maximum tilt angle is ±15. The proposed and manufacturedprototype is shown in Figure 2.13.

• A multi-pole synchronous spherical motor was designed and manufactured by Yano et al.[51]. The actuator consists of a spherical rotor made of a pavement of 260 small cylincricalPermanent Magnet attached to its surface in such a way that north and south poles appearin concentric circles alternatively, and stator with two pairs of air-core electromagnetsplaced perpendicularly to each other. The working area of such actuator is ±45 abouttwo axes, and the maximum torque is 0.49 Nm. The manufactured prototype is shown in

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CHAPTER 2. ACS AND SPHERICAL ACTUATORS OVERVIEW

Figure 2.14: Multi-pole synchronous spherical motor proposed by Yano et al. [51].

Figure 2.15: Polyhedron-based spherical stepper motor proposed by Yano et al. [5254].

Figure 2.14.

• Yano et al. developed a spherical stepper rotor based on a regular polyhedra arrangement ofpoles of rotor and stator [5254]. The design consists of a rotor with 8 Permanent Magnetsplaced on an iron-covered-with-acrylic spherical shell in the positions of the vertex of ahexahedron in such a way that the adjacent poles have opposite polarity. Also 6 iron coresare placed at the faces of the aforementioned inscribed hexahedron in order to x rotationaxis. The stator is composed of 25 air-cored coils attached to an acrylic spherical shell inthe shape of a octahedron (6 at the vertexes, 12 at the center of the edges and 7 ath thecenter of the faces). The proposed design is shown in Figure 2.15.

In this section a brief overview of some of the spherical actuators proposed in history hasbeen done, for further information, please refer to [4], or to the dierent references shown alongthe chapter.

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Chapter 3

Reaction Sphere

In this chapter, the Reaction Sphere concept is going to be introduced. Afterwards the relevantaspects of its modelling will be explained, as well as the development of the current prototypeavailable at CSEM. Finally, the control architecture of the Reaction Sphere is going to be shown,which will introduce the angular velocity estimation explained in the next chapter.

The content of this chapter is essential for understanding the development of the proposedangular velocity estimator, as well as for having a description of the current Reaction Sphereprototype and modelling procedure followed in previous projects related with this actuator atCSEM.

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CHAPTER 3. REACTION SPHERE

3.1 Introduction

During this chapter, the main design choices and modelling process are going to be explained.These concepts are required for understanding the behaviour of the system and the angularvelocity estimation that is proposed in this document.

In the dierent sections, only the results or main equations of the related content are goingto be shown, in order to be able to understand the whole process, but if further details arerequired, please refer to [4], in which the complete design is thoroughly explained. For a less indepth description, [5], [6] or [3] can also be consulted.

The main topics that have to be covered in order to understand the proposed angular velocityestimator design and performance verication with the real prototype are:

• Reaction Sphere design and prototype description. A brief description of the main charac-teristics of the available Reaction Sphere.

• Magnetic Flux Density Model. Model that relates the orientation or conguration of therotor with the generated magnetic ux.

• Force and Torque Models. Model that relates the force and torque applied to the RS's rotoras a function of the input currents to the electromagnets.

• Magnetic State Estimation. Estimation procedure for obtaining the magnetic state, whichcarries on the information about the orientation of the rotor.

• Control Architecture. Brief description of the instrumentation and algorithm used forcontrolling the RS's rotor angular velocity and magnetic bearing.

3.2 Reaction Sphere Concept Overview

In this section, a brief description of the Reaction Sphere model and available prototype is goingto be explained. For further information about the topic, please refer to the references shown atthe beginning of the chapter.

The Reaction Sphere model is based on a 3-DoF synchronous motor with Permanent Mag-nets, whose rotor is magnetically levitated and can be accelerated around any desired axis. Inorder to full this, an 8-pole spherical rotor, composed of Permanent Magnets placed at thevertices of an octahedron, and a stator, with 20 air-core electromagnets located at the verticesof a dodecahedron, are considered. An schematic view of the reaction sphere model can be seenin Figure 3.1.

Two Reaction Sphere's prototypes have been developed and manufactured. In the rst labor-atory prototype, the 8 poles of the RS's rotor were discretised using a mosaic of 728 cylindricalmagnets in order to approximate the desired fundamental harmonic. The second manufactured

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Figure 3.1: Reaction Sphere with 8-pole rotor and 20-pole stator. (Left) Complete rotor and half of thestator is shown. (Right) Half Stator schematics.

prototype consists of 8 bulk permanent-magnet poles, which are parallel-magnetised and havetruncated spherical shape, xed to a back-iron structure which has truncated octahedral shape.The former prototype provides a less distorted magnetic ux density prole compared to theideal 8-pole spherical rotor model, while the latter was designed to improve manufacturabilityand mechanical stability. The preferred alternative, and the one that is currently used, corres-ponds to the second prototype.

In the chosen prototype, dierent sensors and actuators are used in order to be able to knowthe state of the actuator and modify it:

• 20 Air-Core electromagnets that creates the force and torque applied to the RS's rotor.The magnetic bearing and angular velocity are controlled by the 20 input currents.

• 3 laser displacement sensors that measure the position of the rotor inside the stator.

• 15 Hall sensors that measure the magnetic ux at dierent positions of the stator, whichwill give information about the orientation of the rotor.

Further information about the real prototype will be given in Section 3.6, in which thecontrol scheme and design is going to be explained.

3.3 Magnetic Flux Density Model

During this section, the magnetic ux density model is going to be shown, in which a rela-tion between the orientation of the rotor and the magnetic ux generated by it is expressedmathematically.

In order to derive the aforementioned model, an approach based on a Spherical HarmonicDecomposition of the spatial distribution of the magnetic ux density generated by the perman-ent magnet rotor. Spherical Harmonics are a complete set of orthogonal functions dened on the

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CHAPTER 3. REACTION SPHERE

surface of a sphere, which are commonly used to represent certain functions as a sum of harmon-ics, just as Fourier Series are used for expressing for periodic signals. For further informationregarding Spherical Harmonics, please refer to [4].

The magnetic ux density model is required for the formulation of the content of the followingsections, as the force and torque model needed for magnetic bearing and angular velocity control,and for the design optimization of the spherical actuator. This model is derived by using a hybridFEM-analytical approach: analytical expressions, which gives a known structure of the spatialdistribution of the magnetic ux density between the airgap, are combined with Finite ElementMethod (FEM) and measured derived values and boundary conditions.

In order to derive the magnetic ux density model for the Reaction Sphere, the solution ofLaplace's and Poisson's equations is expressed analytically. With the selection of an octupolecubic magnetization of the rotor, the magnetic ux density distribution can be expressed as alinear combination of a nite number of spherical harmonic functions. It is initially expressedin rotor coordinates, and afterwards, a change of coordinate reference frame is performed byexploiting the properties of spherical harmonics under rotation.

The resulting magnetic ux density B at point (rs, θs, φs) within the airgap of the ReactionSphere, expressed in spherical coordinates, can be formulated following the structure:

B(rs, θs, φs) =

Nh∑n=3

n∑m=−n

cmn (α, β, γ)Bmn (3.1)

where cmn (α, β, γ) are the spherical harmonic decomposition coecients of degree n and orderm, dependent on the orientation of the rotor expressed by the ZYZ Euler angles α, β and γ, Bm

n

are the magnetic ux density contributions of each spherical harmonic of degree n and order m,and Nh is the maximum spherical harmonic degree considered in the model.

On the one hand, the magnetic ux density contributions Bmn can be computed analytically

o-line, as shown in [4]. On the other hand, the spherical harmonic decomposition coecientscmn (α, β, γ), by exploiting spherical harmonic properties under rotation, can be expressed as alinear combination of of the spherical harmonic decomposition coecients expressed in rotorcoordinates (immobile rotor) cln,imm with the same degree n:

cmn (α, β, γ) =∑l

Dnm,l(α, β, γ)cln,imm (3.2)

where Dnm,l(α, β, γ) are unitary rotation matrices. The spherical harmonic decomposition

coecients of the immobile rotor cln,imm can be computed either by FEM simulation or bymeasured data.

As it will explained in the following sections, a direct method for estimating the sphericalharmonic decomposition coecients cmn , that for now on they will be called Magnetic State,which carry the information about rotor orientation, without requiring the use of Equation 3.2,will be proposed for real-time control.

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

3.4 Force and Torque Models

Once the magnetic ux density model has been shown, force and torque models for the ReactionSphere are going to be explained. These models relate the applied force and torque to the RS'srotor, as a function of the input currents to the 20 available coils. This relation will also dependon the orientation of the rotor, which will be provided indirectly by the magnetic state, by usinga similar approach to the one used to derive the magnetic ux density model.

The force and torque models are required for closed-loop control of the Reaction Sphere.The term Forward Model is used to specify the model that provides the force and torque appliedto the rotor for a given current vector, while Inverse Model yields the optimal current vector fora desired force and torque inputs.

3.4.1 Force and Torque Forward Models

In order to derive this models, the Lorentz Force law is used, and by exploiting the superpositionprinciple, the force F ∈ <3×1 and torque T ∈ <3×1 applied to the rotor can be expressed as:

F = KF (α, β, γ)i

T = KT (α, β, γ)i(3.3)

where i ∈ <20×1 is the vector of coil input currents, andKF (α, β, γ) ∈ <3×20 andKT (α, β, γ) ∈<3×20 are the force and torque characteristic matrices, respectively, of the spherical actuator.

These characteristic matrices, as it happened with the magnetic ux model, depend on theorientation of the rotor, which can be parametrised by using the spherical harmonic decomposi-tion coecients, or Magnetic State cmn (α, β, γ), following the expressions:

KF (α, β, γ) =

Nh∑n=3

n∑m=−n

cmn (α, β, γ)KmF,n

KT (α, β, γ) =

Nh∑n=3

n∑m=−n

cmn (α, β, γ)KmT,n

(3.4)

where KmF,n and Km

T,n are the force and torque characteristic matrix contributions, respect-ively, of each spherical harmonic of degree n and order m, and where cmn (α, β, γ) values areestimated by using the method explained in Section 3.5. This way, force and torque character-istic matrices are computed as a linear combination of a set of constant and oine-computedmatrices Km

F,n and KmT,n.

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CHAPTER 3. REACTION SPHERE

3.4.2 Force and Torque Inverse Models

For closed-loop control, a desired force or torque is wanted to be applied in order to control themagnetic bearing and angular velocity of the rotor, and the optimal set of currents that appliesthese actuations have to be computed by using the force and torque inverse models.

Starting from the forward model, expression 3.3 can be rewritten in a compact way as

[FT

]=

[KF (α, β, γ)KT (α, β, γ)

]i = KF,T i (3.5)

In order to compute the required control currents, the previous system of linear equationshas to be solved. Note that in this case, the system is underdetermined, as the number ofequations (6) is lower than the number of unknowns (20), thus the least-squares solution is used:

iLS = K>F,T(KF,TK

>F,T

)−1 [FT

]=[K>F K>T

]([KF

KT

] [K>F K>T

])−1 [FT

]=[K>F K>T

] [KFK>F KFK

>T

KTK>F KTK

>T

]−1 [FT

] (3.6)

Due to the rotor symmetry of stator coils and magnetization pattern of the rotor, force andtorque characteristic matrices are mutually orthogonal, and thus KFK

>T = 0, resulting in:

i = MFF +MTT (3.7)

where matrices MF and MT correspond to

MF = K>F (KFK>F )−1, MT = K>T (KTK

>T )−1 (3.8)

and because forces and torques created by the stator coils span a three-dimensional space,the rank of matrices KF and KT is equal to three, thus, MF and MT always exist. Expression3.7 shows that if RS's rotor is perfectly centred, there would be no coupling between the appliedforce and torque, which will make the control of the actuator easier, as both magnetic bearingand angular velocity can be controlled independently.

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

3.5 Magnetic State Estimation

As it has been explained in the previous sections, the information about the orientation of therotor can be parametrised by the so-called Magnetic State, which corresponds to the sphericalharmonic decomposition coecients used for magnetic ux density, and force and torque models.

In Section 3.3, it has been seen that the aforementioned coecients can be computed bytaking advantage of the spherical harmonic properties under rotation, but this approach requiresthe knowledge or measure of the orientation of the rotor. A new approach will be explainedin this section for estimating the Magnetic State without computing explicitly the orientationof the rotor, and thus taking advantage of the computationally-inexpensive spherical harmonicdecomposition for real-time control of the Reaction Sphere.

Due to the available number of sensors and its positioning, the highest considered sphericalharmonic degree is Nh = 3. The Magnetic State of the corresponding spherical harmonic ofdegree 3 is the set of complex parameters dened as

c3 = [c−33 , c−23 , c−13 , c03, c13, c

23, c

33]> (3.9)

By taking into account only the radial component of the magnetic ux density, at positionof Hall sensor k, (Rsens, θk, φk) in spherical coordinates, expression 3.1 can be simplied to

B⊥k =

3∑m=−3

cm3 (α, β, γ)B⊥3,m,k (3.10)

where a simplied notation has been adopted, Brs(Rsens, θk, φk) = B⊥k for the radial com-ponent of the magnetic ux density at sensor k position, and B⊥3,m,k = Bm

rs,3(Rsens, θk, φk) forthe magnetic ux density contributions of degree 3 and order m. By decomposing the magneticstate in its real and imaginary parts as cm3 = am3 + ibm3 , |m| ≤ 3, and applying spherical harmonicproperties, the magnetic ux density can be expressed as a linear combination of real coecients:

B⊥k = a03R03(θk, φk) + 2

3∑m=1

am3 Rm3 (θk, φk) + 2

3∑m=1

Bm3 I

m3 (θk, φk) (3.11)

where Rm3 (θk, φk) = Re[B⊥3,m,k(θk, φk)] and Im3 (θk, φk) = −Im[B⊥3,m,k(θk, φk)]. Then consid-

ering the whole set of Hall sensors, B⊥ ∈ <Nm×1, and writing the previous equation in matrixform, we obtain:

B⊥ = A(Γ)c∗3 (3.12)

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CHAPTER 3. REACTION SPHERE

having the Nm × 7 matrix A(Γ) as

A(Γ)> =

R03(ς1) R0

3(ς2) . . . R03(ςNm)

2R13(ς1) 2R1

3(ς2) . . . 2R13(ςNm)

2R23(ς1) 2R2

3(ς2) . . . 2R23(ςNm)

2R33(ς1) 2R3

3(ς2) . . . 2R33(ςNm)

2I13 (ς1) 2I13 (ς2) . . . 2I13 (ςNm)2I23 (ς1) 2I23 (ς2) . . . 2I23 (ςNm)2I33 (ς1) 2I33 (ς2) . . . 2I33 (ςNm)

(3.13)

where Γ is a parameter set of the position of the set of sensors in spherical coordinates Γ =ς1, ς2, . . . , ςNm, with ςk = (θk, φk) for k = 1, 2, . . . , Nm, c∗3 is the magnetic state decomposition

in real and imaginary part c∗3 =[a03, a

13, a

23, a

33, b

13, b

23, b

33

]> ∈ <7×1. Note that matrix A(Γ) valuesdepend only on Hall sensors position, thus it can be computed analytically o-line.

By solving Equation 3.12, the values of c∗3 can be computed as follows

c∗3 = (A>A)−1A>B⊥ (3.14)

Note that the matrix A>A is non-singular if the sensors are not mutually collinear, and thatthe factor (A>A)−1A> is constant and can be computed o-line.

Once the Magnetic State can be estimated by using expression (3.14), as the format of thecoecients has been changed, Force and Torque Model 's formulation has to be modied in orderto use the purely real coecients c∗3. Proceeding in a similar way to the real and imaginarydecomposition done for the magnetic ux density, force and torque characteristic matrices canbe rewritten in the following form:

KF = a03K0F,R + 2

3∑m=1

am3 KmF,R + 2

3∑m=1

Bm3 K

mF,I

KT = a03K0T,R + 2

3∑m=1

am3 KmT,R + 2

3∑m=1

Bm3 K

mT,I

(3.15)

where KmF,R = Re[Km

F,3], KmF,I = −Im[Km

F,3], KmT,R = Re[Km

T,3] and KmT,R = −Im[Km

T,3] areconstant and computed o-line.

At this point, the theoretical background and model development required for understandingthe operation of the Reaction Sphere have been explained. In the following section the controlarchitecture developed for the system is going to be explained.

26 ETSII-UPM & EPFL

ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

3.6 Control Scheme and Design

The simplied closed-loop control scheme for magnetic bearing and angular velocity control ofthe Reaction Sphere, designed in [4], is presented in Figure 3.2. As it has been mentioned before,the Reaction Sphere is considered to be always in the center of the stator, which allows todecouple the magnetic bearing (levitation) and angular velocity control, and thus consider themas independent control loops.

During this section, only the angular velocity control loop will be explained in more detail,as the current document focuses on angular velocity estimation, thus, if further informationregarding the magnetic bearing control loop is required, please refer to [3], [4], [5], or [6].

On the one hand, the upper control loop controls the position of the rotor inside the stator, byusing a reference position pref , always the origin during the operation of the Reaction Sphere, andan estimated position p. The output of the magnetic bearing controller is the force componentof the coil input currents iF . On the other hand, the lower control loop takes care of the angularvelocity of the rotor, by making use of the angular velocity reference ωref , and the MagneticState c estimated by using the method explained in Section 3.5. The output of the angularvelocity controller is the torque component of the coil input currents iT .

For both control loops, the force and torque characteristic matricesKF andKT are computedby using expression (3.15), which uses real-only coecients c∗3, at each sampling time. Finallythe resulting current applied to the stator coils is the sum of both contributions iF,t = iF + iT .

3.6.1 Angular Velocity Controller

The dynamic model of a rigid body under rotation can be expressed by using Euler's rotationequation:

Jrotd

dtω = T (3.16)

where Jrot is the inertia tensor of Reaction Sphere's rotor, ω is rotor's angular velocity, andT is the torque applied to the rotor. In this case, by design, the rotor inertia tensor is diagonaland with the same inertia value for each main rotation axis x-y-z, thus each axis can be controlledindependently.

The discrete-time open-loop transfer function of the previous expression for each axis wouldbe:

ω(z)

T (z)=Ts/J

z − 1(3.17)

Guzman Borque Gallego 27

CHAPTER 3. REACTION SPHERE

Sp

eed

Estim

ato

r

pre

f

p

v

Contro

llerF

MF

Curren

t

Driv

es

Rea

ction

Sphere

ω p

Positio

n

Estim

ato

r

Positio

n

Sen

sors

Com

pute

KF

andK

T

Magnetic

Sta

te

Estim

ato

r

Hall

Sen

sors

ωre

f

Angula

r

Velo

city

Estim

ato

r

Contro

llerω

TM

T

iFiF

,Ti

B⊥

∆µc

KF

KT

Magnetic

Bea

ring

Contro

ller

Angula

rV

elocity

Contro

ller

iT

Figure

3.2:Main

Contro

lSchem

eformagnetic

bearin

gandangularvelo

citycontro

loftheReactio

nSphere.

Figure

obtained

from

[4].

28 ETSII-UPM & EPFL

ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

where J = 0.0368 kg m2 is the inertia of the rotor around any given axis. The implementedangular velocity controller is a simple proportional controller, that yields the following closed-looptransfer function:

ω(z)

ωref (z)=

Kp Ts/J

z − (1−Kp Ts/J)(3.18)

The controller's gain Kp = 4 is chosen in order to obtain a desired closed-loop bandwidthof 6 Hz.

3.6.2 Prototype Electronics and Development Environment

In this section, the electronics architecture and development environment used for implementingthe control algorithm in the Reaction Sphere's hardware is going to be explained. First, thehardware used for magnetic bearing and angular velocity control has to be explained. In Figure3.3, an schematic block of the Reaction Sphere's control hardware is shown.

Figure 3.3: Main Control Electronics block schematic. Source: [7]

It can be seen that the main control electronics architecture is comprised of four main blocks,which are:

Guzman Borque Gallego 29

CHAPTER 3. REACTION SPHERE

• Central Control System Rack (CCS): centralises the control of the whole system. It receivessensor measurements from the three available laser distance sensors and 15 Hall eectsensors, logs data and results according to conguration done by connected computer, andsets sampling and code execution frequency.

• Power Electronics patch: updates control currents applied to the 20 available coils inthe stator by executing control algorithm using sensor measurements provided at eachinterruption generated by CSS.

• Reaction Sphere Mechanism: mechanical prototype of the Reaction Sphere, with 20 coils(actuators) and 15 Hall eect sensors and 3 displacement sensors. Also 20 coil temperaturesensors are available used for measuring coils' temperature and avoiding possible damageto the actuators.

• Power Ampliers Rack (PA): provides the necessary current and voltage to sensors andactuators in the Reaction Sphere Mechanism.

Then, in general terms, the main process that is executed at each sampling time is as follows:the Central Control System rack receives the sensors' measurements at frequency of 5 kHz, andone out of two, a software interrupt is sent to the Power Electronics patch, along with lastsensors' measurements, at a frequency of 2.5 kHz. Before the next interrupt is received, thecontrol currents are updated by executing the control algorithm shown in Figure 3.2. If thesystems is congured to log data in memory, the required data is sent back to the CCS andstored in memory.

For more information regarding the briey explained electronics, and dierent tests thathave been carried out to calibrate the dierent components, please refer to [7], in which a moredetailed explanation about the design choices and nal architecture can be found.

The general conguration and design was made by making use of dSPACE GmbH productsand solutions. dSPACE GmbH provides hardware and software for developing, testing and cal-ibrating real-time Electronic Control Units (ECUs) for the automotive, aerospace and medicalengineering industries. For further information about the company and ints products and solu-tions, refer to the manufacturer's website in [8].

The software is developed by making use of the Real-Time Interface (RTI) from MATLAB R©

and Simulink, which allows rapid control prototyping for coding and design of new controlstrategies, and it can be directly implemented on the dSPACE's hardware.

The nal Reaction Sphere prototype that is being currently used and that is available atCSEM laboratory is shown in Figure 3.4a. Also the nal set-up of the system, including RSprototype (Reaction sphere, sensors and actuators), power ampliers rack for generating currentto stator coils, and dSPACE ds1005 ECU is shown in Figure 3.4b.

The hardware used in the system as ECU is:

• As Control Unit of the Central Control System Rack, a MicroAutoBox II is used, a robustand stand-alone prototyping unit from dSPACE (Figure 3.5a).

• As Control Unit of the Power Electronics patch, the DS1005 PPC Board is used, a processorboard based on a PowerPC (PPC) processor which provides medium computation powercombined with very fast I/O access (Figure 3.5b).

30 ETSII-UPM & EPFL

ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

(a) Reaction Sphere's current prototype. (b) Reaction Sphere set-up.

Figure 3.4: Reaction Sphere's prototype and laboratory set-up of the system.

(a) MicroAutoBox II. (b) ds1005 PPC board.

Figure 3.5: ECUs manufactured by dSPACE used in the System

3.7 Conclusion

Once the basic concepts about the Reaction Sphere's model developed in [4] which are requiredfor understanding the main content of this thesis, the proposed angular velocity estimator isgoing to be explained in the next chapter.

In the current chapter, the main results concerning the magnetic ux density model, whichspecies the magnetic ux density at the airgap as a function of the magnetic state of the rotorhave been shown. Also the force and torque models, which obtain the force and torque appliedto the rotor as a function of the magnetic state and an input current vector to stator coils andmagnetic state estimation have been explained. Finally, the control scheme for magnetic bearingand angular velocity control have been shown, as well as the Reaction Sphere's electronics anddevelopment environment.

Guzman Borque Gallego 31

CHAPTER 3. REACTION SPHERE

32 ETSII-UPM & EPFL

ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

Chapter 4

Angular Velocity Estimation

In this chapter, a review of the dierent approaches used for angular Velocity estimation of theReaction Sphere's rotor is going to be presented. This will serve as a basis for the development ofthe proposed estimator based on a Linear Parameter-Varying Kalman Filter State-Space observerof the system. The model's equations will be presented, as well as the modelling of the system'snoise. Finally a study of the stability and dynamic behaviour of the closed-loop system will bepresented, which will be used to tune the dierent parameters of the estimator and controller.

The theoretical development of the State-Space observer is going to be explained during thischapter, whereas the implementation and the experimental results obtained with the proposedalgorithm will be shown in the next chapter.

Guzman Borque Gallego 33

CHAPTER 4. ANGULAR VELOCITY ESTIMATION

4.1 Previous Work on Angular Velocity Estimation

As it has been previously explained, the angular velocity estimation plays an important role inthe performance and overall behaviour of the Reaction Sphere, and in this section, the previousapproaches developed for angular velocity estimation are going to be briey explained, as it willhelp the reader to understand the the development of the proposed observer.

These aforementioned estimators are:

• Back-EMF-based Angular Velocity Estimation: The estimator that has been usedin the closed-loop system until now due to its computation speed, and it is based on theestimation of the voltages induced in the coils of the stator.

• Extended Kalman Filter for Angular Velocity Estimation: An Extended KalmanFilter state-space observer that estimates orientation and angular velocity.

4.1.1 Back-EMF-based Angular Velocity Estimator

This approach is based on the back-EMF estimation for the complete set of coils as a functionof the magnetic state's derivative, and due to energy conservation, the coil back-EMF voltagescan be related to the angular velocity.

As there is no direct measure of the back-EMF voltages induced in the stator coils, thisvalue has to be estimated. In this section only the results and main equations are going to beshown in order to maintain the simplicity of the text, so for further information, refer to [4],where the whole development can be found.

Back-EMF Estimation

The procedure presented in [4] derives the back-EMF by studying the interaction between therotor magnetic ux density and an independent stator coil, and by considering the superpositionprinciple, the back-EMF in the complete set of coils is computed in matrix form.

The Faraday's law of electromagnetic induction models the voltage induced in an electriccircuit. The average back-EMF induced in a specic coil, can be computed by multiplying theaverage back-EMF of a single turn of a coil, and multiplying afterwards by the number of turns:

uemf,k = −Ntd

dtΨav,k (4.1)

As done for Force and Torque models and magnetic ux density model of the reaction spheredeveloped in Chapter 3, the average ux linkage Ψav,k can be parametrised by using spherical

34 ETSII-UPM & EPFL

ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

harmonic coecients cmn that carry all the information relative to rotor's orientation. If onlythe fundamental component of degree n = 3 is considered, the back-EMF expression 4.3 can belinearised as follows:

uemf,k = −Nt

3∑m=−3

Ψmav,k

(d

dtcm3

)(4.2)

where Ψmav,k is the rotor ux linked by the spherical harmonic of degree 3 and order m for

coil k, and Nt the number of turns of the coil.

Then, considering the whole set of stator coils, the back-EMF voltage induced can be ex-pressed as:

uemf = −Nt

3∑m=−3

[Ψmav,1 Ψm

av,2 · · · Ψmav,20

]T ( d

dtcm3

)= Φ

d

dtc3 (4.3)

where uemf = [uemf,1 uemf,2 · · · uemf,20]T ∈ <20 is a vector that contains the induced

voltage in each coil, c3 =[c−33 c−23 · · · c33

]T ∈ <7 the spherical harmonic coecients of degree3, and Φ ∈ <20×7 contains all the magnetic ux linkage for each coil and each spherical harmonicorder:

Φ = −Nt

Ψ−3av,1 Ψ−2av,1 · · · Ψ3

av,1

Ψ−3av,2 Ψ−2av,2 · · · Ψ3av,2

......

. . ....

Ψ−3av,20 Ψ−2av,20 · · · Ψ3av,20

which is a constant matrix and can be computed o-line.

Angular Velocity Estimation

Due to energy conservation, the induced voltages induced in each coil can be expressed as afunction of the rotor's angular velocity:

uemf = KTT ω (4.4)

where KTT is the torque characteristic matrix of the rotor for a given orientation. Then, the

angular velocity can be estimated by using the following equation:

Guzman Borque Gallego 35

CHAPTER 4. ANGULAR VELOCITY ESTIMATION

ω =(K>T)+

uemf =(K+T

)>uemf = M>T uemf (4.5)

where MT denotes the pseudo-inverse of the torque characteristic matrix, which is alreadyavailable from the inverse torque model necessary for force and torque closed-loop control.

By substituting the expression for uemf of Eq. 4.3 in the previous equation, the inducedvoltage of the whole set of coils can be expressed as a linear combination of the time-derivativeof the magnetic state:

ω = M>T Φd

dtc3 (4.6)

To sum up, the main characteristics of this approach are:

• The Angular Velocity estimator is non-iterative, linear and fast to compute.

• MT matrix is already available for torque control, and Φ matrix can be computed o-line.

• The angular velocity estimation deteriorates for higher rotation speeds, due to the deriv-ative approach which amplies the noise.

• Higher order spherical harmonics contribute to non-modelled errors that increase oscilla-tions in the angular velocity estimations.

This method is the one used for closed-loop control, due to its simplicity and fast computa-tion, but its aforementioned issues limit the maximum speed achievable with the current set-up.A new approach has been developed in order to try to overtake this limitations and it will beexplained in the following section.

4.1.2 Extended Kalman Filter

In order to overcome the limitations of the previous method, a study has been carried on for im-plementing an Extended Kalman Filter for angular velocity estimation, which will give smootherestimations, and if the system is modelled correctly, closer to the real rotor speed.

As only an overview of the approach is going to be shown, for further information and a fulldevelopment of this method and the obtained results in simulation, please refer to [9].

The studied approach made use of the kinematic and Dynamic behaviour of the sphere, asa rigid solid, in order to model the evolution of the system, which considers both the orientationand angular velocity of the rotor.

36 ETSII-UPM & EPFL

ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

In this case, the system behaviour is expressed in a State-Space form, as it is always foundfor Kalman Filter approaches. The state-space vector x, input u, and output y of the systemare:

x = [ω, q]T ∈ <7

y = B⊥ ∈ <15

u = Tcmd ∈ <3

(4.7)

where w ∈ <3 is the angular velocity of the rotor expressed in stator coordinates, q ∈ <4 isthe quaternion 1 that expresses the orientation of the rotor, B⊥ ∈ <15 are the radial componentsof the magnetic ux density, measured by the Hall sensors, and Tcmd ∈ <3 is the commandTorque generated by the controller of the closed-loop system.

The equations of the continuous-time state-space model will be:

x(t) = f(x(t),u(t),w(t)) ⇒ω = J−1Tcmd + w1

q = 12Ωq + w2 = 1

2GTω + w2

y(t) = h(x(t),u(t),v(t)) ⇒ B⊥ = h3(q) + v

(4.8)

where Ω and GT matrices relate current angular velocity and quaternion with the latter'sderivative:

Ω =

0 −ωx −ωy −ωzωx 0 −ωz ωyωy ωz 0 −ωxωz −ωy ωx 0

GT =

−q1 −q2 −q3q0 q3 −q2−q3 q0 q1q2 −q1 q0

The noise modelling of the state-space system can be reviewed in [9]. The most import-

ant and conditioning aspect of the noise modelling is the selection of the measurement noisecovariance Rk, which was done for better consider the non-modelled higher harmonics:

Rk = diag(abs(h11(qk)− h3(qk)

))(4.9)

The main limitations of this approach are:

• Hall sensor measurement estimation by rotating spherical harmonics is heavy to compute,and highly non-linear, which requires linearisation.

1The considered quaternion has the following structure: q = (q0, q1, q2, q3)T , where q0 denotes the scalar

component, and (q1, q2, q3) the vector components.

Guzman Borque Gallego 37

CHAPTER 4. ANGULAR VELOCITY ESTIMATION

• The use of an Extended Kalman Filter requires linearisation at each sampling time, whichrequires a considerable time to compute.

• The selection of a measurement noise covariance matrix as in Eq. 4.9 is computationallyexpensive, mainly due to the rotation of spherical harmonics up to degree 11.

• The use of a constant and simpler measurement noise covariance matrix did not givesuciently satisfactory results.

• Due to the previous aspects the algorithm is dicult to implement in real-time.

4.2 Linear Parameter-Varying Kalman Filter Development

At this point, the development of the current Master's Thesis takes place. The two approachespreviously explained did not reach the desired performance for the former, or were not imple-mentable in real-time, for the latter.

The proposed algorithm tries to combine the advantages of both methodologies, by consid-ering the magnetic state for carrying the information about the rotation of the rotor, which willallow faster computations, and by making use of a Kalman Filter estimator that would smoothand reduce the oscillations in the angular velocity estimations.

4.2.1 State-Space Model Equations

As it has been previously said, the system's equations will make use of the magnetic stateapproach, for estimating the angular velocity. Due to the need of both the angular velocity andthe magnetic state for the closed-loop controller, a rst approach was tried by including them inthe state-space vector, but observability and stability issues prevent this approach to work.

First the continuous-time model of the system is derived, followed by the discretisationrequired for implementation in the controller.

Continuous-Time Model

The proposed method will include as state-space vector x, input u, and output y the followingvariables:

x = ω ∈ <3

y = c∗ ∈ <7

u = Tcmd ∈ <3

(4.10)

38 ETSII-UPM & EPFL

ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

where, as explained before, ω is the angular velocity of the rotor, Tcmd the command torquegenerated by the controller, and c∗ is the magnetic state time-derivative of the rotor expressedin real coecients, in the same format used for least-squares estimation of the magnetic state:

c∗ = [a03, a13, a

23, a

33, b

13, b

23, b

33]T

cm3 = am3 + jbm3 , m = −3,−2,−1, 0, 1, 2, 3(4.11)

In order to model the system, as state-space equation, the dynamic equation of a rigidsolid is used as state-space equation, which relates the evolution of the angular velocity withthe applied torque, and as output equation, the back-EMF estimation is used for relating theangular velocity with the system's output, the magnetic state time-derivative:

x(t) = Ax(t) +Bu(t) + w(t)

y(t) = Cx(t) +Du(t) + v(t)

=⇒

ω = J−1Tcmd + w(t)

c∗ = Φ+KTT (c∗)ω + v(t)

(4.12)

where J−1 is the inverse of the rotor's inertia matrix, Φ+ is the pseudo-inverse of the averageux linkage coecient matrix related to each coil and spherical harmonic, and KT

T (c∗) thetranspose of the torque characteristic matrix of the rotor, dependent on the magnetic state, asit was expressed in Section 4.1.1.

For the output equation, the magnetic state is estimated at each sampling time by using theleast-squares projection shown in Section 3.5, which is used for estimating the torque character-istic matrix, and its derivative as output vector of the system.

This conguration of the model results in a linear equation, which varies with time as afunction of the magnetic state, which corresponds to a Linear Parameter-Varying (LPV) Model.This fact will have an impact in the controller and stability analysis, which will be explained atthe end of this chapter.

Then, the continuous-time Linear Parameter-Varying (LPV) Model can be ex-pressed in the following set of linear equations:

x(t) = Bu(t) + w(t)

y(t) = C(t)x(t) + v(t)

=⇒

ω = J−1Tcmd + w(t)

c∗ = Φ+KTT (c∗)ω + v(t)

(4.13)

which correspond to the next state-space model matrices: A = 03×3, B = J−1, C =Φ+KT

T (c∗), and D = 03×3. Note the time-dependence of matrix C.

Guzman Borque Gallego 39

CHAPTER 4. ANGULAR VELOCITY ESTIMATION

Model Discretisation

In order to be able to use the model in a computer-based controller, it has to be discretised.Considering the general solution of the state-space dierential equation:

x(t) = eA(t−t0)x(t0) +

∫ t

t0

eA(t−τ)Bu(τ)dτ (4.14)

Letting t = kTs + Ts and t0 = kTs, where Ts is the sampling period, and considering theinput constant between each sampling time, which is reasonable as the signal is generated by adigital controller, the previous equation can be reformulated as:

x(kTs + Ts) = eATs)x(kTs) +

∫ kTs+Ts

kTs

eA(kTs+Ts−τ)Bu(τ)dτ

= x(kTs) +

∫ kTs+Ts

kTs

Bu(τ)dτ

= x(kTs) + TsBu(kTs)

(4.15)

Furthermore, the output equation does not need discretisation, as it relates the angularvelocity at each instant with the output of the system. but on the other hand, in order tocompute the time-derivative of the magnetic state, a rst-order approximation is used.

The resulting set of equations of the discrete-time Linear Parameter-Varying (LPV)Model are:

xk+1 = Fxk +Guk + wk

yk = Hkxk + vk

=⇒

ωk+1 = ωk + TsJ−1Tcmd,k + wk

c∗k = Φ+KTT (c∗k)ωk + vk

(4.16)

where F = I3×3, G = TsB = TsJ−1 and Hk = Φ+KT

T (c∗k) are the discrete-time state-spacematrices of the system.

4.2.2 Linear Parameter-Varying Kalman Filter

The Kalman Filter bases its optimal state estimation on the correct model of the system, whichalso includes de noise and covariance propagation.

The LPV-KF estimator, during each sampling period, predicts the state estimation by usingthe available data of the previous sampling time, which is going to be called State Prediction,and afterwards, as soon as the measurements are available, it corrects these predictions andcovariance matrices, or Measurement Update.

40 ETSII-UPM & EPFL

ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

For further information regarding Kalman Filter and optimal state estimation, please referto [10], an extensive overview of Kalman ltering is included, and the same notation is used inthis document.

In this section, the covariance propagation of the states is going to be derived, and the useof the measurements will be shown, which cover the state prediction and measurement updatesteps of the KF, respectively.

State Prediction

As it has been done in [10], lets dene x−k as the a priori estimation at instant k, while x+k as

the a posteriori state estimation an instant k. Then, by using the discrete-time system model,the a priori estimation will be obtained by computing:

x−k = x+k−1 +Guk−1 (4.17)

Note the dierence between u and u: the former corresponds to the desired torque computedby the controller, and the latter to the true torque applied to the rotor, which is not available.

The covariance of the state is stored in P matrix, and its evolution along time can bestudied by considering the denition of a covariance matrix, and comparing the real state withthe estimation:

Pk = E[(xk − xk)(xk − xk)

T]

=

= E[(Fxk−1 +Guk−1 + wk−1 − xk) (· · · )T

]=

= FPk−1FT +Qk−1

(4.18)

where Qk = E[(xk − E[xk])(xk − E[xk])

T]

= E[wkw

Tk

]is the covariance matrix of the

discrete-time model noise. Then, the following equation is used for computing the propagationof covariances after an a priori estimation:

P−k = P+k−1 +Qk−1 (4.19)

which follows the same notation as before, by using the super-index + for a posteriori values,and − for a priori data.

Guzman Borque Gallego 41

CHAPTER 4. ANGULAR VELOCITY ESTIMATION

Measurement Update

Once the measurement data is available, the estimation is corrected by comparing the expectedoutput for the a priori state with the actual output value. In order to weight this correction, theobserver gain matrix Lk is computed by considering the uncertainties of states and measurements:

Lk = P−k HTk (HkP

−k H

Tk +R)−1

x+k = x−k + Lk(yk −Hkx

−k )

(4.20)

Note that matrixHk is not xed, but it changes at each sampling time as a linear combinationof the magnetic state, estimated by using the approach explained at 3.5. This situation impliesthe introduction of a Linear Parameter-Varying Kalman Filter. For this case, a Sampled-DataSystem is considered, thus, the covariance propagation after measurement update for this typeof systems should be used:

P+k = (I − LkHk)P

−k (I − LkHk)

T + LkRLTk (4.21)

where R = E[(yk − E[yk])(yk − E[yk])

T]

= E[vkv

Tk

]is the covariance matrix of the

measurement noise and I a 3× 3 identity matrix. For further information about the derivationof the previous equations, [10] can be consulted.

4.2.3 Final Algorithm

The resulting algorithm, based on a Linear Parameter-Varying Kalman Filter state-space estim-ator can be summarised in the following steps:

1. Kalman Filter Initialisation, initial state estimation x+0 and covariance matrix P+

0 , fork = 0:

x+0 = E[x0]

P+0 = E

[(x0 − x+

0 )(x0 − x+0 )T

] (4.22)

2. For k = 1, 2, 3 . . . compute:

(a) Magnetic State estimation by least-squares projection of Hall Sensors Measurementsand compute required values:

c∗k = HprojB⊥k

c∗k =c∗k − c∗k−1

Ts

KT,k =

7∑m=1

KmT c∗k,m

(4.23)

being c∗k,m the m-th component of the magnetic state at instant tk.

42 ETSII-UPM & EPFL

ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

Alngular Velocity Estimator

Kalman

Filterz−1z

cEstimation

B⊥ c

˙c ω

T cmd

Figure 4.1: Graphical Representation of Angular Velocity Estimator.

(b) A priori estimation of the system's state and covariance matrix, by making use of Eq.4.17 and Eq. 4.19:

x−k = x+k−1 +Guk−1

P−k = P+k−1 +Qk−1

(4.24)

(c) Observer gain matrix Lk computation, and a posteriori update of state estimation x+k

and covariance matrix P+k :

Lk = P−k HTk (HkP

−k H

Tk +R)−1

x+k = x−k + Lk(yk −Hkx

−k )

P+k = (I − LkHk)P

−k (I − LkHk)

T + LkRLTk

(4.25)

The aforementioned process can be represented graphically as shown in Figure 4.1. Thedierent blocks represented in the gure correspond to the steps expressed before: Magneticstate estimation, by using least-squares projection of the Hall sensor measurements, discrete-time derivative of the magnetic state, and the Linear Parameter-Varying Kalman Filter, whichupdates state estimates and covariance matrices.

4.3 Noise Model

In order to obtain a good performance and reliable estimations with a Kalman Filter, the system'snoise has to be correctly modelled. In this section the theoretical development of the noise modelis going to be explained.

The main noise models that have to be derived are the one related to the state-space equationQk, and the second related to the output equation Rc∗ . In the following sections the models foreach component will be shown.

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CHAPTER 4. ANGULAR VELOCITY ESTIMATION

Qk Model Derivation

In Kalman Filter literature, as it is the common situation for almost every system, the input isperfectly known, and thus, no uncertainty related to the input is carried on by the covariancematrix of the states P, which is not the case here. In order to identify the impact of theseuncertainty, the propagation of covariances has to be revised.

For this particular case, as the input to the system cannot be perfectly known, due to thethird-degree approximation of Spherical Harmonics used for computing the torque characteristicmatrix KT , which will include associated errors, the uncertainty associated with the input hasto be included.

Starting from the denition of the covariance matrix of the states, and studying its propaga-tion after a prediction, the relation between the uncertainty related to the input and the onecarried on by the estimations:

Pk = E[(xk − xk)(xk − xk)

T]

=

= E[(Fxk−1 +Guk−1 + wk−1 − xk) (· · · )T

]=

= FPk−1FT +GE

[(uk−1 − uk−1)(uk−1 − uk−1)

T]GT +Qfix,k−1 =

= Pk−1 +GQT,k−1GT +Qfix,k−1 =

= Pk−1 + J−2T 2sQT,k−1 +Qfix,k−1 =

= Pk−1 +Qk−1

(4.26)

where Qk is the covariance matrix of state propagation for the discrete-time system, whichcan be decomposed in two: QT,k = E

[(uk−1 − uk−1)(uk−1 − uk−1)T

]is the covariance matrix

of the input, and Qfix,k a xed term of the covariance matrix.

As it has been done in [9] and in [55], the uncertainty in the applied torque can be approx-imated as a percentage of the modulus of the desired torque. After some preliminary simulationsit is estimated as a 10%, resulting in:

QT,k = 0.1‖Tcmd,k‖2 =⇒ Qk = J−2T 2s 0.1‖Tcmd,k‖2 +Qfix,k (4.27)

The selection of the Qfix,k matrix values is going to be chosen, on the one hand, for prevent-ing null covariance when the command torque is 0, and on the other hand, for placing the polesof the estimator in a desired position, i.e. for choosing a good compromise between estimatorspeed and noise rejection.

A more in-depth study of its selection will be explained in Section 4.4, as it is related withthe analysis of the closed-loop system.

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Rc∗ Model Derivation

As the noise in the time-derivative of the magnetic state, the output of the system's model, isdicult to measure and quantify directly, its model is going to be derived by starting from thesensor's measurement noise, which can be directly measured.

Considering a given hall sensor measurement noise, modelled by the covariance matrix RB⊥ ,and the magnetic state estimation, which relates the aforementioned sensor measurements withthe magnetic state, the covariance propagation after a linear combination of random variablesresults in the covariance matrix of the magnetic state estimation Rc∗ :

c∗k = HprojB⊥k =⇒ Rc∗ = HprojRB⊥H

Tproj (4.28)

At this point, a rst-order approximation of a time-derivation is computed, thus, consideringthat the measurements at two consecutive sampling times are independent one from the other,the covariance matrix of the time-derivative of the magnetic state R is:

c∗k =c∗k − c∗k−1

Ts= fs(c

∗k − c∗k−1) =⇒ Rc∗ = 2f2sRc∗ (4.29)

Thus, the nal relation for modelling the magnetic state time-derivative is:

Rc∗ = 2f2sRc∗ = 2f2sHprojRB⊥HTproj (4.30)

In order to specify the measurement noise of the Hall sensors, instead of directly consideringtheir noise, due to the third-degree approximation of the spherical harmonics, also the modellingerror is included. For that purpose the error between the real measurement for dierent rotororientations and its estimations using the magnetic state is considered, and its variance, σ2, iscomputed.

The standard deviation σ of the sensor measurements is computed by using the data obtainedfrom the experiment shown for the model validation in Appendix B. For the known evolutionof the orientation of the rotor, the estimation of all the Hall sensor measurements is computedby using up to degree 3, and it is substracted from the real Hall sensors measurements, givingan evolution of the error for the 15 sensors. Then, the standard deviation of this evolution iscomputed for each sensor and the resulting values are averaged between all the sensors, andincluded in a 15× 15 diagonal matrix RB⊥ = σ2I15×15.

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CHAPTER 4. ANGULAR VELOCITY ESTIMATION

4.4 Controller and Closed-loop Design

The estimator behaviour cannot only be studied independently from the rest of the system, thus,the combination of the controller and estimator has to be taken into account.

The particularity of the considered system is the presence of a Linear Parameter-VaryingModel, which increases the complexity of the stability and performance analysis of the closed-loop system in an analytical way. Dierent approaches have been proposed for controller designof a Linear Parameter-Varying Model, generally they can be classied in two main groups:

• Gain-Scheduling: dierent controllers are designed for dierent operating points, andthen the controller parameters are interpolated. This approach has been studied in [56]or [57] among others. This approach is characterised by faster and easier computations ofthe controllers, but stability is dicult to guarantee, specially if the scheduling parametersvary fast, as in this case.

• Parameter-Dependent Lyapunov Function (PDLF): restrictions derived from Lya-punov's stability constraints are imposed, as done, for example, in [58] or [59]. In this case,the design of the controller is more complex, but stability is guaranteed.

The main diculty that can be found while trying to implement any of these approaches, isthat, the fact of using a Kalman Filter for estimating the states of the system (angular velocity)generates a non-constant observer gain matrix L, and additional dynamics are added to theclosed-loop, thus it has also to be considered in the analysis, which increases the complexity ofthe process.

For this thesis the stability of the Linear Parameter-Varying Kalman Filter and of theclosed-loop system including the controller is not proven analytically, but as it will be analysedin the next chapter, the experimental results of the observer and the overall system show thestability and good performance of the proposed estimator, being able to reduce considerably theoscillations and maximum error compared to the previous method.

In order to analyse the closed-loop behaviour of the Reaction Sphere, the approach proposedin [11] for state-space design is going to be studied. In this case, the closed-loop system willcontain the plant (Reaction Sphere), the state observer, as no direct measure of the state isavailable, and the controller. In Figure 4.2, the angular velocity control loop is shown.

For the closed-loop analysis of the state-space system, four independent steps are going tobe followed:

1. Control-law design, by assigning a set of pole locations for the closed-loop system.

2. As the full state is not directly known, the dynamics of the estimator have to be considered.

3. Combination of the control law and estimator in the closed-loop system, which is alsoknown as compensation.

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

Compensator

Controller Plant

Observer

r e u y

x

-

δu δy

Figure 4.2: Angular Velocity Control Loop.

4. Adding the reference signal.

As explained before in the current document, the input of the plant cannot be perfectlyknown. This uncertainty is considered as a disturbance δu. Also, the measurement noise andhigher order non-modelled components of the magnetic ux are considered in the output dis-turbance δy.

As shown in [11], if a linear system is considered, represented by its state-space model denedby the equations:

x = Ax +Bu

y = Cx(4.31)

where, as before, x represents the state-space vector of the system, u the input, and y theoutput vector. For this system a Full State Feedback controller (that in this case corresponds toa Proportional controller) is going to be studied.

As the state of the studied system is composed of the three components x-y-z of the angularvelocity, the resulting gains of the controller will be the same for the 3 states, as the desiredperformance of the closed-loop system for each component is the same.

4.4.1 Full State Feedback Controller

The rst chosen controller is the same proposed in [4], a proportional controller whose actuationat instant k follows the next equation:

up,k = Kp(xref,k − xk) = Kpek (4.32)

having Kp as the gain of the proportional controller, which will be xed in order to achievea desired level of noise rejection and tracking performance, and xref is the reference signal of thestate, the desired value for the state of the system.

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CHAPTER 4. ANGULAR VELOCITY ESTIMATION

1. Introducing this control law to Equation (4.31), considering a reference signal of zeroamplitude xref = 0, and a perfect knowledge of the system's state x = x, the dynamics ofthe state can be expressed as:

x = Ax +Bup = Ax−BKpx (4.33)

Then the characteristic equation of this closed-loop system, only considering the controllaw, would be as follows:

det [sI − (A−BKp)] = αc(s) = 0 (4.34)

2. In order to analyse the estimator's behaviour, the error in the estimation is dened asx = x− x, and its evolution over time is expressed as:

˙x = x− ˙x

= [Ax +Bup]− [Ax +Bup + L(y − Cx)]

= (A− LC)x

(4.35)

Then the characteristic equation of this closed-loop system, only considering the controllaw, would be as follows:

det [sI − (A− LC)] = αe(s) = 0 (4.36)

In this case the matrix L is computed as shown previously in this chapter, by making useof the Kalman Filter gain computation of optimal estimation. This approach will makethe matrix vary over time, and thus, the linear analysis will only be considered as a rstapproximation.

3. If The control law in Eq (4.33) is combined with a full order estimator, as it is the case forthe Reaction Sphere, the overall system dynamics in state form, can be represented as:

[x˙x

]=

[A−BKp BKp

0 A− LC

] [xx

](4.37)

Then the characteristic equation of the closed-loop system can be expressed as follows:

det [sI − (A−BKp)] det [sI − (A− LC)] = αc(s)αe(s) = 0 (4.38)

which corresponds to the complete set of poles made by the union of control poles andestimator poles. Thus, if the system is linear, the design of the control law and the estimatorcan be carried out independently and when combined, their poles stay unchanged.

For the studied system of the Reaction Sphere, the only matrix that is parametric-dependentis the output matrix C for the continuous-time system, or H for the discrete-time one, thus itis the observer's dynamics the ones that are going to be aected by this variations of the LPV

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

system. As a Kalman Filter is used as an observer, the gain matrix of the estimator L also varieswith time, but it has been proven that, as the parameters c cannot take any possible value, andthey are bounded inside the hypercube [−0.5, 0.5]7 ∈ <7, the variability of matrix C or H islimited, and the poles of the estimator vary always between a xed interval.

The gain of the controller will be chosen in such a way that the poles xed by the control-ler are slower (closer to the unit circle for discrete-time systems, or to the imaginary axis forcontinuous-time systems) than the ones xed by the estimator. Thus, experimentally, the polesof the characteristic polynomial of the estimator are computed, either by using experimentalor simulated data, and the gain of the controller is xed by pole placement, choosing slowerdynamics. The process can only be considered as a rst approach, as the system itself is notlinear, and the variations of matrix C or H are fast (directly related with the angular velocityof the rotor), and then they have to be tested in simulation, and further ne-tuned in the realsystem.

As it will be shown in Section 5.4, when using a proportional controller with the proposedestimator, only in the experimental set-up with the real prototype, a permanent error is presentin all of the components, thus a controller with integral action should be also considered forfuture work in order to reject this permanent error in the system.

4.5 Conclusion

In this chapter, the proposed angular velocity estimator has been developed, also comparedwith previous approaches for angular velocity estimation that has been tried in the past. Thetheoretical derivation of the reaction sphere and estimator models has been shown, by making useof a state-space representation and a Linear Parameter-Varying Kalman Filter as an observer.Also the chapter has been concluded by showing the selection of the controller used for closed-loopcontrol of the angular velocity, which correspond to a full-state feedback controller.

Thus, the main contribution of this chapter is the theoretical development of a LinearParameter-Varying Kalman Filter, a novel approach for state-space estimation based on a KFobserver for LPV systems. No theoretical stability have been proven analytically, but results innext chapter will show its good performance and convergence to the real state for the analysedconditions.

In the following chapter the results of the implementation of the proposed estimator aregoing to be shown and analysed, as well as the description of the simulation model used fortesting the estimator and control laws.

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Chapter 5

Simulation and Experimental Results

This chapter will show the results obtained by implementing the proposed angular velocityestimator both in simulation and in the real prototype, thus the analysis will be split in Simulationand Experimental Results.

First the developed Simulink model of the Reaction Sphere has to be developed and val-idated, and the set-up used for this process will be shown. Afterwards the results obtainedby combining the dierent levels of abstraction of the Reaction Sphere model and the angularvelocity estimator are explained.

The experimental results obtained with the real prototype are classied in open-loop andclosed-loop experiments. In open-loop analysis, the proposed estimator is run in parallel withthe original closed-loop structure, while in closed-loop experiments, the previous angular velocityestimator is substituted by the proposed Linear Parameter-Varying Kalman Filter.

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5.1 Introduction

Once the proposed estimator has been designed, which includes not only the system's model,but also its noise propagation, the performance of the aforementioned design has to be tested,and the results of this procedure will be shown in the following sections.

In order to check if the results that are going to be obtained in simulation are going tobe close to reality, the model used for this purpose will be developed and validated. Then, theexperiments that has been carried on in order to study the behaviour of the estimator are goingto be classied in two main groups:

• Open-loop Experiments: the proposed estimator is executed in parallel by taking thetorque that wants to be applied to the system, and the Hall sensor measurements generatedby simulation or the real system. For the real system, as it always requires a closed-loopcontrol for working, the previous approach based on back-EMF is used for angular velocityestimation that is used by the CL controller.

• Closed-loop Experiments: the proposed estimator is included in the closed-loop system,substituting the previous method for angular velocity estimation.

In both cases, simulated and experimental results will be included, as this mixed approachhas been the one used for the development and tuning of the angular velocity estimator.

5.2 Simulation Models Description

Before implementing the angular velocity estimator, a truthful model of the Reaction Sphere hasto be developed and validated. For this purpose, the model initially designed in [9] is taken as abasis. First, the dierent levels of abstraction of the simulation model are going to be explained:

• Reduced Model: at this level of abstraction, the Reaction Sphere's model takes as inputdirectly the torque, and as output the angular velocity and Hall sensors' measurements.This model allows faster simulation but less accurate simulations, as no electromagneticinteraction between stator and rotor is considered for applying the desired force and torque.

• Complete Model: in this case, the input of the system is the set of currents applied tothe stator's coils, and the outputs are, as before, the angular velocity and Hall sensors'measurements. This model is heavier, computationally speaking, but is able to simulatethe force and torque applied to the rotor as a function of the input currents. Thus theerror between desired and real torque applied is taken into account.

For both models, the reaction sphere is considered in the center of the stator, thus, positionand angular velocity control problems are independent one from each other, allowing decoupledcomputations, even though in reality this condition is not always satised.

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5.2.1 Reduced Model

This model simulates, rst the kinematics and dynamics of the rigid solid that constitutes theReaction Sphere's rotor in order to simulate the orientation (quaternion) and angular velocityevolution. Afterwards, at each sampling time, the quaternion is used for rotating the desirednumber of spherical harmonics whose linear combination will yield the radial magnetic uxdensity at each sensor position. In Figure 5.1 the block scheme of the aforementioned model isshown.

Reaction Sphere

Reaction Sphere

Kinematics & DynamicsB⊥

Estimation

T bias

T cmd

q B⊥

ω

Figure 5.1: Reaction Sphere's Reduced Model for simulation.

The approach proposed for this Reaction Sphere's Reduced Model is based on the same setof dierential equations proposed in Section 4.1.2 as follows:

ω = J−1T = J−1(Tcmd + Tbias)

q =1

2Ωq =

1

2GTω

(5.1)

where T cmd is the command torque, or desired torque to apply to the sphere, and T bias thetorque disturbance at the input of the system. This disturbance is added in order to simulatethe existing error between the command torque and the real torque applied to the sphere, dueto the approximation used for compute the torque characteristic matrix.

The evolution of the orientation and angular velocity is computed by Kinematics and Dy-namics block, while the Hall sensor measurements estimation is executed by B⊥ Estimationblock.

The procedure of estimating sensor measurements is equivalent to the one proposed in [9],by rotating spherical harmonics using the orientation of the rotor, but with the dierence thatin this case, depending on the desired level of accuracy in the estimation, the maximum degreeof spherical harmonics Nmax can be chosen.

In order to estimate the aforementioned measurements, the coecients for spherical har-monics decomposition are computed by using the previously computed orientation to rotatethe spherical harmonic coecients cmn,imm, as expressed in Section 3.3, where the magnetic uxdensity model was derived. Then, the radial component magnetic ux density B⊥i = Bi,rs isevaluated at the dierent Hall sensor's position, specied by its spherical coordinates (rs, θs, φs),as a linear combination of the rotated spherical harmonic coecients:

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CHAPTER 5. SIMULATION AND EXPERIMENTAL RESULTS

Bi,rs(rs, θs, φs) =∑n∈=n

n≤Nmax

n∑m=−n

cmn (α, β, γ)Bmi,rs,n (5.2)

where cmn (α, β, γ) denotes the rotated spherical harmonic coecients as a function of the ori-entation of the rotor, expressed using Euler angles α, β and γ. Due to rotor symmetry, the numberof spherical harmonics hat are considered are included in the set =n = 3, 5, 7, 9, 11, 13 . . . .

5.2.2 Complete Model

At this level of abstraction, the torque applied to the Reaction Sphere and Hall sensor meas-urements are simulated by using Electromagnetic Model block, where the torque characteristicmatrix and spherical harmonics rotation is done. Afterwards, this torque is introduced in theKinematic and Dynamic Model block, which computes orientation and angular velocity of therotor. In Figure 5.2 a graphical representation of this model is shown.

Reaction Sphere

Electromagnetic

Model

Kinematic & Dynamic

Model

icmd T ω

q

B⊥

Figure 5.2: Reaction Sphere's Complete Model for simulation.

As before, the orientation and angular velocity of the rotor are modelled by using thedynamics of a rigid solid, and the Hall sensor measurements by rotating the spherical harmonicsup to degree Nmax, as shown in Eq. 5.1.

For this model, the reduced Reaction Sphere model is completed by adding the torquecharacteristic matrix estimation, which is computed as expressed in Section 3.4, i.e. by a linearcombination of the force characteristic matrices associated to each spherical harmonic of degreen and order m:

KT (c) =

Nmax∑n=0

n∑m=−n

cmn (α, β, γ)KmT,n ⇒ T = KT (c)icmd (5.3)

Then, the torque applied to the sphere is computed as expressed in the previous equation:by multiplying the torque characteristic matrix KT and the current generated by the controllericmd.

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In order to prevent an algebraic loop in the simulation, due to the interdependence ofElectromagnetic Model andKinematic and Dynamic Model blocks, as for computing the evolutionof the orientation of the rotor, the applied torque is required, and for computing the latter, theorientation is needed, an unitary delay is added at the orientation (quaternion) feedback.

Before being able to use the developed model, a validation is required, in order to verify howclose to the real system the model is. The process of validation of the reduced model is includedin Appendix B.

5.3 Open-loop Experiments

Once the simulation model has been validated, it is expected that the results obtained by usingthis model are accurate enough for analysing the behaviour of the proposed Kalman Filter. Inthis section the behaviour of the proposed observer is going to be studied for an open-loopconguration in simulation, and running o-line for the experiments with the Reaction SpherePrototype.

First the simulation results are going to be showed, both for the reduced and completemodel of the Reaction Sphere. Afterwards the experimental results will be analysed for dierentexperiments.

5.3.1 Simulation Results

As expressed before, the analysis will be dierentiated for the reduced and complete model of theReaction Sphere. This will allow to study in an incremental level of complexity the behaviour ofthe proposed angular velocity estimator.

Reduced Model

This model will be used to test the general behaviour of the observer and check the inuence ofthe disturbances available at the input and output of the plant, e.g. the error between the desiredand applied torque simulated by Tbias and Hall sensors disturbances. For each experiment, theset-up showed in Figure 5.3 will be used to execute the simulation in Simulink.

As it can be seen in Table B.1, the dierence in results between the available Nmax is notbig for the reduced model, thus, a value of Nmax = 3 is chosen for improving computer eciencyduring simulations, as the execution time is reduced considerably. The experiments that hasbeen carried on by using the reduced model in simulation can be seen in Table 5.1.

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CHAPTER 5. SIMULATION AND EXPERIMENTAL RESULTS

Reaction Sphere

Alngular Velocity Estimator

Reaction Sphere

Kinematics & DynamicsB⊥

Estimation

cEstimation

Kalman

Filterz−1z

T bias

T cmd

q B⊥

ω

c

˙c ω

Figure 5.3: Simulation Set-Up for Reduced Model of the Reaction Sphere.

No. fs Tcmd Tbias Nmax Description

1 2.5 kHz None None 3 Experiment for verifying the speed of con-vergence of the estimation to the real an-gular velocity.

2 2.5 kHz Step None 3 Experiment that shows the response of astep input to the estimator.

3 2.5 kHz None Sinusoidal 3 Experiment that shows the inuence of thetorque disturbance in the estimations.

Table 5.1: Simulation Experiments for Reduced Model.

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

0 5 · 10−2 0.1 0.15 0.20

50

100

150

Time (s)

AngularVelocity

(rpm)

Angular Velocity

Real ωy

Estimated ωy

0 5 · 10−2 0.1 0.15 0.2

0

50

100

150

Time (s)AngularVelocity

(rpm)

Angular Velocity Error (Real-Estimation)

Figure 5.4: Experiment 1 Results. Comparison between the real angular velocity of the RS's model inblue and the KF's estimation in red (left). Error of the estimation: error = real − estimation (right)

• Experiment 1: Constant angular velocity (no external torque applied) with incorrectinitialisation of Angular Velocity estimator. Standard deviation of white noise in measure-ments 0.1 mT.

In Figure 5.4, the obtained results for this initial experiment are shown. The estimator isinitialised with a angular velocity of 0 rpm around all the axis, while the Reaction Sphere'smodel with 300 rpm around axis [1, 1, 1]T . Only the y-component of the angular velocityis shown, as the rest of them are equivalent.

In this experiment it can be seen that the observer's estimation converges to the real valueof the angular velocity in 0.02 seconds approximately. The speed of convergence and theamount of residual noise will depend on the choice of the covariance matrices of the sensormeasurements and of the state propagation, R and Q respectively, which will be xed forthe experimental set-up.

For the following experiments, only the steady state is going to be shown, skipping theinitial convergence to the real initial angular velocity, in order to show more clearly thedesired results of the experiments.

• Experiment 2: Step command torque without disturbance in the torque (Tbias = 0.Standard deviation of white noise in measurements 0.1 mT.

This experiment is carried on in order to analyse the inuence of the input of the systemwhen there is no torque error. As input, a step of 0.1 Nm is applied to axis x at instant0.2s of simulation, with a zero bias torque. The results of this experiment can be observedin Figure 5.5.

It can be seen that the estimation follows the real angular velocity with a small delayand a non-ltered part of the measurement noise. The maximum error obtained for thisexperiment is below 5 rpm. The increase of the magnitude of the torque applied to theReaction Sphere will increase also the maximum error obtained in the experiment, as thesystem's angular acceleration will be higher.

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CHAPTER 5. SIMULATION AND EXPERIMENTAL RESULTS

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

5

10

·10−2

Time (s)

Torque(N

m)

Command and Applied Torque along X-axis

Command Tx

Applied Tx

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−100

−50

0

50

Time (s)

AngularVelocity

(rpm)

Angular Velocity along X-axis

Real ωx

Estimated ωx

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−4

−2

0

2

4

Time (s)

AngularVelocity

(rpm)

Angular Velocity Error along X-axis (Real-Estimation)

Figure 5.5: Experiment 2 Results. Command and applied torque (top). X-component of real andestimated angular velocity (middle). X-component of angular velocity estimation error: error = real −estimation (bottom).

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0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7−1

−0.5

0

0.5

1

Time (s)

Torque(N

m)

Command and Applied Torque along X-axis

Command Tx

Applied Tx

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7−40

−20

0

20

40

60

Time (s)

AngularVelocity

(rpm)

Angular Velocity along X-axis

Real ωx

Estimated ωx

Figure 5.6: Experiment 3 Results. Command and Applied Torque (top). Real and Estimated angularvelocity (bottom).

This experiment shows that the estimator is able to correctly estimate the angular velocityif the input is perfectly known.

• Experiment 3: Constant desired angular velocity (no command torque applied) withsinusoidal torque disturbance (Tbias) of 1 Nm of amplitude and 10 Hz of frequency. Standarddeviation of white noise in measurements 0.1 mT.

The results of this experiment can be seen in Figure 5.6. It is possible to see that even if noinput command torque is applied to the observer, it is still able to follow the real angularvelocity by using the information provided by the Hall sensors. In this case there is ahigher delay that in the previous experiment mainly due to the lack of a priori informationregarding the torque applied to the reaction sphere, thus, the observer has to correct theestimations a posteriori.

This experiments shows that the estimator is able to correct errors at the input torque ofthe system by using sensors' information. A better knowledge of the input it will makethe estimations converge faster to their real values, but it will still converge otherwise. A

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CHAPTER 5. SIMULATION AND EXPERIMENTAL RESULTS

good compromise between convergence speed and noise rejection has to be found in theclosed-loop analysis of the system.

Once the behaviour of the proposed estimator is analysed by using the most basic model,more factors have to be taken into account by using the complete model, which will be shown inthe next section.

Complete Model

In this section, the behaviour of the Angular Velocity Estimator is going to be analysed by usingthe complete model of the Reaction Sphere. By using this model, shown in Figure 5.7, thedisturbance and error existing at the input (torque) is modelled more closely to reality by usingthe electromagnetic model of the sphere that computes the torque characteristic matrix.

Reaction Sphere

Alngular Velocity Estimator

Control

Current

Estimation

Electromagnetic

Model

Kinematic & Dynamic

Model

Kalman

Filterz−1z

cEstimation

T cmd icmd T ω

q

B⊥

B⊥ c

˙c ω

Figure 5.7: Simulation Set-Up for Complete Model of the Reaction Sphere.

Furthermore, in order to be able to run the model, the command torque has to be convertedto command currents of the stator coils. In order to do this, the same approach as in the real-time controller is used: by estimating the torque characteristic matrix only using up to sphericalharmonic of degree 3.

In this case, the inuence of the chosen value for Reaction Sphere's Model Nmax is import-ant, as it will increase the disturbance generated at the input due to the consideration of higherharmonics when computing the applied torque. For this reason, a value of Nmax = 11, whichis the maximum available value, is preferred for this set of experiments. The experiments thathave been carried out are enumerated and briey explained in Table 5.2.

1Nmax value only used for Reaction Sphere Model. For control current estimation Nmax = 3 is always used.

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No. fs Tcmd Nmax1 Description

4 2.5 kHz None 11 Experiment with constant angular velocitywithout command torque.

5 2.5 kHz Step 11 Experiment that shows the response of astep in the command torque.

6 2.5 kHz Sinusoidal 11 Experiment with sinusoidal commandtorque.

Table 5.2: Simulation Experiments for Complete Model.

• Experiment 4: Constant angular velocity (no command torque applied) with incorrectinitialisation of Angular Velocity estimator. Standard deviation of white noise in measure-ments 0.1 mT.

This experiment is equivalent to Experiment 1 shown for the reduced model. In this caseas the disturbance torque cannot be controlled, a more realistic situation is simulated, eventhough not perfect yet, as no gravity nor positioning of the rotor is considered. In Figure5.8, the results of the experiment can be seen.

0.2 0.4 0.6

−60

−40

−20

0

Time (s)

AngularVelocity

(rpm)

Angular Velocity

Real ωy

Estimated ωy

0.2 0.4 0.6

−5

0

5

Time (s)

AngularVelocity

(rpm)

Angular Velocity Error (Real-Estimation)

Figure 5.8: Experiment 4 Results. Comparison between the real angular velocity of the RS's model inblue and the KF's estimation in red (left). Angular velocity estimation error: error = real− estimation(right)

As shown in Table 5.2, the use of higher order harmonics, NMAX = 11, has an impact in theestimations obtained by the observer, which results in oscillations in the aforementionedvalues around the real value of angular velocity. As no torque is applied, the inuence ofthe approximation in the torque characteristic matrix cannot be studied in this experiment.

The observer is able to estimate the real value of angular velocity with the presence of oscil-lations around this value due to the non-modelled higher harmonics. Maximum deviationaround 5 rpm for this set-up.

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• Experiment 5: Experiment in which the command torque is a pulse of 0.1 Nm of amp-litude and a duration of 0.3 seconds. Standard deviation of white noise in measurements0.1 mT.

This experiment shows the inuence of the higher harmonics that have not been consideredin the observer and in the control current estimator. In Figure 5.9 the results of thesimulation can be observed.

This set-up combines the inuence of higher harmonics in the hall sensors' measurementsand in the computation of the real torque applied to the Reaction Sphere. In the top plotof the gure, the dierence between the real applied torque and the command torque canbe observed, which is reected in slight variations in the angular acceleration of the rotor.The overall performance of the estimator is comparable to the previous case in Experiment4.

• Experiment 6: Sinusoidal command torque applied to all the axes. The amplitude of thetorque is 1 Nm and the frequency 5 Hz. Standard deviation of white noise in measurements0.1 mT.

In Figure 5.10, the results of the experiment are shown for component Z, being the othersequivalent to this one. It can be seen that the maximum error is higher than in experiment5, due to the higher torque applied to the rotor, which generates a faster variation of theangular velocity.

The observer, as happened in Experiment 3, is able to follow the real angular velocity,but with certain delay, due to the dynamics of the Kalman Filter, which requires time toconverge to the nal value. Thus, the higher the torque applied is, the higher the maximumerror will be. Then a compromise between speed and noise rejection have to be selected innext section, by using the experimental values of the real prototype.

5.3.2 Experimental Results

As explained before, in this section the proposed estimator is going to be analysed by usingavailable experimental data from the real prototype, thus, the estimator is computed o-line,or in parallel with the closed-loop estimator, by making use of the aforementioned data. TheReaction Sphere has to be controlled in closed-loop always, as it is actively levitated, the positionand angular velocity have to be controlled to avoid any collision of the rotor with the stator.

In Table 5.3, the experiments that have been carried out by using the real prototype andthe proposed estimator in open-loop are enumerated and briey detailed. The main purposeof these experiments is comparing the results of the two available angular velocity estimators:Back-EMF and Kalman Filter.

• Experiment 7: Angular velocity reference signal of the controller is incremented step bystep until reaching a norm of 300 rpm approximately. The torque and hall sensor valuesare saved in memory during the execution of the control loop. The proposed observer iscomputed o-line by using available data.

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0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

0

5

10

·10−2

Time (s)

Torque(N

m)

Command and Applied Torque along X-axis

Command Tx

Applied Tx

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

−140

−120

−100

−80

−60

Time (s)

AngularVelocity

(rpm)

Angular Velocity along X-axis

Real ωx

Estimated ωx

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7−10

−5

0

5

10

Time (s)

AngularVelocity

(rpm)

Angular Velocity Error along X-axis (Real-Estimation)

Figure 5.9: Experiment 5 Results. Command and applied torque (top). X-component of real andestimated angular velocity (middle). X-component of angular velocity estimation error: error = real −estimation (bottom).

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1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7

−1

0

1

Time (s)

Torque(N

m)

Command and Applied Torque along Z-axis

Command Tz

Applied Tz

1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7−100

0

100

Time (s)

AngularVelocity

(rpm)

Angular Velocity along Z-axis

Real ωz

Estimated ωz

1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7−20

−10

0

10

20

Time (s)

AngularVelocity

(rpm)

Angular Velocity Error along Z-axis (Real-Estimation)

Figure 5.10: Experiment 6 Results. Command and applied torque (top). X-component of real andestimated angular velocity (middle). X-component of angular velocity estimation error: error = real −estimation (bottom).

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No. fs ωref Computation Description

7 2.5 kHz Stepped O-line Angular velocity reference is incremen-ted step by step up to 300 rpm for eachcomponent.

8 1.25 kHz Stepped On-line Angular velocity reference is incremen-ted step by step up to 40 rpm for eachcomponent.

9 1.25 kHz Sinusoidal On-line Sinusoidal angular velocity reference ofamplitude 25 rpm and 60 seconds ofperiod for each component.

Table 5.3: Experiments in Real Prototype with open-loop estimator.

The main goal of this experiment is being able to study the possible performance of theproposed estimator, compared to the previous proposed approach. Even though the resultswill not be denitive, it can show the capabilities of the estimator at dierent speeds. InFigure 5.11, the results for the X-component of the angular velocity of the experiment canbe observed. The other components are equivalent as the angular velocity reference wasapplied around axis ω = [1, 1, 1]T .

In the top plot of the gure, it can be seen that the estimation generated by the KalmanFilter follows the reference signal, but also has a considerable level of noise or oscillations.In the bottom part, the Back-EMF estimation and the Kalman Filter estimations arecompared, in which a considerable decrease in the noise can be seen by using the latter,specially at high rotation speeds. On the contrary at lower speeds, the dierence is notnoticeable.

As the Back-EMF is used for closed-loop control, the oscillations in the estimation of theKalman Filter does not only depend on the level of noise in the Hall sensor measurements,but it is also a result of real oscillations of the rotor, due to the use of noisy estimationsfor control.

• Experiment 8: Angular velocity reference signal of the controller is incremented step bystep until reaching a norm of 40 rpm approximately. The torque and hall sensor valuesare saved in memory during the execution of the control loop. The proposed estimator iscomputed on-line, in parallel to the closed-loop control system.

This experiment is similar to the previous one, with the exception of computing the al-gorithm on-line and in parallel to the control loop. This extra computation of the twoestimators in parallel requires a reduction in the sampling frequency, which is xed at 1.25kHz. The results obtained for this experiment are shown in Figure 5.12.

It can be seen that in this case using the Kalman Filter for estimating the angular velocitydoes not improve the performance when compared to the Back-EMF estimator. This is dueto the maximum speed that has been reached is considerably low, and as it also happenedin Experiment 7, at lower speeds, both approaches present a similar level of oscillations.The maximum rotational speed that could be achieved was mainly limited by the reducedsampling frequency of 1.25 kHz.

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Figure 5.11: Experiment 7 Results. X-component of reference angular velocity and Kalman Filter'sestimation (top). X-component of angular velocity estimation by using Back-EMF and Kalman Filter(bottom).

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Figure 5.12: Experiment 8 Results. X-component of reference angular velocity and Kalman Filter'sestimation (top). X-component of angular velocity estimation by using Back-EMF and Kalman Filter(bottom).

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Figure 5.13: Experiment 9 Results. Reference angular velocity and Kalman Filter's estimation (top).X-component of angular velocity estimation by using Back-EMF and Kalman Filter (bottom).

It can also be seen that the sense of the angular velocity reference does not aect theestimations, obtaining a similar behaviour in both cases.

• Experiment 9: Angular velocity reference signal of the controller is generated as a si-nusoidal signal of amplitude 25 rpm and 60 seconds of period for every component. Thetorque and hall sensor values are saved in memory during the execution of the control loop.The proposed estimator is computed on-line, in parallel to the closed-loop control system.

In this experiment, instead of varying the reference signal in a stepped way, a sinusoidalsignal is generated. The results can be seen in Figure 5.13. On the top gure, the referencesignal, in dashed black for all components, and the Kalman Filter estimations, in red, greenand blue for x, y and z components respectively, are compared. It can be seen that theestimations follow the reference signals for all the components, with an average error of 5rpm during the experiment.

On the bottom part of the gure, the estimations obtained by using the Kalman Filter,in blue, and by using the Back-EMF approach, in red, are compared for the X-component

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of the angular velocity. As it happened in the previous experiments, due to the maximumrotation speed reached in the experiment, the back-emf approach obtains a similar orslightly less noisy estimations.

In this section, the back-EMF approach for estimating the angular velocity and the pro-posed estimator are compared in dierent conditions. In lower rotation speeds, both approachesbehave in a similar way, but at higher speeds the Kalman Filter approach shows promising res-ults, reducing considerably the magnitude of the oscillations and noise in the angular velocityestimations. Also it is worth to point out that the magnitude of the error for the open-loop KFestimations does not increase with the angular velocity reference signal. Also the fact of havingused the back-EMF estimations for the closed-loop control induces the oscillations available inthe estimations to the real angular velocity of the rotor, so further reduction could be obtainedby using the Kalman Filter's estimation in the controller.

Once the open-loop analysis of the estimator has been performed, and after obtaining prom-ising results, the closed-loop analysis of the proposed observer will be performed in next section.

5.4 Closed-loop Experiments

In this section, the behaviour of the proposed estimator and the closed-loop system will beanalysed. In order to follow the same structure than the one used with the open-loop experiments,the simulation results will be shown rst, and the experimental ones right after, but in thedesign process of the system a mixed development have been used, combining simulation andexperimental results alternatively.

As explained in Section 4.4, the poles of the controller will be placed in such a way thatthe controller's dynamic is slower than the observer's one, in order to reduce the oscillationsgenerated if this condition is not satised. In order to do that, the order of magnitude of thecontroller's gain will be obtained in simulation, and nally it will be ne-tuned by using the realsystem. This procedure has to be repeated several times in order to nd a good compromisebetween noise rejection of the lter and convergence speed.

The controller used for these experiments is the one described in Section 4.4, whose gainhas to be established during this section for obtaining a reasonable performance.

5.4.1 Simulation Results

The developed model for analysing the closed-loop behaviour of the system is shown in Figure5.14, which corresponds to the complete model of Figure 5.7 with a controller that generatesthe torque input as a function of the desired angular velocity and the estimated one. The sameschema could be used for simulating the previous set-up by substituting the proposed estimatorby the one based on Back-EMF.

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Angular

Velocity

Controller

Control

Current

Estimation

Reaction

Sphere

Model

Angular

Velocity

Estimator

ωref

T cmd icmd

ω

B⊥

B⊥

ω

ω

Figure 5.14: Simulation Set-Up for Closed-loop analysis of the Reaction Sphere control system.

No. fs ωref Nmax Qfix Kcont Description

10 2.5 kHz Stepped 3, 11 0.01I 0.01I Angular velocity reference is incre-mented step by step up to 300 rpmfor each component.

11 2.5 kHz Sinusoidal 3, 11 0.01I 0.01I Sinusoidal angular velocity refer-ence of amplitude 25 rpm and fre-quency of 1 Hz for each compon-ent.

Table 5.4: Closed-Loop experiments in Simulation.

As in the previous cases, only the angular velocity control loop is considered, without takinginto account the position control loop. If the reaction sphere is centred in the stator, the positionand angular velocity systems can be decoupled.

In Table 5.4, the simulation experiments carried out for dening the correct combination ofestimator and controller gain are shown. The experiments carried out try to reproduce the samesituations as in Experiments 7, 8 and 9, in order to compare the behaviour of the closed-loopsystem when using the proposed estimator.

• Experiment 10: Angular velocity reference signal of the controller is incremented stepby step until reaching a norm of 300 rpm approximately. The experiment is executed forNmax = 3, 11.The set-up executed in this experiment is the equivalent to Experiment 7. Initially, byusing a faster simulation with Nmax = 3, the covariance matrix Qfix of the observer isselected for desired convergence speed of the observer and noise rejection, and afterwards,with Nmax = 11, the controller gain is chosen for minimising oscillations in the angularvelocity.

For simulation, the steps are applied more frequently in order to reduce the simulationtime, specially for Nmax = 11. For both cases, Nmax = 3 and Nmax = 11, the noise

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Figure 5.15: Experiment 10 Results (Nmax = 3). X-component of reference angular velocity and KalmanFilter's estimation (top). X-component of error in angular velocity along x-axis (bottom).

covariance xed matrix used is Qfix = 0.01I, where I ∈ <3×3 is the identity matrix. Thecontrollers gain is initially xed at K = 0.01I ∈ <3×3, and it will be further tuned in thefollowing experiments.

For Nmax = 3, the experiment results are shown in Figure 5.15. The standard deviationof the white noise applied to the Hall sensor estimation is 2.5 mT, which corresponds tothe standard deviation of the angular velocity error between the real measurements andthe estimations using Nmax = 3, measured in model validation experiment. In the upperplot, a comparison between the reference and the estimation can be seen, which apparentlyfollows properly the desired angular velocity. In the lower plot, the error is shown, whichnot only includes estimation errors, but also errors due to the chosen controller.

For Nmax = 11, the experiment results can be seen in Figure 5.16. The standard deviationof the white noise applied to the Hall sensor estimation is 1 mT. Compared to the previouscase, no

• Experiment 11: Angular velocity reference signal of the controller is generated as asinusoidal signal of amplitude 25 rpm and 60 seconds of period for every component. The

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Figure 5.16: Experiment 10 Results (Nmax = 11). X-component of reference angular velocity and KalmanFilter's estimation (top). X-component of error in angular velocity along x-axis (bottom).

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Figure 5.17: Experiment 11 Results (Nmax = 3). Reference angular velocity and Kalman Filter's estim-ation (top). X-component of error in angular velocity along x-axis (bottom).

experiment is executed for Nmax = 3, 11.

This experiment tries to imitate Experiment 9 from the previous section. As it has beendone with the previous experiment, the results obtained for Nmax = 3 and Nmax = 11are going to be compared. A controller's gain of Kcont = 0.01I, and observer's covariancematrix of Qfix = 0.01I was used. As the frequency of the applied sinusoidal wave isconsiderably high in order to reduce the simulation time, the variations in angular velocityare too fast for the previous conguration tried in Experiment 10, and the system becomeshighly oscillatory, and thus, the parameters had to be modied. In Figure 5.17, the resultsfor Nmax = 3 are shown.

For Nmax = 11, the results of the simulation can be seen in Figure 5.18. Compared to theprevious case, it can be seen that the maximum error is higher in this case, surpassing 10rpm, while for the former it was bounded below 10 rpm. In this case the sinusoidal wave ismore noticeable in the error plot, which shows the dierence between the reference signaland the angular velocity estimation.

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Figure 5.18: Experiment 11 Results (Nmax = 11). Reference angular velocity and Kalman Filter'sestimation (top). X-component of error in angular velocity along x-axis (bottom).

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No. fs ωref Qfix Kcont Description

12 2.5 kHz Stepped 0.01I 0.01I Angular velocity reference is incremen-ted step by step up to 100 rpm angularvelocity norm.

13 2.5 kHz Stepped 0.01I 0.1I Angular velocity reference is incremen-ted step by step up to 160 rpm angularvelocity norm.

14 2.5 kHz Stepped 0.002I 0.01I Angular velocity reference is incremen-ted step by step up to 200 rpm angularvelocity norm.

15 2.5 kHz Square wave 0.002I 0.01I Square wave angular velocity referenceof 160 rpm angular velocity norm ofamplitude.

16 2.5 kHz Sinusoidal 0.002I 0.01I Sinusoidal angular velocity reference ofamplitude 25 rpm and 60 seconds ofperiod for each component.

17 2.5 kHz Stepped 0.002I 0.01I Angular velocity reference is incremen-ted step by step up to maximum pos-sible angular velocity (300 rpm norm).

Table 5.5: Closed-Loop experiments in the real prototype.

5.4.2 Experimental Results

In this section, the results obtained by using the proposed estimator in the closed-loop systemare going to be analysed. These experiments are using for ne-tuning the parameters of theestimator and controller, starting from the rst approximation obtained by simulation.

The experiments that have been carried out are summarised in Table 5.5, where a briefdescription is included. Experiments 12, 13 and 14 are run in order to x the most optimal para-meters of the closed-loop system, and all of them try to imitate the set-up shown in Experiment7, in order to compare the performance and behaviour of the system. Finally Experiment 15puts the system in a dierent situation, in which the angular velocity vector is modied not onlyin amplitude, but also in orientation.

• Experiment 12: Angular velocity reference signal of the controller is incremented step bystep until reaching a norm of 100 rpm. The experiment is executed by using the proposedestimator in the real system's control loop. The covariance matrix of the estimator used isQfix = 0.01I, and controller's gain Kcont = 0.01I.

In this experiment, the performance of the estimator for dierent angular velocities isgoing to be tested. In Figure 5.19 the results are shown. It can be seen that the use of theestimator in the closed loop system generates a permanent error in the angular velocitydepending on the studied axis, which did not happen in the previous experiments.

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Figure 5.19: Experiment 12 Results. Stepped reference angular velocity and Kalman Filter's estimation(top). Angular velocity error: ωerr = ωref − ωest (bottom).

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On the one hand, this oset shows that the proportional controller is not capable of rejectingthe steady-state permanent error when the observer is used. Note that the x-componentdoes not have the aforementioned oset, whereas the other components have it. Also thisbias depends on the angular velocity reference, as it is not constant for the dierent appliedsteps.

On the other hand, as it happened in previous experiments, the performance of the estim-ator is better for higher speeds than for smaller angular velocities, as it can be seen afterthe rst step, in which the maximum error is obtained, due to the resulting oscillations.

Furthermore, the frequency of the oscillations in the angular velocity estimation and actualrotor's angular velocity increases with the magnitude of the rotor's speed, and its amplitude,for this experiment, is progressively reduced step after step. Further analysis is going tobe carried out in order to check the behaviour of the system for dierent conditions.

• Experiment 13: Angular velocity reference signal of the controller is incremented step bystep until reaching a norm of 160 rpm. The experiment is executed by using the proposedestimator in the real system's control loop. The covariance matrix of the estimator used isQfix = 0.1I, and controller's gain Kcont = 0.01I.

In this case, a similar experiment is carried out, but using a dierent conguration of theestimator, xing Qfix = 0.1I, which makes the observer faster, but it is able to reject asmaller level of noise in the estimations. The results obtained for this experiment can beseen in Figure 5.20. As it happened in the previous experiment, an oset in the angularvelocity estimation is also visible in this case, dierent for each component, with a biasdependent on the angular velocity, and oscillations that increase in frequency and reducein amplitude with the reference angular velocity.

For this experiment, the resulting bias for each component is dierent compared to Exper-iment 12. The closest component to 0 oset is the Y-axis instead of X-axis. This situationcan be due to dierent initial conditions of the rotor for each experiment, that leads to adierent steady state.

It can be seen that the noise still present in the estimations is higher both in magnitudeand frequency, which can be expected after having increased the value of Qfix.

• Experiment 14: Angular velocity reference signal of the controller is incremented step bystep until reaching a norm of 200 rpm. The experiment is executed by using the proposedestimator in the real system's control loop. The covariance matrix of the estimator used isQfix = 0.002I, and controller's gain Kcont = 0.01I.

For this experiment, the covariance matrix of the Kalman Filter has been reduced, whichwill imply a slightly slower estimator, but a better noise rejection, as it can be seen inFigure 5.21. Compared to the previous experiments, the convergence of the estimator toinitial condition (0 rpm) requires a bigger amount of time, and the behaviour at low speedis deteriorated, where the highest error is obtained. However, at higher speed, the noise isconsiderably reduced in amplitude and frequency.

This reduction in the estimation noise, makes the oset more noticeable, specially forcomponent Z. Also, a drift in the estimations from the reference signal can be seen in theerror plot, while keeping the same oset between the 3 components of the angular velocity.The reduced amount of noise makes this combination of parameters the most promising

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Figure 5.20: Experiment 13 Results. Stepped reference angular velocity and Kalman Filter's estimation(top). Angular velocity error: ωerr = ωref − ωest (bottom).

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Figure 5.21: Experiment 14 Results. Stepped reference angular velocity and Kalman Filter's estimation(top). Angular velocity error: ωerr = ωref − ωest (bottom).

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Figure 5.22: Experiment 15 Results. Square Wave reference angular velocity and Kalman Filter's estim-ation (top). Angular velocity error: ωerr = ωref − ωest (bottom).

one for higher rotation speeds. In Experiment 17, the maximum achievable speed is goingto be tested under this conditions.

• Experiment 15: Square wave Angular velocity reference signal of the controller of amp-litude 200 rpm. The experiment is executed by using the proposed estimator in the realsystem's control loop. The covariance matrix of the estimator used is Qfix = 0.002I, andcontroller's gain Kcont = 0.01I.

In this experiment, the behaviour of the closed-loop system under more aggressive changesin angular velocity reference is going to be tasted. In order to do that, a square wave signalis going to be applied, making the Reaction Sphere's rotor pass from 160 rpm to -160 rpm.The results of this experiment can be seen in Figure 5.22

• Experiment 16: Sinusoidal angular velocity reference of amplitude 25 rpm and 60 secondsof period. The experiment is executed by using the proposed estimator in the real system'scontrol loop. The covariance matrix of the estimator used is Qfix = 0.002I, and controller'sgain Kcont = 0.01I.

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Figure 5.23: Experiment 16 Results. Sinusoidal reference angular velocity and Kalman Filter's estimation(top). Angular velocity error: ωerr = ωref − ωest (bottom).

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Figure 5.24: Experiment 17 Results. Stepped reference angular velocity and Kalman Filter's estimation(top). Angular velocity error: ωerr = ωref − ωest (bottom).

As it was done in experiments 6,9 or 11, the closed-loop system is going to be tested fora sinusoidal angular velocity reference, which makes the angular velocity vector vary withtime. The amplitude of the sinusoidal signal is increased in a stepped way, having applied60, 120 and 180 rpm. The results of this experiment is shown in Figure 5.23.

In this experiment, at lower amplitude of the sinusoidal wave, some oscillations can beobserved, as it happened in previous closed-loop experiments, obtaining the maximumerror close to 40 rpm in these conditions. The performance of the estimator and controllerimproves when the amplitude of the sinusoidal reference signal is augmented.

• Experiment 17: Angular velocity reference signal of the controller is incremented stepby step until reaching a the maximum possible angular velocity. In this case was a norm of200 rpm. The experiment is executed by using the proposed estimator in the real system'scontrol loop. The covariance matrix of the estimator used is Qfix = 0.002I, and controller'sgain Kcont = 0.01I.

The main goal of this experiment is to test the highest reachable speed by using the

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Speed Range Back-EMF ErrorLPV-KF Error

(w o-set)

LPV-KF Error

(wo o-set)

Low Speed (≤ 50 rpm) 6.63 rpm 14.78 rpm 14.78 rpm

High Speed (> 50 rpm) 29 rpm 23.6 rpm 11.05 rpm

Table 5.6: Angular Velocity Estimators comparison. Maximum absolute error for each estimator: ωAE =|ωref − ωest|.

proposed estimator in the closed-loop system. The results of this experiment are shownin Figure 5.24, in which it can be seen that the system follows a similar behaviour as inprevious experiments: angular velocity-dependent oset in estimation and actual angularvelocity and deteriorated performance for lower angular velocities.

On the contrary, the system starts to oscillate in a sinusoidal way, in which the amplitudeincreases with the angular velocity. The system reaches a maximum reference speed of 300rpm, the same value obtained with the Back-EMF estimator, but obtaining considerablylower frequency and amplitude of the oscillations. Further ne-tuning of the controller-observer parameters can improve the maximum obtained speed and reduce oscillations.

The error shown in the rst seconds of the experiment is originated because the angularvelocity is not operating in closed-loop yet. At 60 s of simulation the system is alreadyworking in closed-loop, thus, the results were analysed from this point.

5.5 Back-EMF and LPV Kalman Filter Comparison

In this section, a comparison between the previous method and the proposed one for angularvelocity estimation of the Reaction Sphere's rotor is going to be shown, in order to quantifythe performance dierence between both methods. The third method studied for estimatingthe angular velocity based on a Extended Kalman Filter, shown in Section 4.1.2, cannot becompared, as it has not been implemented in the real prototype, as its execution time did notallow to implement it in real-time

In order to compare both approaches, equivalent experiments with each estimator are goingto be studied. For this purpose, Experiment 7, shown on Figure 5.11, and 17, on Figure 5.24,are going to be considered. In both experiments, the reference angular velocity, around direction[1, 1, 1] is incremented step by step until reaching a norm of 300 rpm.

As it has shown before, the performance experienced by the proposed Linear Parameter-Varying Kalman Filter depends on the norm of the reference angular velocity. This situation isgoing to be studied by splitting the analysis in two sections: low and high speeds. The formeris considered when the reference speed norm is lower than 85 rpm (each component lower than50 rpm), and the latter when higher. The main factor to study will be the maximum error forboth experiments in the studied speed interval, and in Table 5.6 these values are shown.

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CHAPTER 5. SIMULATION AND EXPERIMENTAL RESULTS

Figure 5.25: Experiment 17 Results with Back-EMF Estimator. Stepped reference angular velocity andBack-EMF's estimation (top). Angular velocity error: ωerr = ωref − ωest (bottom).

For convenience, the error obtained in Experiment 7 by using the Back-EMF estimator isshown in Figure 5.25. In order to compare the estimators, the maximum absolute error on anycomponent is considered, and for the LPV-KF, only the error after closing the control loop istaken into account.

Due to the o-set induced by the LPV-KF, the oscillations and error in the angular velocityhave a bias, thus the maximum error of the estimator does not show properly the variability of theestimations due to the controller and observer. In order to measure the maximum deviation, thelast column of the table has been included, in which the oset has been cancelled by measuringthe center of the sinusoidal signal, and subtracting its value to the o-set signal after the step inthe reference signal.

On the one hand, for lower values of reference angular velocity, the proposed LPV-KF es-timator obtains considerable worse performance than the Back-EMF, due to the high oscillationsobtained with the former, which could be improved by modifying the parameters of the control-ler and observer, as happened in Experiment 12, shown in Figure 5.19, in which the maximum

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

absolute error stays bounded in 10 rpm, including oset, during the transients, and bounded in5 rpm during steady state.

On the other hand, for higher values of angular velocity reference, the improvement of usingthe proposed estimator, instead of the Back-EMF-based one, is considerable. As shown in Table5.6, by also considering the oset in the angular velocity, the maximum absolute error is inferiorto the one obtained by using Back-EMF. If the oset is compensated, the oscillations are reducedin less than a half compared to the previous method for angular velocity estimation.

5.6 Conclusion

During this chapter two simulated models of the Reaction Sphere with dierent level of abstrac-tion have been explained and developed. Furthermore the validation process of one of the modelsis available in B in order to preserve the continuity of the chapter.

The behaviour of the proposed Linear Parameter-Varying Kalman Filter has been shown bycarrying out a total of 17 dierent experiments, using both the developed simulated models andthe real prototype available at CSEM. These experiments show a promising performance for theclosed-loop system.

Compared to the previous method used for angular velocity estimation based on Back-EMF estimation, the rotor experiences considerably lower oscillations, having being reducedtheir amplitude by less than a half, at high rotation speeds and with much less aggressivevibrations. For lower angular velocities the experienced performance is deteriorated, but bymodifying controller's and observer's gains, a similar performance to the Back-EMF methodcan be obtained. Also the improvement of the controller by including for example an integralactuation will compensate the oset available with the full state feedback controller implementedfor these experiments.

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CHAPTER 5. SIMULATION AND EXPERIMENTAL RESULTS

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

Chapter 6

Conclusion and Future Work

This chapter will explain the dierent contributions of this thesis, comparing the proposed LinearParameter-Varying Kalman Filter estimator with the one previously used based on Back-EMFestimations. Thus, ther results obtained in the previous chapter will be summarised for makingthe comparison in performance and stability between both methods.

Furthermore the suggested future work required for continuing with the improvement of thestudied Reaction Sphere's system is going to be proposed. Due to the complexity of the RS, andthe focus of this thesis on angular velocity estimation, its contributions can be used for furtherimproving the other components of the closed-loop system, adapting some of them to the newestimator, mainly the control part of the RS.

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CHAPTER 6. CONCLUSION AND FUTURE WORK

6.1 Conclusion

In Chapter 5, the results of the implementation of the proposed Angular velocity estimator basedon a Linear Parameter-Varying Kalman Filter have been shown, analysed and compared withthe previous method used in the system based on Back-EMF estimation. In this section, theconclusions extracted from the aforementioned experiments are going to be explained.

Furthermore, the fullment of the objectives xed for the development of the current Mas-ter's Thesis shown in Chapter 1 is going to be studied, which will give the level of completenessof the initial proposal.

The main purpose of the current Master's Thesis is the development of a new angular velocityestimator of a Reaction Sphere's rotor intended for satellite attitude control. Having this in mind,after having developed and analysed the behaviour of the proposed Angular Velocity Estimator,the following conclusions can be extracted:

• A new angular velocity estimator based on a Linear Parameter-Varying Kalman Filterobserver have been developed and tested, both in simulation and by using the real prototypeof the Reaction Sphere.

• The proposed Linear Parameter-Varying Kalman Filter estimator has been implementedin real-time, with small impact on the execution time of the control algorithm comparedto the previously available method.

• Making use of the spherical harmonics decomposition allows faster computations for imple-menting a Kalman Filter in real-time, which was not possible by making use of quaternions,as tried in [9].

• By using the proposed observer in the closed-loop control system, at low rotational speedsof the rotor (smaller than 50 rpm for each component), a similar level of amplitude ofoscillations is experienced compared to the previous method based on Back-EMF estimationused for estimating its angular velocity.

• For higher rotational speeds of the rotor (higher than 50 rpm for each component), aconsiderably lower level of oscillations is obtained compared to the Back-EMF-based es-timator. The amplitude of the maximum error is reduced by a factor of two at 300 rpmof norm (maximum reached angular velocity for both methods), and the vibrations arealmost completely rejected.

• The best performance for each range of speeds, is obtained with dierent parameter valuesof the controller gain and observer conguration, thus an adaptive closed-loop compensatorshould be implemented for good performance for all speeds.

• By substituting the estimator by the Linear Parameter-Varying Kalman Filter, an oset inthe angular velocity appears, and the error becomes sinusoidal, which can be reduced byadding an integral actuation in the controller, and ne tuning the controller's parameters.

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

• Closed-loop stability of the system with the proposed Linear Parameter-Varying KalmanFilter is not guaranteed analytically, but the experiments carried out with the real proto-type show the convergence of the estimations to the real angular velocity values and thestability of the system is maintained by choosing the appropriate controller gain.

Now, for convenience, the objectives xed for the current thesis are included below, withthe corresponding level of fullment for each objective:

• Objective 1 : Develop and algorithm, implementable in Real-Time, capable of improvingthe estimations of the angular velocity of a reaction sphere's rotor by using the availablesensors and controller, and thus, improving the performance of the current system withoutincreasing its complexity and cost.

The proposed angular velocity estimator based on a Linear Parameter-Varying KalmanFilter have been developed in Chapter 4, and its performance studied in Chapter 5, havingobtained an improvement in performance.

• Objective 2 : Improve and validate the Simulated Reaction Sphere model already imple-mented using Simulink, which will be required for algorithm tests.

The developed simulation models for the Reaction Sphere have been explained in Section5.2, with their dierent levels of abstraction, and the validation studied in Appendix B.

• Objective 3 : Test the algorithm using simulation and available experimental data in orderto analyse its behaviour before its implementation.

The proposed observer has been tested by making use of the available simulated models,obtaining the results showed in Chapter 5 for both open-loop and closed-loop simulationset-ups.

• Objective 4 : Implement, if possible, the developed algorithm in the real system and compareits behaviour and performance with the current estimations.

The proposed estimator has been implemented by using the same set-up shown in Section3.6, and obtaining the results exposed in Section 5.4.

To sum up, all the xed objectives have been accomplished in the current Master's Thesis,but the performance of the overall system can be further improved. In the next section theproposal of future work is going to be exposed in order to overcome some limitations of thecurrent set-up, and improving the behaviour of the closed-loop system.

6.2 Future Work

Even if the work developed during this Thesis has accomplished the objectives xed for thisproject, the system can be further improved by completing and carrying out several suggestedtasks. After having obtained promising results with the proposed estimator, it is considered thatthe observer should be used as a starting point for the future work, which will try to improveand x some issues still available in the system, which are marked in the following points:

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CHAPTER 6. CONCLUSION AND FUTURE WORK

• Issue 1 : Sinusoidal oscillations available in the resulting angular velocity of the rotor by us-ing Linear Parameter-Varying Kalman Filter and Full-State Feedback controller, dependenton the angular velocity reference and controller's gain.

This could be reduced by ne-tuning the full-state feedback controller gain, or even adapt-ing the gain for dierent angular velocities.

• Issue 2 : Oset and bias available in the real prototype while using the proposed LinearParameter-Varying Kalman Filter in the closed-loop system.

This oset could be compensated by implementing a Full-State Feedback controller withIntegral actuation. Proportional and integral actuations of the controller should be alsone-tuned again after the modication.

• Issue 3 : Best performance at low and higher speeds obtained with dierent values ofcontroller gain and observer conguration by using the proposed Linear Parameter-VaryingKalman Filter in the closed-loop system.

An adaptive controller could be studied for implementation in the real prototype in whichthe controller gain, and observer parameters, can be modied as a function of the desiredangular velocity of the rotor.

• Issue 4 : Available repetitive disturbances due to non-modelled higher harmonics in theoutput equation of the State-Space Model and in the command torque estimation, due tothe third order approximation of spherical harmonics considered in the control loop.

Study implementing a notch lter or a generalised notch lter, as done, for example, forunbalance compensation in magnetic bearings explained in [12], in order to reduce theseperturbations.

• Issue 5 : For studying the angular velocity control loop, no interaction between bearing andangular velocity is considered, which is only true if the rotor is perfectly centred inside thestator. This supposition is not true as observed in the dierent experiments with the realprototype, as the bearing controller performance is deteriorated when the angular velocityis increased.

Thus, implementing a more sophisticated and improving the bearing controller should bestudied, in order to reduce its inuence on the angular velocity control loop.

By studying this aforementioned issues the performance and behaviour of the overall Reac-tion Sphere system can be improved, allowing the rotor to achieve higher maximum rotationalspeeds and reducing vibrations and oscillations by using the available hardware, and thus notincreasing considerably the investment on the system.

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ANGULAR VELOCITY ESTIMATION OF A REACTION SPHERE ACTUATOR

Appendix A

Time Planning and Budget

As specied in [60], this appendix will introduce the time planning xed for developing thecurrent Master's Thesis, including work packages decomposition and its in a Gantt Diagram.Finally the budget required for the project is going to be included.

Work Packages Decomposition

The list of work packages required for developing the current Master's Thesis is:

1. Specify objectives of the internship according to CSEM needs and student preferences.

2. Dene the material available for the project.

3. Literature overview about the reaction sphere.

4. Literature overview about angular velocity estimation.

5. Selection of angular velocity estimating methodology.

6. Develop system's equations.

7. Analytical development of angular velocity estimator and state-space observer.

8. Study existing Reaction Sphere simulation model in MATLAB and Simulink.

9. Complete and modify Reaction Sphere's models according to needs.

10. Validate simulation model using available experimental data.

11. Implement observer using MATLAB and Simulink.

12. Study behaviour and debug proposed estimator using simulated model of Reaction Sphere.

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APPENDIX A. TIME PLANNING AND BUDGET

13. Study available Reaction Sphere's prototype and available hardware.

14. Study available control routine for Reaction Sphere operation.

15. Implement proposed observer in control hardware.

16. Verify estimator's implementation in open loop.

17. Substitute previous estimator with proposed one.

18. Study behaviour of proposed estimator in real prototype.

19. Master's thesis document writing.

20. Correction and verication of developed work and document.

21. Print and send nal document.

Gantt Diagram

In this section, the Gantt diagram obtained by introducing the previous work packages in theappropriate software, in this case ProjectLibre has been used, is going to be shown. The timeinterval in wich the current Master's Thesis was developed was since February 8th of 2016 untilthe nal deadline for submission, September 12th.

92 ETSII-UPM & EPFL

F S S M8 Feb 16

T WT F S S M15 Feb 16

T WT F S S M22 Feb 16

T WT F S S M29 Feb 16

T WT F S S M7 Mar 16

T WT F S S M14 Mar 16

T WT F S S M21 Mar 16

T WT F S S M28 Mar 16

T WT F S S M4 Apr 16

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T WT F S S M18 Apr 16

T WT F S S M25 Apr 16

T WT F S S M2 May 16

T WT F S S M9 May 16

T WT F S S M16 May 16

T WT F S S M23 May 16

T WT F S S M30 May 16

T WT F S S M6 Jun 16

T WT F S S M13 Jun 16

T WT F S S M20 Jun 16

T WT F S S M27 Jun 16

T WT F S S M4 Jul 16

T WT F S S M11 Jul 16

T WT F S S M18 Jul 16

T WT F S S M25 Jul 16

T WT F S S M1 Aug 16

T WT F S S M8 Aug 16

T WT F S S M15 Aug 16

T WT F S S M22 Aug 16

T WT F S S M29 Aug 16

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T WT F S S M12 Sep 16

T WT F S S M19 Sep 16

T WT F S S M26 Sep 16

T WT F S S M3 Oct 16

T WT F S S M10 Oct 16

T WT F S1 Specify objectives of the internship according to CSEM needs and student ... 5 days 08/02/16 08:00 12/02/16 17:002 Define the material available for the project. 5 days 08/02/16 08:00 12/02/16 17:003 Literature overview about the reaction sphere. 5 days 15/02/16 08:00 19/02/16 17:00 1;24 Literature overview about angular velocity estimation. 5 days 22/02/16 08:00 26/02/16 17:00 35 Selection of angular velocity estimating methodology. 5 days 29/02/16 08:00 04/03/16 17:00 46 Develop system's equations. 3 days 07/03/16 08:00 09/03/16 17:00 57 Analytical development of angular velocity estimator and state-space obse... 7 days 10/03/16 08:00 18/03/16 17:00 68 Study existing Reaction Sphere simulation model in MATLAB and Simulink. 5 days 21/03/16 08:00 25/03/16 17:00 79 Complete and modify Reaction Sphere's models according to needs. 12 days 28/03/16 08:00 12/04/16 17:00 8

10 Validate simulation model using available experimental data. 11 days 13/04/16 08:00 27/04/16 17:00 911 Implement observer using MATLAB and Simulink. 3 days 28/04/16 08:00 02/05/16 17:00 1012 Study behaviour and debug proposed estimator using simulated model of ... 10 days 03/05/16 08:00 16/05/16 17:00 1113 Study available Reaction Sphere's prototype and available hardware. 3 days 17/05/16 08:00 19/05/16 17:00 1214 Study available control routine for Reaction Sphere operation. 6 days 20/05/16 08:00 27/05/16 17:00 1315 Implement proposed observer in control hardware. 7 days 30/05/16 08:00 07/06/16 17:00 1416 Verify estimator's implementation in open loop. 7 days 08/06/16 08:00 16/06/16 17:00 1517 Substitute previous estimator with proposed one. 5 days 17/06/16 08:00 23/06/16 17:00 1618 Study behaviour of proposed estimator in real prototype. 15 days 24/06/16 08:00 14/07/16 17:00 1719 Master's thesis document writing. 25 days 15/07/16 08:00 18/08/16 17:00 1820 Correction and verification of developed work and document. 15 days 19/08/16 08:00 08/09/16 17:00 1921 Print and send final document. 5 days 08/09/16 17:00 15/09/16 17:00 20

Name Duration Start Finish Pred...

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APPENDIX A. TIME PLANNING AND BUDGET

Budget

In order to compute the required budget for developing the current Master's Thesis, only theresources exclusively used for this thesis are going to be considered, as the majority of theelements and devices that have been used were already available due to previous projects, suchas the reaction sphere's prototype and all its associated control hardware. This way, exclusivelythe resources required for this project are studied, and a better understanding of its cost can beachieved.

In the following Table, the expenses for the development of the current Master's Thesis areshown. Due to the origin of the work, for which all the required material was already available,the only expenses are the required equipment for the human work (laptop, MATLAB license andmanpower costs). All the expenses are expressed in e.

Cost for developing the current Master's Thesis

Element Unitary Price Units Total Cost(e/ut.) (ut.) (e)

Laptop 1000 1 1000MATLAB Student license1 700 1 700

Human Resources (Working hours) 20 300 6000TOTAL 7700

Table A.1: Cost for developing the current Master's Thesis.

1Exclusively the required toolbox for this thesis are included in MATLAB's license price: symbolic toolbox

and MATLAB coder.

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Appendix B

Simulation Model Validation

Once the developed models for the reaction sphere are derived and presented, in order to beused in simulation with desirable similarity to reality, its behaviour must be veried. This is thepurpose of the model validation process explained in this section.

For the validation of the model, already available experimental data is going to be used.In the experiment, the rotor is attached to a guiding axis, which rotates at constant angularvelocity (orientation and modulus) thanks to the action of a electric motor.

In this case, the rotor rotates at 15 rpm while the Hall sensors measure the magnetic uxdensity during the process. The set-up of the experiment is shown in Figure B.1. The initialorientation of the rotor is required for the validation, which can be computed as the rotor isattached to the guiding axis, and taking the zero-crossing of any sensor measurement as origin.

The conguration of the set-up only allows to validate the reduced model of the reactionsphere, and more specially, the Hall sensor measurement estimation. This is due to the factthat the rotor is attached to an external axis, which prevents the usage of the coils for rotatingthe sphere. In order to simulate the set-up, no external torque is applied to the Simulinkmodel and an initial angular velocity equal to the one generated by the motor is specied. Thesimulation is executed for dierent values of maximum spherical harmonic Nmax, consideringNmax = 3, 7, 9, 11.

For graphically observing the similarities between the experimental data and the simulatedone, the comparison between the rst six sensors is shown in Figure B.2 for a value of Nmax = 3,which corresponds, theoretically, to the fastest and less accurate approximation.

It can be seen that the results obtained in simulation are similar to the experimental ones.As the simulations execute exactly the same experiment, in order to compare the accuracy foreach Nmax = 3, 7, 9, 11, the average error of all sensors is used. The results generated foreach Nmax are divided by the average error obtained for Nmax = 3, which considers the relativeaverage error for the dierent simulations en

e3. Also, the relative execution time is included tn

t11,

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APPENDIX B. SIMULATION MODEL VALIDATION

Figure B.1: Reaction Sphere's set-up for model validation.

Nmax n = 3 n = 7 n = 9 n = 11

Average Error [en (T)] 3 7 9 11

Relative Average Error[ene3

]1 0.725 0.663 0.614

Execution Time [tn (s)] 123.6 1830.8 4286.8 8647.0

Relative Execution Time[tnt11

]0.0143 0.2117 0.4958 1

Table B.1: Reaction Sphere Model Validation Comparison for Nmax = 3, 7, 9, 11.

computed as the division between the execution time for Nmax = 3, 7, 9, 11 and Nmax = 11,which corresponds to the longest simulation. This comparison is shown in Figure B.3.

The simulated time for all cases is 2 seconds, which correspond to half a turn of the sphere,and due to the symmetry of the rotor poles, a complete period in the Hall sensor measurements.As expected, the error between the experimental data and the simulated one reduces as Nmax

increases, and the execution time increases as the higher considered spherical harmonic increases.Then, for faster and less accurate simulations, Nmax = 3 is going to be used, while for accuratesimulations Nmax = 11 will be executed. For further detail, the exact values obtained during themodel validation process are shown in Table B.1.

The execution time shown in the aforementioned table depends not only on the computa-tional requirement for executing the model, but also on the hardware in which is run or otherprocesses that could be running in the background while simulating. If a more in-depth analysisof the execution time is required, several executions should be performed, but in this case, aqualitative result is sucient, as the time required for executing the model is not crucial.

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0 0.5 1 1.5 2−10

−5

0

5

10·10−2

Time (s)

Magnetic

FluxDensity

(T)

Sensor 1

ExperimentalSimulated

0 0.5 1 1.5 2−4

−2

0

2

4·10−2

Time (s)

Magnetic

FluxDensity

(T)

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ExperimentalSimulated

0 0.5 1 1.5 2−10

−5

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FluxDensity

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ExperimentalSimulated

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−5

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FluxDensity

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ExperimentalSimulated

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−5

0

5

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Time (s)

Magnetic

FluxDensity

(T)

Sensor 5

ExperimentalSimulated

0 0.5 1 1.5 2−4

−2

0

2

4·10−2

Time (s)

Magnetic

FluxDensity

(T)

Sensor 6

ExperimentalSimulated

Figure B.2: Reaction Sphere Model Validation. Experimental (blue) and Simulated (red) results com-parison for sensors from 1 to 6.

The complete model cannot be validated, as the initial orientation of the rotor cannotbe known nor measured directly, and also the exact current applied to the stator coils is alsounknown as there is no measure of the current applied by the current control loop.

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APPENDIX B. SIMULATION MODEL VALIDATION

3 7 9 11

0

0.2

0.4

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1

0

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0.730.66

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RelativeAverageError

e n e 3

1

0.730.66

0.61

0.01

0.21

0.5

1

RelativeExecutionTim

et n t 11

Simulation Accuracy Comparison

Average Error Execution Time

Figure B.3: Reaction Sphere Model Validation. Relative average error and execution time obtained forNmax = 3, 7, 9, 11.

102 ETSII-UPM & EPFL

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