Spacecraft Dynamics and Control: The Embedded Model Control Approach

By Enrico Canuto, Carlo Novara, Donato Carlucci and 2 more

Spacecraft Dynamics and Control: The Embedded Model Control Approach provides a uniform and systematic way of approaching space engineering control problems from the standpoint of model-based control, using state-space equations as the key paradigm for simulation, design and implementation.

The book introduces the Embedded Model Control methodology for the design and implementation of attitude and orbit control systems. The logic architecture is organized around the embedded model of the spacecraft and its surrounding environment. The model is compelled to include disturbance dynamics as a repository of the uncertainty that the control law must reject to meet attitude and orbit requirements within the uncertainty class. The source of the real-time uncertainty estimation/prediction is the model error signal, as it encodes the residual discrepancies between spacecraft measurements and model output. The embedded model and the uncertainty estimation feedback (noise estimator in the book) constitute the state predictor feeding the control law. Asymptotic pole placement (exploiting the asymptotes of closed-loop transfer functions) is the way to design and tune feedback loops around the embedded model (state predictor, control law, reference generator). The design versus the uncertainty class is driven by analytic stability and performance inequalities. The method is applied to several attitude and orbit control problems.

• The book begins with an extensive introduction to attitude geometry and algebra and ends with the core themes: state-space dynamics and Embedded Model Control
• Fundamentals of orbit, attitude and environment dynamics are treated giving emphasis to state-space formulation, disturbance dynamics, state feedback and prediction, closed-loop stability
• Sensors and actuators are treated giving emphasis to their dynamics and modelling of measurement errors. Numerical tables are included and their data employed for numerical simulations
• Orbit and attitude control problems of the European GOCE mission are the inspiration of numerical exercises and simulations
• The suite of the attitude control modes of a GOCE-like mission is designed and simulated around the so-called mission state predictor
• Solved and unsolved exercises are included within the text - and not separated at the end of chapters - for better understanding, training and application
• Simulated results and their graphical plots are developed through MATLAB/Simulink code

• Language: English
• Publisher: Elsevier Science
• Release date: Mar 8, 2018
• ISBN: 9780081017951

Enrico Canuto

Enrico Canuto has designed the drag-free control of the successful European GOCE spacecraft. Recently he has studied and proved the integrated formation, drag-free and attitude control of the European Next Generation Gravity Mission. He also contributed to conception, design and implementation of on-ground test facilities for space qualification, among them the thrust stand Nanobalance. He has published several journal papers on the spacecraft control, based on the Embedded Model Methodology, which developed and applied to several applications. He has taught a course on Aerospace modelling and control at Politecnico di Torino.

Book preview

Spacecraft Dynamics and Control

The Embedded Model Control Approach

Enrico Canuto

Politecnico di Torino, Turin, Italy

Carlo Novara

Politecnico di Torino, Turin, Italy

Luca Massotti

RHEA for ESA - European Space Agency, ESTEC, Noordwijk, The Netherlands

Donato Carlucci

Politecnico di Torino, Turin, Italy

Carlos Perez Montenegro

Politecnico di Torino, Turin, Italy

Table of Contents

Cover image

Title page

Copyright

Dedication

Chapter 1. Introduction

1.1. Objectives and Rationale

1.2. Notation Rules and Tables

1.3. Abbreviations

Chapter 2. Attitude Representation

2.1. Objectives

2.2. Vectors and Matrices

2.3. Matrices

2.4. Unit Quaternions

2.5. Space and Time Coordinates

2.6. Representations of Rigid Body Attitude

2.7. Infinitesimal and Error Rotations

Chapter 3. Orbital Dynamics

3.1. Objectives

3.2. The Two-Body Problem

3.3. Free Response of the Restricted Two-Body Problem

3.4. Orbit Propagation

3.5. Analysis of Orbital Trajectories

3.6. Stability of Orbit

Chapter 4. The Environment: Perturbing Forces and Torques

4.1. Objectives

4.2. Gravity Forces and Torques

4.3. Electromagnetic Radiation Forces and Torques

4.4. Aerodynamic Forces and Torques

4.5. Atmospheric Density

4.6. Planetary Magnetic Field Torques

4.7. Internal Forces and Torques

4.8. Embedded Model of Disturbances

Chapter 5. Perturbed Orbital Dynamics

5.1. Objectives

5.2. Perturbed Orbits

5.3. Dynamics of the Orbital Elements

5.4. From n-Body System to Three-Body System

5.5. Hill–Clohessy–Wiltshire Equation

5.6. Restricted Three-Body Problem

Chapter 6. Attitude Kinematics: Modeling and Feedback

6.1. Objectives

6.2. Attitude Matrix and Vector Kinematics

6.3. Euler Angle Kinematics

6.4. Quaternion Kinematics

6.5. Error Quaternion Kinematics

6.6. Feedback Implementation and Pole Placement

Chapter 7. Attitude Dynamics: Modeling and Control

7.1. Objectives

7.2. Attitude Dynamics

7.3. Attitude Dynamics and Feedback

7.4. Torque-Free Rigid Body Attitude

7.5. Attitude Dynamics Under Gravity Gradient and Aerodynamic Torques

7.6. Simple Control Laws

7.7. Attitude Dynamics and Control With Internal Rotating Masses

Chapter 8. Orbit and Attitude Sensors

8.1. Objectives

8.2. Sensor and Measurement Error Models

8.3. Inertial Navigation Sensors

8.4. Accelerometers

8.5. Gyroscope Sensors

8.6. Global Navigation Sensors

8.7. Sun Sensors

8.8. Horizon Sensors

8.9. Star Trackers

8.10. Magnetic-Field Sensors

Chapter 9. Orbit and Attitude Actuators

9.1. Introduction

9.2. Propulsion Systems

9.3. Propulsion Geometry

9.4. Momentum Exchange Actuators

9.5. Reaction Wheels

9.6. Control Momentum Gyros

9.7. Magnetic Torquers

Chapter 10. Attitude Determination

10.1. Objectives

10.2. The Measurement Errors

10.3. Two-Axis Static Attitude Determination Methods

10.4. Static Attitude Determination: The Problem of Wahba. Fundamentals

10.5. Static Attitude Determination: The Problem of Wahba. Quaternion Algorithms

10.6. Angular Rate Determination From Direction Measurements

Chapter 11. Orbital Control and Prediction Problems

11.1. Objectives

11.2. Drag-Free Control

11.3. Orbital Quaternion Prediction

Chapter 12. Attitude Control: A Case Study

12.1. Objectives

12.2. Passive and Active Attitude Control

12.3. Control Modes and Requirements

12.4. Block Diagram of a Generic Attitude Control Subsystem

12.5. Case Study: Drag-Free and Attitude Control of a Scientific Mission

Chapter 13. Introduction to Dynamic Systems

13.1. Objectives

13.2. State-Space Representation

13.3. Stability Concepts and Criteria

13.4. Controllability and Observability

13.5. Ideal Control Law: State Feedback and Disturbance Rejection

13.6. State Predictor and Real Control Law

13.7. Random Processes, State Estimation, and Prediction

Chapter 14. Introduction to Embedded Model Control

14.1. Objectives and EMC Principles

14.2. Models and Uncertainty

14.3. State Predictor Design Versus Uncertainty

14.4. Control Law Design

Index

Copyright

Butterworth-Heinemann is an imprint of Elsevier

The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom

50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States

Copyright © 2018 Elsevier Ltd. All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions.

This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

Library of Congress Cataloging-in-Publication Data

A catalog record for this book is available from the Library of Congress

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library

ISBN: 978-0-08-100700-6

For information on all Butterworth-Heinemann publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Mara Conner

Acquisition Editor: Sonnini R. Yura

Editorial Project Manager: Ana Claudia A. Garcia

Production Project Manager: Sruthi Satheesh

Designer: Matthew Limbert

Cover image: Artist's view of the European Satellite GOCE (Gravity field and steady state Ocean Circulation Explorer, 2009-2013). Courtesy of the European Space Agency (ESA).

Typeset by TNQ Books and Journals

Dedication

To our parents and families

Chapter 1

Introduction

Abstract

This introductory chapter explains the reasons for adopting the embedded model control methodology in the design of the spacecraft orbit and attitude control algorithms addressed by the book. Precedence among the book chapters is outlined with the help of a block diagram. The content of each chapter and the links between them are presented in agreement with the precedence rules. Omitted topics are also mentioned. The chapter ends with the main book notations, their rules and abbreviation tables.

Keywords

Abbreviations; Audience; Authorship; Book objectives; Embedded model control; Notations; Topics

1.1. Objectives and Rationale

1.1.1. History and Audience

This book is a significant extension and revision of the lecture notes for a course on Aerospace Modelling and Control given by one of the authors to graduate students of Politecnico di Torino (first and second year of the Master of Sciences in Aerospace and Computer Engineering). In turn, these lecture notes were an elaboration of brief notes prepared by G. Sechi, ESA/ESTEC, on the fundamentals of the attitude control.

The book includes both fundamental and advanced topics. Fundamental topics can be found in several classical and recent textbooks, such as Battin [1], De Ruiter and coauthors [2], Greenwood [3], Hughes [4], Kaplan [5], Markley and Crassidis [6], Montenbruck and Gill [7], Schaub and Junkins [8], Sidi [9], Vallado [10], Wakker [11], Wertz [12], and Wie [13]. Here, the effort is to emphasize state equation formulation and feedback design, as a preparation to more specific control topics such as the design of uncertainty-based state predictors and model-based control laws. As an example, the classical topics of the attitude kinematics are extended in Chapter 6 to include feedback design, closed-loop stability properties, and the design of an attitude state predictor driven by gyroscope and star tracker data.

For these reasons, the book may address different categories of readers. PhD students and researchers in aerospace and control sciences are eligible readers. Because most of the design procedures can be repeated manually or with the aid of simple computer programs, aerospace and control engineers may find analytic and first-trial procedures, which guide them from requirements to state predictor and control design parameters. Last but not least, several sections may fit graduate and undergraduate studies, as fundamental concepts and their formulations, detailed derivations of intermediate and final results, and in-text exercises and simulations are provided.

1.1.2. Why the Embedded Model Control Methodology?

The goal of the book is the application of a model-based control design to orbit and attitude control systems, the model being expressed in terms of state equations. To this purpose, an introduction to dynamic systems is provided in Chapter 13, and models are always derived in terms of state equations. Among model-based design methodologies, the authors have selected the embedded model control (EMC) methodology, due to their experience with the method. An introduction with a single-input single-output (SISO) case study is available in Chapter 14. With the term embedded, not to be confused with the embedded systems of computer and electrical engineering, we mean that any control unit (not only for space applications) is built around a discrete-time (DT) state equation of the plant to be controlled, the embedded model. An alternative term would have been internal, but the EMC methodology and implementations are somewhat different from the internal model control (IMC) approach [14,15].

State equations enable the implementation of control units around a real-time model of the plant (the embedded model) running in parallel and synchronous with the plant. This is the first EMC principle, which implies that exactly the same digital command dispatched to a plant (spacecraft in the book) is dispatched to the embedded model. Second, the model should not only include the command-to-output dynamics (controllable dynamics, in brief), but must always be completed with the disturbance dynamics. These are a set of equations, capable of storing, in their state variables, all the past discrepancies between plant and model that need to be canceled on the real plant. These discrepancies can only be revealed by the measured model error, that is by the difference between the plant measurements and homologous model outputs. Such discrepancies are given the collective name of uncertainty, but they may be only partly uncertain/unknown, because they may have been dropped from the embedded model because of their complexity. The disturbance dynamics synthesis is an essential step of the embedded model design and takes advantage of the so-called design model that defines and makes explicit all the expected uncertainty classes (initial state, causal uncertainty or random sources, parametric uncertainty, neglected dynamics).

The concept and practice of disturbance dynamics give rise to the second principle. How to update the relevant state? The problem has been given the name of noise design [16] and estimation because the input samples of the disturbance dynamics must be completely arbitrary in their values to capture the most challenging uncertainty sources. If a stochastic framework is adopted, arbitrary input signals are formulated as white noise processes. This is the second EMC principle. Noise design and estimation means to decide where noise enters the embedded model, and to design and tune a real-time mechanism for the noise estimation. The only way for a continuous retrieval of the input noise is the noise correlation with the previously defined measured model error. The model error is formally, but not substantially, the same signal as the Kalman filter innovation, and also the same as the driving feedback signal of the IMC approach. The correlation algorithm (noise estimator), either static or dynamic, must be designed as an output-to-noise feedback capable of stabilizing, in the presence of the whole uncertainty class, the closed-loop error system that consists of the difference between design and embedded model and of the noise estimator. The ensemble, embedded model and noise estimator, performs the functions of a closed-loop state predictor. We say that the state predictor design is uncertainty based and the prediction error (true minus embedded model variables) is affected by the uncertainty sources. The noise estimator term does not imply that the noise estimate is optimal for some criterion, but that is correlated with the current model error and respects the possible singularity of the noise layout. No noise is injected to the state variables not directly affected by an uncertainty source.

The third point concerns the control law design, which is model based, because the embedded model and reference state variables are the law ingredients. In principle, a model-based design should ignore uncertainty, only aiming at performance achievement as in the classical separation theorem. In reality, because the design is constrained by the state predictor dynamics, it should be developed without degrading the state predictor performance, which entails precedence between state predictor and control law design. A sequential design of this kind is referred to as the standard design and possesses the verifiable property (also during operations) that the measured tracking errors (embedded model minus reference variables) are numerically zero.

The last point addresses the design procedure. In both cases (state predictor and control law, and also in the reference generator design) the procedure is a closed-loop pole placement, which exploits the asymptotic sensitivity and complementary sensitivity (CS) of the state predictor and of the overall control loop. The method that may seem naïve and rather approximated with respect to the classical robust control design of J.C. Doyle and coauthors [17] takes advantage of the error equation linearity and of appropriate upper bounds on the output signals of the uncertainty sources. Because the size of the closed-loop poles is usually small and can be further reduced by coordinate decoupling, the first-trial design can be refined by analytic, simulated, and in-field optimization.

Most of the EMC models and algorithms outlined in the book have been tested on ground instruments and prototypes (references are in Chapter 14). The all-propulsion drag-free and attitude control of the GOCE satellite (Gravity Field and Steady-State Ocean Circulation Explorer, see the artist's view on the book cover [18]), which was conceived during the Mission Phase A/B, was designed as explained in Chapter 12 [19]. Recently, an EMC design has been employed during the feasibility study of the European Next Generation Gravity Mission (NGGM) [20].

1.1.3. Logical Reading Sequence and Book Contents

Fig. 1.1 shows the logical sequence of the book chapters. The reading rule is from top to down. Black nodes are derivation points. A pair of summing nodes simplifies the block diagram. Any chapter, which is required to be read before another chapter (the precedence is denoted by an up-down link)—for instance, Chapter 2 should be read before Chapter 3—is also to be read before the lower chapters—for instance, Chapter 4 should be read after Chapter 2. The diagram splits into three columns: (1) the left column is mainly concerned with orbital dynamics and control, (2) the central column deals with sensor/actuator models and performance, and attitude determination algorithms, (3) the right column is mainly concerned with attitude modeling and control.

Precise cross-reference between chapters is given (which may annoy some readers). An advantage is to limit usage of the analytic index.

The greater part of simulated examples and exercises refers to a hypothetical GOCE satellite, because (see, for instance, the case study of Chapter 12) the adopted GOCE parameters, sensors, actuators, and environment differ from the real mission. Only simulated data are provided and discussed.

Introductory Chapters

Chapter 2 in the left column begins with a summary of the vector, matrix, and quaternion algebra, then defines spatial (Cartesian frames) and time coordinates, and proceeds to the classical attitude representations (rotation matrices, Euler angles and quaternions). Rodrigues and modified Rodrigues parameters have been omitted; they can be found in Ref. [6]. The chapter ends with the infinitesimal and error rotations, one of the prominent mathematical tools throughout the entire book. The right column starts with a summary of dynamic systems (Chapter 13). The first part is devoted to the fundamentals of state equations and their properties: stability, controllability, and observability. Time-varying periodic systems are briefly mentioned. The core of the chapter is the derivation of the linear time-invariant (LTI) error equations. To this end, the whole set of errors is defined and summarized in a table. The Z-transform of the error equation points out the classical closed-loop transfer functions, sensitivity and CS. Their asymptotic approximation is widely employed in the book for the pole placement of state predictors and control laws. In this chapter, pole placement takes advantage of the classical separation theorem because only the causal uncertainty (independent of prediction and tracking errors) is assumed to be effective. The chapter ends with the fundamentals of random processes, as they are essential in the synthesis of the disturbance dynamics. Passivity concepts and average dynamics, though employed in the book, have been omitted. The definition and properties of Fourier, Laplace, and Z-transforms are assumed to be known [21,22]. The same assumption applies to the basic concepts of the probability theory [23].

Figure 1.1  Logical reading sequence among the book chapters.

Chapter 13 is preparatory to Chapter 14 (Introduction to EMC), where the whole uncertainty class is deployed: initial state, causal and parametric uncertainty, neglected dynamics. Upper bounds of the uncertainty sources are formulated and included in the Z-transform equations that relate errors to sensitivity and CS. The H-infinity norm and small-gain theorem enable the derivation of two stability inequalities (one due to neglected dynamics and the other due to parametric uncertainty) and four performance inequalities, which are driven by the ingredients of the causal uncertainty (random disturbances, sensor and actuator noise). The use of asymptotic approximations in the frequency domain makes the closed-loop poles explicit and converts the aforementioned inequalities into simple design tools. At first sight, such inequalities may seem approximate and naïve with respect to classical robust control methods [17], but they may be refined and solved by analytic, simulation, and in-field optimization. A SISO case study accompanies all the chapter development. Chapters 13 and 14 are confined to the end of the book, being outside the classical sequence of the orbit and attitude topics.

Orbital Models and Control

Chapters 3–5 repeat the topics of classical textbooks, but with some extension. Chapter 3, which is dedicated to the two-body problem, Kepler's equation, and orbital parameters, ends with a study of the orbital stability by means of the Lyapunov direct method. The study enables a first derivation of the Hill–Clohessy–Wiltshire (HCW) state equation, which is extensively studied in Chapter 5. The study of the environmental perturbations in Chapter 4 combines forces and torques, provides a brief introduction to the internal forces and torques due to flexible appendages and liquid sloshing, and to the method of the disturbance dynamics in terms of DT stochastic equations. The method is illustrated with the state equations of the aerodynamic forces/torques of the GOCE satellite. In Chapter 5, the classical topics of the Gauss and Lagrange planetary equations are completed with the definition and conditions of the frozen orbits. The HCW state equation and the relevant feedback stabilization are studied to some extent. The conditions for a decoupled feedback between longitudinal and radial motion are derived and proven by simulation. The chapter ends with an introduction of the restricted three-body problem. The orbital control in Chapter 11 is the most abridged topic in the book. Of the originally planned topics, (1) orbital drag compensation (referred to in the book as drag-free control), (2) orbital quaternion prediction and (3) altitude control, only the first two topics were maintained, being rarely, if never treated in textbooks. Drag-free control is preceded by a short historical and scientific introduction. No mention is given of launch and landing control problems as they deserve specific treatment: only a brief mention of the orbit transfer is given in Chapter 3 (namely Lambert's problem and gravity assist). Orbit estimation algorithms are ignored (see Ref. [7]).

Attitude Models and Control

Chapters 6 and 7 are the core of the book. They are organized in a rather unusual way, as they include both modeling and control topics. Chapter 6 starts with the three classical methods of the rigid-body attitude kinematics, attitude matrix, Euler angles and quaternions. As already mentioned, Rodrigues parameters are omitted. Euler kinematics is used to anticipate concepts and equations of precessing bodies, which are commonly treated together with the rigid-body attitude dynamics. The second part of the chapter is entirely devoted to the quaternion error kinematics (continuous and discrete time) and to several kinds of stabilizing feedback systems. We start from the proportional feedback, then we move to the PI feedback where the integrative part plays the role of first-order disturbance dynamics, to the PID feedback where the derivative part is implemented by a quaternion feedback and to a feedback that consists of an integrative chain. The derivative feedback is employed in the attitude state predictors, which are typical of the EMC methodology. As suggested by the relevant literature, the direct Lyapunov method is extensively used to find out closed-loop stability conditions and to prove the asymptotic convergence of the closed-loop error quaternion kinematics to LTI state equations. These methods and results are employed to design a quaternion state predictor driven by gyroscope and star tracker data. The simulated performances are compared with those provided by a classical Kalman filter.

The three parts of Chapter 7 are (1) the classical Euler's equation of rigid bodies and the whole closed-loop attitude dynamics, (2) the rigid-body attitude under torque-free conditions and under gravity-gradient and aerodynamic perturbations, (3) the classical attitude control problems (namely active nutation damping, spacecraft detumbling by magnetic torquers, reaction wheel and magnetic torque attitude control). The attitude closed-loop dynamics, in the first part, is obtained by endowing the quaternion kinematics and Euler's equations with a PD feedback (namely attitude and angular rate feedback) enriched by disturbance rejection. Closed-loop stability is studied with the direct Lyapunov method. Attitude and angular rate may be provided by two different state predictors driven by: (1) gyroscope and star tracker data as in Chapter 6, (2) star tracker data alone. The second predictor is employed in the attitude control systems of the chapter. Both state predictors are uncertainty-based designed with the method of Chapter 14. In the second part of Chapter 7, the study of the attitude dynamics affected by gravity-gradient and aerodynamic torques is undertaken. The topic is peculiar of the GOCE satellite design, whose nominal attitude was open-loop unstable, because the minimal inertia axis was required to be aligned with the orbital track. The attitude control in the third part assumes a spacecraft, which is actuated by reaction wheels and magnetic torquers (for desaturating the wheels). Control design and simulation prove that the disturbance dynamics of the embedded model and the relevant prediction enable a complete decoupling of the nonrotating (S/C) and rotating masses (reaction wheels) controllers, and the overall disturbance cancellation without an explicit model, but only driven by a parameter-free disturbance dynamics.

Chapter 12 is devoted to the S/C attitude control during a complete mission: the hypothetical mission of a GOCE-like scientific satellite. The case study is preceded by a discussion and formulation of the attitude requirements, which takes into account international standards. The mission attitude control design takes advantage of a unique mission state predictor, which fits the different control modes by switching between input data (namely measurement and commands) and noise estimator gains. Five different control modes are accounted for. The first one is the coarse pointing mode that enables S/C despin and Sun pointing by means of magnetic torquers and a coarse Earth and Sun sensor. The intermediate modes are the two successive fine pointing modes, which employ star tracker data. The last two modes, a preparatory mode and the normal operating mode, are the drag-free modes, which aim to achieve challenging drag-free and attitude requirements. With respect to the real GOCE mission that just employed magnetic torquers as attitude actuators, an assembly of electric microthrusters is assumed to be available in addition to a redundant pair of electric mini ion-thrusters. The all-propulsion orbit and attitude control system was conceived during the Phase A/B of the GOCE mission, but was partly abandoned due to a poor level of maturity of the electric microthruster technology in the early 2000s.

Technology and Attitude Determination

Chapter 8 is devoted to orbit and attitude sensors. The chapter starts with the model and discretization of the measurement errors. Discretization assumes that random errors are either affected by drift components or their measurements are integrated as occurs in inertial measurement sensors (accelerometers and gyroscopes). The first part of the chapter concerns inertial sensors, accelerometers and gyroscopes. The dynamic model of 6D electrostatic accelerometers (linear and angular measurements) is dealt with some details because it is the class of the fine accelerometers mounted on the GOCE satellite. Mention of the accelerometer stabilizing controller is given, but no simulated data are provided. The classical spinning-mass gyroscopes are carefully treated due to their historical and practical importance in the spacecraft attitude control. As an exercise of the EMC methodology, the closed-loop control of the rate-integrating gyro is designed with the help of embedded model, state predictor with dynamic feedback, and control law with disturbance rejection. The second part of Chapter 8 describes global navigation sensors and specifically the NAVSTAR GPS. The contents are standard. The third part is concerned with the four classical types of attitude sensors: Sun sensors, Earth horizon sensors, star trackers, and three-axis magnetometers. Calibration and initialization algorithms are omitted.

Chapter 9 deals with three classical S/C actuators: linear momentum actuators (also known as propulsion systems or thrusters), angular momentum actuators (namely reaction wheels and control moment gyros), and magnetic torquers. Energy dissipation actuators are not treated (see Refs. [5,6]). The chapter starts with a few generalities about reliability, which are marginally employed throughout the book. The first part is dedicated to the propulsion systems. After the well-known Tsiolkovsky's equation, the concept of Delta-V maneuver and a brief exposition of current technologies, the problem of designing minimum-size propulsion assemblies, capable of satisfying force and torque requirements, is approached. The problem is solved with the help of Walsh vectors. Besides the layout design, the optimal and constrained distribution law among different thrusters is found, the constraint being imposed by a nonnegative lower thrust bound. Optimality is defined in terms of the Euclidean norm of the thrust vector. Optimality in terms of propellant consumption is addressed and solved in the case of the abandoned GOCE all-propulsion design, but is kept alive for the NGGM design. The second part of Chapter 9 addresses the reaction-wheel geometry, distribution law and disturbances. The exposition is classical. The second part ends with a brief mention of the control moment gyros. The third part concerns magnetic torquers and is the pursuit of the magnetic torque introduction of Chapter 4.

Chapter 10 treats the classical problem of the static attitude determination from a set of simultaneous direction measurements. The problem has been already cited in Chapter 2 when treating the frames of reference, because the set of observed directions defines an observational frame. The chapter starts with the statistical properties of the direction errors, which requires the Gaussian distribution on the sphere. The first part of the chapter is dedicated to the two-axis determination methods: the cone intersection and the three-axis attitude determination (TRIAD) adopted by the earlier space missions. The important TRIAD error covariance is derived. The second part deals with the classical problem of Wahba, which admits an arbitrary finite number of observed directions. Partly following Ref. [6], the problem is formulated as a maximum likelihood estimation and then solved with the singular value decomposition to provide the estimate of the body-to-inertial transformation matrix. The statistical efficiency of the solution and the comparison with the TRIAD error covariance in the case of two observations are investigated. The exposition proceeds to the modern solution methods, such as the q-method, the quaternion estimation (QUEST) method, and the ESOQ improvement, as they directly provide the quaternion estimate. A Monte Carlo comparison between TRIAD and QUEST methods closes the second part. The third part addresses the problem of the angular rate determination from direction measurements.

1.1.4. Omitted Topics

A few significant topics of the orbit and attitude modeling and control have been omitted, mainly because of time and space restrictions.

In Chapter 4, no mention of the Sun eclipse conditions for Earth satellites is done. There is a brief mention of the internal perturbations caused by flexible appendages and liquid sloshing, but no systematic treatment is provided of the relevant attitude control problems. Only the SISO case study of Chapter 14 assumes a somewhat unusual flexible appendage, which is rather challenging for the attitude control design. A classical and broad treatment of the topic can be found in Refs. [9] and [13].

Off-line sensor calibration (magnetometers, star trackers) and initialization (lost-in-space algorithms) are ignored. Some topics can be found in Ref. [6]. Online prediction and cancellation of gyroscope and accelerometer bias/drift is treated in Chapters 6, 7 and 12. GPS legacy algorithms are mentioned in Chapter 8, but no explicit use of them is made. Only aggregate range and rate error properties are employed by the orbital quaternion prediction of Chapter 11. ON–OFF propulsion is detailed in Chapter 9, but no explicit usage is made.

Chapter 10, about the attitude determination from observed celestial directions, does not mention the recursive methods based on extended Kalman filters. They would deserve a specific chapter.

Another important topic of the attitude control, which is only partly treated, is the attitude reference generation. Out of the three main kinds of attitude reference, namely (1) orbital frames, (2) scanning trajectories such as those of the Hipparcos satellite (see Chapter 6), (3) arbitrary pointing directions toward sky objects and planetary sites, only the orbital frames are treated in Chapter 11. A brief mention to the second and third class is made by the SISO case study of Chapter 14. The design of a supervisor system, which schedules and switches the different orbit and attitude control modes, is not addressed. In Chapter 12, switching between control modes is managed by a single logic variable. Last but not least, fault detection and isolation recovery is not treated.

1.1.5. Authorship and Acknowledgments

Every author contributed to the book chapters in a different way and to a different extent. The concept and architecture of the book are mainly due to E. Canuto, who also wrote the draft of Chapters 1–3, 6–8, 10–12, and 14. C. Novara was responsible for the dynamic system theory and the relevant theorems, and he wrote the draft of Chapters 5 and 13. L. Massotti was responsible for the whole manuscript, sensor/actuator modeling and performance, space environment, and he wrote the draft of Chapters 4 and 9. C. Perez Montenegro and D. Carlucci were responsible for the simulated runs, their explanation, and discussion. The authors are grateful to the anonymous reviewers of the book proposal for their helpful remarks and suggestions. All the simulation results and their graphical plots have been obtained with an academic license of MATLAB/Simulink. The authors will greatly acknowledge any readers’ correction, criticism, remark and suggestion.

Since the early preparation of the European Hipparcos mission, E. Canuto was introduced to spacecraft modeling and control by F. Donati, Politecnico di Torino, Turin, Italy. F. Donati is the author with M. Vallauri of the seminal paper [24] on model-based control that inspired the EMC methodology. This is the right circumstance to express sincere gratitude for their guidance in life and science.

EMC was applied by E. Canuto and his students to prototypes, instruments, and studies led by Thales Alenia Space Italia (TASI), Turin, within ESA-funded projects and under research contracts with Politecnico di Torino. The topics and results of the cooperation with the TASI control system staff have been a precious source for this book. Acknowledgment is expressed to G. Sechi, now at ESA/ESTEC, S. Cesare, A. Bacchetta, P. Martella, M. Buonocore, S. Dionisio, F. Cometto, and M. Parisch.

Part of the book was written by E. Canuto during two research periods at the Centre for Gravity Experiments, Huazhong University of Science and Technology (HUST), Wuhan, China. Their warm hospitality and a lively research staff cannot be forgotten. Special thanks to J. Luo, leader of the center, now President of the Sun Yat-Sen University, Guangzhou, China, and to H–C. Yeh, Z-B. Zhou, and J. Mei.

Many topics and examples of this book were developed by E. Canuto together with his PhD students from Politecnico di Torino, A. Rolino, D. Andreis, F. Musso, J. Ospina, W. Acuña-Bravo, A. Molano-Jimenez, L. Colangelo, and M. Lotufo and two PhD students from HUST, Wuhan, China, Y. Ma and H. Li.

Last but not least, the work of E. Canuto was made possible by the patience of his wife Maria Angela.

1.2. Notation Rules and Tables

1.2.1. Notation Rules

Free 3D vectors , coordinate vectors are written in bold as v, and the symbol of the coordinate frame of reference may be appended as a subscript. For instance, vi . The coordinates of v are written unbolded with a numeric subscript vj, j  =  1, 2, 3. A generic coordinate of vi is denoted by vij, j  =  1, 2, 3, where reference takes precedence over numeric subscripts. The coordinate vectors are column vectors, but when they are inserted in a text line, the nonstandard inline notation v  =  [v1, v2, v3] represents 3D column vectors and v  =  [v1, … , vj, … , vn] n-dimensional column vectors.

The problem of finding a simple font for quaternions that is different from q. The quaternion coordinates are ordered as per the Hamilton representation, which gives precedence to the real part q0 as in complex numbers. Unfortunately, the Hamilton notation is different from most of the aerospace textbooks, but fits C language, whose vector components are indexed from zero. Thus, quaternions are denoted by

(9.1)

the last bracket showing the inline notation.

As a baseline, the same symbol is used to represent different elaboration outcomes of the same variable. They are distinguished by marks as follows:

1. The true variable (known mathematically or by simulation) has no mark, like y(t);

;

;

4. The estimate of the true variable x(t;

5. The prediction of x(t.

Other marks and their meaning can be found in Table 1.2. The same notations apply to vectors and quaternions (see Table 13.2 in Section 13.5.1 for the error notations and meaning).

A set or sequence , if n is known. Round brackets are mostly reserved to functions and diagonal matrices.

Passing to the notations of random variables and processes in Table 1.2, the (unilateral) power spectral density (PSD) of a stationary process d(t. The superscript 2, denoting square, is appropriate though not standard, because any PSD is nonnegative. The advantage is to simply denote the PSD square root with Sd(f)  ≥  0. Because Sd(f) is commonly used, it is given the name of spectral density (SD, no standard name seems available).

1.2.2. Notation Tables

The main vector and matrix notations are collected in Table 1.1.

The main notations for random variables and variable elaboration are collected in Table 1.2.

Table 1.1

Table 1.2

1.3. Abbreviations

The main abbreviations from letter A to G are listed in Table 1.3, from letter H to P in Table 1.4, from letter Q to Z in Table 1.5.

Table 1.3

Table 1.4

Table 1.5

References

[1] Battin R.H. An Introduction to Mathematics and Methods of Astrodynamics. New York: AIAA Education Series, AIAA; 1987.

[2] De Ruiter A.H.J, Damaren C.J, Forbes J.R. Spacecraft Dynamics and Control. An Introduction. Chichester, UK: J. Wiley & Sons, Ltd; 2013.

[3] Greenwood D.T. Principles of Dynamics. Englewood Cliffs: Prentice-Hall; 1965.

[4] Hughes P.C. Spacecraft Attitude Dynamics. New York: Dover Publications, Inc.; 2004.

[5] Kaplan M.H. Modern Spacecraft Dynamics & Control. New York: John Wiley & Sons; 1976.

[6] Markley F.L, Crassidis J.L. Fundamentals of Spacecraft Attitude Determination and Control. New York: Springer Science; 2014.

[7] Montenbruck O, Gill E. Satellite Orbits: Models, Methods, Applications. Berlin: Springer-Verlag; 2000.

[8] Schaub H, Junkins J.L. Analytical Mechanics of Space Systems. second ed. Reston, VA: AIAA Education Series, AIAA; 2009.

[9] Sidi M.J. Spacecraft Dynamics and Control. A Practical Engineering Approach. Cambridge Univ. Press; 1997.

[10] Vallado D.A. Fundamentals of Astrodynamics and Applications. New York: McGraw-Hill; 1997.

[11] Wakker K.F. Fundamentals of Astrodynamics, Institutional Repository Library. Delft, the Netherlands: Delft University of Technology; 2015.

[12] Wertz J.R. Spacecraft Attitude Determination and Control. Dordrecht: D. Reidel Pu. Co.; 1978.

[13] Wie B. Space Vehicle Dynamics and Control. Reston: AIAA Education Series, AIAA Inc.; 1988.

[14] Francis B.A. The internal model principle of control theory. Automatica. 1976;12(5):457–465.

[15] Morari M, Zafiriou E. Robust Process Control. Englewood Cliffs, NJ: Prentice-Hall; 1988.

[16] Canuto E, Molano A, Massotti L. Drag-free control of the GOCE satellite: noise and observer design. IEEE Transactions on Control Systems Technology. March 2010;18(2):501–509.

[17] Doyle J.C, Glover K, Zhou K. Robust and Optimal Control. first ed. Englewood Cliffs, NJ: Prentice-Hall; 1996.

[18] European Space Agency, The GOCE Satellite Site. http://www.esa.int/export/esaLP/goce.html.

[19] Canuto E. Drag-free and attitude control for the GOCE satellite. Automatica. July 2008:1766–1780.

[20] Canuto E, Colangelo L, Lotufo M, Dionisio S. Satellite-to-satellite attitude control of a long-distance spacecraft formation for the Next Generation Gravity Mission. European Journal of Control. September 2015;25:1–16.

[21] Papoulis A. The Fourier Integral and its Applications. New York: McGraw-Hill; 1962.

[22] Franklin G.H, Powell J.D, Workman M. Digital Control of Dynamic Systems. third ed. Half Moon Bay, CA: Ellis-Kagle Press; 1998.

[23] Papoulis A. Probability, Random Variables, and Stochastic Processes. third ed. New York: McGraw-Hill; 1991.

[24] Donati F, Vallauri M. Guaranteed control of ‘almost-linear’ plants. IEEE Transaction on Automatic Control. 1984;29(1):34–41.

Chapter 2

Attitude Representation

Abstract

This chapter, which is introductory to the textbook, briefly recalls the basic elements of linear algebra, vectors, matrices, and quaternions. A special class of matrices is the class of the proper orthogonal matrices as they perform the change of basis (or coordinates) in a vector space. The ‘proper’ attribute restricts the group of orthogonal matrices to the ‘proper’ rotations, endowed with a positive unitary determinant. In a three-dimensional (3D) space, the algebraic concept of basis materializes into the geometric and observational concept of the Cartesian frames of reference, where 3D coordinate vectors are defined. Conversion between 3D coordinates is performed by 3D proper orthogonal matrices that consequently bear the essential task of representing the attitude/orientation of rigid bodies. The chapter shows that attitude may be represented by an alternative algebraic entity, known as quaternion, which has the advantage of a smaller number (four) of elements than the nine elements of rotation matrices. A key property of rotation matrices is their factorization into elementary Euler rotation matrices, each defined by a single angular rotation, known as Euler angle, about an appropriate independent direction. Since the minimum number of independent Euler angles is three (the same dimension of a 3D space), a 3D set of independent Euler angles constitute the minimum dimension attitude representation. The four-dimensional unit quaternions have a geometrical interpretation, which descends form the Euler rotation theorem, and is related to rotation matrices via the well-known Rodrigues formula. Rotation matrices, unit quaternions, and Euler angles are the three main attitude representations. Conversions between them are duly explained.

Keywords

Attitude representations; Conversions between representations; Euler angles; Linear algebra; Quaternion; Rotation matrix; Space and time coordinates

2.1. Objectives

This chapter is concerned with the orientation of a rigid body in a three-dimensional (3D) space. Spacecraft and celestial objects, such as planets can be well approximated as rigid bodies. The mathematical formulation of the orientation is indicated as attitude. Attitude representation is closely related to Cartesian frames of reference (frames, in brief) and to the vector coordinates in such frames. The attitude formulation requires the definition of two frames: an observer's frame, and a body frame which is representative of the rigid body under study. Attitude is defined as the orientation of the body frame with respect to the observer's frame. The observer's frame is not necessarily inertial, although attitude is usually represented and measured with respect to inertial frames. When their orientation changes over time, the frames become function of the time variable.

Throughout this textbook, and also in practice, attitude may be represented in several ways, each way possessing different properties and usage. The starting point is the cosine direction matrix (or attitude matrix), which has two interpretations: as the coordinate transformation, for example, from the body to the observer's frame, and as the rotation of one frame into another. Any attitude matrix can be constructed by composing elementary rotations around the Cartesian axes of the observer's, body and intermediate frames. The Euler angles are the degrees of freedom (DoFs) of such compositions, and Euler showed that their minimum dimension is three. Euler angles, like spherical coordinates, are subject to singularities (the well-known gimbal lock phenomenon) that prevent their use for efficient and robust numerical computations. Unit quaternions, a subset of the quaternions invented by W.R: Hamilton in 1843, represent the modern alternative. The four quaternion components are the algebraic representation of the four Euler parameters (rotation angle and axis) that come out from the fundamental Euler rotation theorem. As well as Euler and Hamilton, the French mathematician O. Rodrigues made a decisive and illuminating contribution to the field in the 19th century. In recent years, the field has been organized by M.D. Shuster (see Refs. [16,17]) and F.L. Markley (see Ref. [10]).

The chapter begins with an extensive review of vectors and matrices, giving emphasis to 3D vectors and their operations, as they will be extensively used in the textbook. The matrix review is centered on orthogonal matrices, whose class attitude matrices belong to, and on coordinate transformations (change of basis) between Cartesian frames, operated by orthogonal matrices. The review ends with the introduction of quaternions and their operations. The second part of the chapter discusses the materialization of Cartesian frames, with the aim of defining and constructing the most significant frames where the linear and angular motion of spacecraft can be represented. Frames are classified as body, celestial, trajectory, and observational, according to their different uses and constructions. The third part of the chapter is concerned with attitude representations. The chapter ends with the formulation of attitude errors by means of infinitesimal rotations.

2.2. Vectors and Matrices

2.2.1. Three-Dimensional Vectors

A 3D Cartesian frame (or frame of referencedeparting from the origin Ois known as the poleis the reference or observer's planefrom O to the point P as in Fig. 2.2. The dot indicates scalar products, to be defined in Section 2.2.2. The scalar vk is the k. The coordinates can be collected in a column vector v: the coordinate vector or simply vector. The unusual inline coordinate notation v  =  [v1, v2, v3] will be employed to denote a column vector in the running text. Bracketed symbols like {v1,…} denote a set or sequence of elements, also in the case of a pair {v1, v2}. Brackets drop when the set becomes the argument of a function like f(v1, v2, v3). We write dimv = 3 or vcan be written as follows:

(2.1)

. The mutual orthogonality of the frame axes is expressed by the scalar product of the vectrix being equal to the identity matrix I:

(2.2)

may be selected to be right handed or left handed. Let us denote any of the triads 123, 231 and 312 with ijk, where the symbol  ×  denotes the cross product to be defined in . The sign in front of the following table of cross products does the same:

(2.3)

along the thumb as in Fig. 2.1.

The coordinate notation v  =  [v1, v2, v3] may be replaced by v  =  [x, y, z], and the Cartesian axes may be referred to as x-axis, y-axis, and z-axis. Each symbol of {x, y, z} may become a subscript instead of k  =  1, 2, 3. The coordinate vk is the cosine of the angle αk , and for this reason is known as the kth direction cosine . The Euclidean norm or length v . This allows vk to be written as vk  =  vcosαk. , the angle αk , and the planes where the projections lie.

Figure 2.1  A right-handed Cartesian frame.

Figure 2.2 .

lying on a sphere of radius r . The coordinate vector r may be represented by spherical coordinates, which consist of the radius r, azimuth α, and polar angle β as follows:

(2.4)

With the help of Fig. 2.3, the azimuth α on the reference plane; the polar angle β with the previous identity proves Eq. (2.4). The hemisphere r3  >  0 is the North hemisphere and the hemisphere r3  <  0 is the South hemisphere.

Often the polar angle β is replaced by the elevation δ  =  π/2  −  β . The coordinates in Eq. (2.4) are replaced by

(2.5)

The order of the spherical components in Eq. ((x(y-axis). By denoting the coordinate order in Eq. (2.4) and (2.5) with 123, the order becomes 312 if the x-axis is chosen as the pole, and changes to 231 if the y-axis is selected.

Figure 2.3  Spherical coordinates.

The spherical coordinates become singular at the poles of the sphere, where β  =  {0, π} and δ  =  ±π/2, because the azimuth α becomes arbitrary owing to r1  =  r2  =  0. The singularity is known as two-dimensional (2D) gimbal lock.

Two-Dimensional Gimbal Lock

To understand practical implications, let us place, in the observer's frame origin O, an antenna which tracks a target point P . The antenna can only scan the North hemisphere rhas two DoF, azimuth −π  ≤  αa  <  π, and declination 0  ≤  δa  <  π. Let the width of the antenna field-of-view be ϕ  <<  π. When the antenna points toward δa  =  π, and whichever be the azimuth αa, the pointing direction cannot change: one degree of freedom is lost. Now, let us assume that when the target P reaches the zenith, selects an arbitrary direction of motion, which means an arbitrary azimuth α  ≠  αa. To track P, the antenna azimuth should perform a jump α  −  αasince the antenna cannot track the jump instantaneously.

Exercise 1

Prove that the equation of a great circle passing through two points P1 and P2 on a sphere of radius r, with coordinates r1 and r2, is given by

(2.6)

.□

By replacing φ in Eq. (2.6) with φ  =  γΩ, 0  ≤  γ  <  1, the great circle arc between r1 and r2 is obtained. The arc is the shortest-distance path between two points P1 and P2 on a sphere, the distance d, known as the orthodromic distance, being equal to d  =  rΩ. Eq. (2.6) with φ  =  γΩ, 0  ≤  γ  <  1 can be interpreted as a spherical linear interpolation (SLERP, in Shoemake [14]). It tends to become a linear interpolation as soon as Ω→0.

2.2.2. Vector Operations

with coordinate vectors v1 and v2, basic operations are the vector sum, the dot or scalar or inner product, already mentioned, and the vector or cross product. The same expressions and identities apply to the coordinate vectors v1 and v2. The length of the vector sum satifies the triangle inequality

(2.7)

Exercise 2

Prove the left-hand-side inequality of Eq. (2.7). □

The dot product can be written in terms of the angle α between the component vectors:

(2.8)

which proves the Cauchy–Schwarz inequality,

(2.9)

Two vectors are orthogonal when their scalar product is zero, they are aligned when the equality holds in Eq. (2.9), and they are linearly independent can be decomposed into the sum of the orthogonal projection. By using coordinates, the following series of identities is found:

(2.10)

where

(2.11)

P and Π in Eq. (2.11) are square matrices known as projection matrices and are the first examples of a transformation between coordinate vectors. P is a particular case of the outer productbetween v1 and v2.

Sometimes complex vectors like v  =  a  +  jb , which is a simple case of a dual vector (see Luenberger [9]). Given two complex vectors v1  =  a1  +  jb1 and v2  =  a2  +  jb2, their dot product is defined by

(2.12)

implies orthogonality. A complex unit vector v .

Exercise 3

Given a complex vector v  =  a  +  jb, or which is the same vTv = 0, implies a·b . □

The cross product, whose result is a vector, can be rewritten as a vector transformation, upon definition of the cross-product matrix V× of v as follows:

(2.13)

An alternative notation of V× is just v×.

Exercise 4

With the help of Eq. (2.13) prove that

(2.14)

Exercise 5

With the help of Eq. (2.14) prove that v  ×  v  =  0 and that v1·(v1  ×  v2)  =  v2·(v1  ×  v2)  =  0. In other terms, the cross product of two aligned vectors is zero, and the cross-product vector v1  ×  v2 is orthogonal to v1 and v2. □

Exercise 6

Prove that the scalar triple product v1·(v2  ×  v, that is

(2.15)

Exercise 7

Prove that the adjugate can be written in terms of cross products as follows:

(2.16)

where (−1)i+jdetVij is the cofactor of the ith row and jth column entry vij of V. □

The term adjugate allows us to retain the term adjoint for the conjugate transpose of a matrix. Let us assume that rankV  =  2, where V appears in Eq. (2.15). In other terms, we assume that VT , which is defined by λVTn  =  0, with nTn  =  1 and λ real scalar. Any vector vj  ×  vh, j  ≠  h  =  1, 2, 3, which is a row of adjV in Eq. (2.16) (a column of adj(VT)  =  (adjV)T, that is, VTvj  ×  vh  =  0. The dual result is that the rows of adj(VT) (the columns of adjV. This simple and interesting result can be extended to a generic n  ×  n square matrix A with (Aij)  =  aij and rankA  <  n, in other terms with detA  =  0. The extension is done through the property Aadj(A)  =  det(A)In. In the rank-deficient case, all the columns of adj(A) are orthogonal to the rows of A and span the nullspace of A. If rankA  =  n  −  1, any nonzero column of adj(A. This property will be employed in Section 10.5.2 to find out a singularity-free solution of the attitude determination problem known as Wahba's problem [12]. The diagonal identities of Aadj(A)  =  det(A)In correspond to the Laplace expansion of detA, that is to

(2.17)

is the ith row of A, and (adjA)i is the ith column of adjA.

Sometimes the vector triple products u1  =  v1  ×  (v2  ×  v3) and u2  =  v2  ×  (v1  ×  v3) become useful. The product uk, which is orthogonal to vk, k  =  1, 2, can be written as a linear combination of vj, j = 1,2, j≠k, and v3, as follows:

(2.18)

Exercise 8

With the help of Eq. (2.18), prove that

(2.19)

Comparison of the first row in is a projection matrix.

Exercise 9

Prove that a unit vector v, satisfies

(2.20)

Exercise 10

With the help of Eq. (2.19), prove the following identity:

(2.21)

A collection of