Aalborg University Institute of Electronic Systems

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1 Aalborg University Institute of Electronic Systems TITLE Attitude Control System for AAUSAT-II PROJECT PERIOD IAS8, Febuary 2 nd - June 2 nd, 2004 PROJECT GROUP 830a GROUP MEMBERS Daniel René Pedersen Jacob Deleuran Grunnet Jesper Abildgaard Larsen Karl Kaas Laursen Ewa Kolakowska Isaac Pineda Amo SUPERVISOR Roozbeh Izadi-Zamanabadi Rafal Wisniewski Number of reports printed 10 Number of pages in report 140 Number of pages in appendix 26 Total number of pages 169 ABSTRACT The second student Cubesat satellite developed at Aalborg University, AAUSAT-II, has a scientific mission that requires precision attitude control. For this purpose an attitude control system is designed which, by the use of magnetic and mechanical actuation, is capable of detumbling and correcting the attitude of a CubeSat. Magnetorquers and momentum wheels have been selected resulting in an over actuated system facilitating three axis control of the spacecraft. A simulation library containing tools for simulating the orbit and attitude including disturbances has been implemented in Simulink. ADCS is integrated in the library including sensors, actuators and implementation of the attitude determination and control algorithms. A supervisory controller is developed to manage the attitude controllers to allow autonomous behavior such as momentum wheel bias control and target tracking. Classical pole-placement and optimal control is utilized to tune a state space controller for momentum wheel attitude control. The B-dot algorithm is used for detumbling using only electromagnetic actuation and a custom algorithm for momentum wheel desaturation and bias control has been developed. Simulations of the different controllers have indicated that the controllers satisfy all accuracy requirements. Further testing of the momentum wheel control will be carried out on a parabolic flight campaign.

2 Contents Nomenclature Preface v xi 1 Introduction 1 2 Mission analysis Mission objectives In-orbit operation Satellite structure Requirements Mission Definitions Functional requirements Budgets System configuration Actuator strategy Sensor analysis Configuration of momentum wheels Hardware configuration Software configuration Perspectives Actuators Magnetorquers design

3 5.2 Design of Momentum Wheels Fault Detection and Isolation system Summary Modelling Coordinate system definitions Model structure Orbit and magnetic field modelling Ephemeris model Disturbance modelling Spacecraft dynamics and kinematics Modelling of actuators Summary Perspectives Controllers Control strategies Control supervisor Detumbling controller based on B-dot Momentum wheel attitude controller Desaturation of momentum wheels Perspectives Test Introduction Feel free - feel zero-g Test strategies Conclusion Project objectives Study objectives Perspectives Bibliography 130

4 iv Attitude Control System for AAUSAT-II A Quaternion 135 A.1 Axioms and definitions A.2 The time derivative of a quaternion B Implementation of simulation 139 B.1 IGRF model B.2 Orbit model B.3 Solar radiation B.4 Atmospheric drag B.5 Gravitational disturbances B.6 Magnetic residual B.7 Spacecraft dynamics and kinematics B.8 Magnetorquers B.9 Momentum wheels C Interface Control Document 151 C.1 System modes C.2 External Interfaces C.3 Internal Interfaces

5 Nomenclature Common terms Attitude The orientation of a spacecraft given as a rotation between two coordinat systems. Ecliptic plane Mean plane of the Earth s orbit around the Sun. Eclipse Transit of the Earth in front of the Sun, blocking all or a significant part of the Sun s radiation. Failure The inability of an actuator, sensor, subsystem or system to accomplish its required function. Fault A change in the characteristics of a component, such that it influences the operation mode or performance of the component in a undesired way. Geostationary A spacecraft travels above Earth s equator from west to east at an altitude of approximately 35900km and at a speed matching that of the rotation of the Earth, thus remaining stationary in relation to the Earth. Latitude The angular distance on the Earth measured north or south of the equator along the meridian of a satellite location. Longitude The angular distance measured along the Earth s equator from the Greenwich meridian to the meridian of a satellite location. Vernal Equinox The point where the ecliptic crosses the Earth s equatorial plane going from south to north. Acronyms ACS Attitude Control System ADCS Attitude Determination and Control System ADS Attitude Determination System

6 vi Attitude Control System for AAUSAT-II BGA Ball Grid Array CDH Command Data Handling COM Communication system ECI Earth Centered Inertial coordinate system ECEF Earth Centered Earth Fixed coordinate system EKF Extended Kalman Filter EPS Electric Power Supply system FMEA Failure Mode and Effect Analysis FOV Field Of View GEO GEostationary Orbit GPS Global Positioning System GRB Gamma Ray Burst GS Ground Station IGRF International Geomagnetic Reference Field INSANE INternal Satellite Area NEtwork LEO Low Earth Orbit MECH Mechanical system OBC On Board Computer P/L Payload system RMS Root Mean Square SCT Spacecraft Control Toolbox SGP4 Simplified General Perturbation version 4 TLE Two-Line Element

7 Group 830a vii Notation Physics parameters There are a number of physical parameters for which standard symbols are used. 1 n n The n dimensional identity matrix a Acceleration A Area I Inertia tensor. Al Altitude B Magnetic field vector c Velocity of light eu Engineering Unit (ie. a unit less scalar) F Force G Gravitational constant H, h Angular momentum h mw Angular momentum for momentum wheels h sat Angular momentum for rigid body part of the satellite I Moment of inertia I xx, I yy, I zz Moments of inertia I xy, I xz, I yz, I yx, I zx, I zy Products of inertia JD Julian Date JD E Julian Date since defined epoch La Latitude Lo Longitude M, m Mass N Torque q Quaternion

8 viii Attitude Control System for AAUSAT-II R Position r Radius r earth Mean radius of the Earth T, t Time V Velocity Φ Solar flux µ earth Gravitational constant times the mass of the Earth. θ Rotation angle S( ) Skew symmetric matrix (3 3), vectors cross product function ω Angular velocity vector x State vector e Unit rotation vector R n n dimensional real space P Payload axis Coordinate systems Coordinate systems, also known as frames, are abbreviated as below. The definition of the frames can be found in section 6.1 on page 41. E Earth centered earth fixed. I Earth centered inertia. S Spacecraft fixed. P Spacecraft fixed principal axes. O Orbit fixed attitude reference.

9 Group 830a ix Objects To ease the typesetting of equations, the following objects are abbreviated as below: sc Spacecraft s Sun e Earth m Moon Vectors, matrices and rotations Coordinate vectors are typed as E V subscript with the reference frame of the coordinates superscripted in front of the vector. Vector components are typed as E V x which denotes the x component of the V vector in the ECEF frame. Matrices are typed like vectors I M sc The above matrix M relates to the sc object (the spacecraft) in the ECI frame. Rotations (quaternions) are typed like vectors, but with a source (bottom) and destination (top) indices; I Eq m The quaternion q rotates the moon, m, from the ECEF frame to the ECI frame Subranges of eg. vectors and matrices use the following notation E V [1:3] This subrange signifies element 1, 2 and 3 of the first row of the vector V in the ECI frame. In other words, subranges selects elements in the first row. If columns are to be selected, the vector or matrix will first be transposed. Unit vectors are typed like vectors but hatted, ie. EˆV subscript

10 x Attitude Control System for AAUSAT-II Matlab notation In diagrams from Simulink for Matlab, typesetting is not possible to the same extent as in the report. Due to this, the following notation is used: Vectors are typed as V(I) where the letter in parenthesis is the reference frame. Rotations are typed like q(i E) which denotes a rotation from the ECI to the ECEF frame. Note that scalars are written in small caps, eg. i_xx

11 Preface This report documents the development of an Attitude Control System (ACS) for the AAUSAT- II which is a 8 th semester project in Intelligent Autonomous Systems specialization at Aalborg University, Department of Control Engineering. The project began on the 2nd of February 2004 and was finished on the 2nd of June Throughout the report, figures and tables are numbered consecutively according to the chapters. References to literature are done like this: [30, p. 24], which refers to Space Mission Analysis and Design by J. R. Wertz, page number 24. The bibliography can be found on page 131. The attached CD-ROM contains the Simulink models used by the ACS. It also contains the datasheets for the motors and solar panels. The nomenclature, which contains the notations used throughout the report, can be found on page v. Since this project is closely related to the Attitude Determination System project, the first part of the report has been written in conjunction with the project group which has been responsible for this subsystem (group 04gr830b). The parts that are common for the entire Attitude Determination and Control System are the mission analysis in chapter 2, system requirements in chapter 3, system configuration in chapter 4, part of the modelling in chapter 6 and the Interface Control Document in appendix C Aalborg University June 2 nd Daniel René Pedersen Jacob Deleuran Grunnet Jesper Abildgaard Larsen Karl Kaas Laursen Ewa Kolakowska Isaac Pineda Amo

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13 CHAPTER Introduction This project is part of the ongoing process at Aalborg University to design and build a picosatellite named AAUSAT-II. The satellite will be constructed by students without professional help. AAUSAT-II is part of the CubeSat concept [28] that defines a standard for pico-satellites which makes it possible to both construct and launch satellites into space at a relatively low cost. The cheap launch is achieved by sending it into space as a secondary or tertiary payload using a standardized deployment system known as a P-pod, developed by California Polytechnic State University. The primary aspects of the CubeSat concept are that the satellite must be a cube with side lengths of 10 cm 10 cm 10 cm and weigh no more than 1 kg. The low cost of the satellite has made it possible for Aalborg University to participate in the construction of satellites. On the 30 th of June 2003 the first satellite from AAU, the AAU-CubeSat, was launched, fulfilling several of its main success criteria. The goal of the satellite was to take photographic images of the Earth using its on board camera. However due to weak radio signals and failure of the batteries on board the satellite did not succeed in taking pictures. On the 22nd of September 2003 the AAU-CubeSat project announced end of operations. AAUSAT-II was started in September 2003, and a preliminary launch date is set for late The goals of the AAUSAT-II mission are to measure gamma-rays and X-ray from the Sun. Therefore the satellite has to point towards the Sun when it is required. The secondary goal include testing an attitude control system using momentum wheels. The primary goal of the AAUSAT-II project is to educate students in working with satellite construction. Several groups, spread across different semesters and specializations, are involved in designing the different parts of the satellite. Such a distributed development process makes it necessary to have a system steering committee which consists of representatives from every group involved. The system steering committee is in charge of maintaining consistency in the satellite project, and approving changes in one part of the satellite that might affect other parts of the satellite. It is also up to the system steering committee to derive the basic requirements to the different parts of the satellite. These requirements are power usage, mass, size, interfaces and the goals of the satellite mission. This project will deal with the development of the Attitude Control System (ADCS). The goal of this subsystem is to control the satellite attitude continuously, such that the satellite points in the desired direction.

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15 CHAPTER Mission analysis This chapter will describe the background of the overall mission requirements for the AAUSAT- II satellite. These include the primary objectives, the satellite life cycle and briefly describe which subsystems the satellite is composed of. 2.1 Mission objectives The AAUSAT-II satellite is known as a student satellite. That is, the satellite is built and operated by students with help and advise from supervisors. The AAUSAT-II development team consists of several groups working on one or more specific subsystems. The Steering Committee, which can be regarded as the management group, has the overall responsibility for the AAUSAT-II team. The Steering Committee consists of one student from each of the project groups, the project manager, the technical manager, the system engineering manager and finally a representative from Aalborg University Space Center. The Steering Committee has defined the goals of the AAUSAT-II project, of which the most important have been listed below in order of priority. Detailed information about the structure of the AAUSAT-II development team can be found in the AAUSAT-II user requirement document [15] or on the official AAUSAT-II homepage, Please note that these requirements are not related to the physical satellite itself. 1. Education of students and staff The whole project should contribute to the education of the students involved in the project and assigned members of the staff. It should produce a greater insight into space engineering in general and enhance the competence of the students in the relevant engineering disciplines on which it has its foundation. 2. Primarily be built by students The project should be carried out primarily by students. This means that everything from system engineering decisions to the acquiring of electronic components should be done by students, and that the staff members assigned to the project should function only as advisors and final arbitrators. 3. Gathering of telemetry The satellite should be able to gather and provide telemetry so that internal temperatures, voltages and power consumption can be monitored from the Earth. The satellite should be able to continuously collect and store these data and send them to earth, so the performance of the satellite can be analyzed.

16 4 Chapter 2 Mission analysis Attitude Control System for AAUSAT-II 4. Conduct a scientific experiment As the payload of the satellite, AAUSAT-II should have a scientific experiment which should be conducted and the telemetry sent to earth. The actual experiment has been provided by the Danish Space Research Institute (DSRI) and features a combined Gamma Ray Burst (GRB) and X-Ray detector. 5. Active control utilizing momentum wheels A feasibility study of active control utilizing momentum or reaction wheels should be conducted to see if it is possible and/or feasible to control a pico-satellite using a electro-mechanical system. As this report is part of the Attitude Determination and Control System (ADCS), mission objective 5 is what is treated in this report. 2.2 In-orbit operation Both before, under and after the satellite is launched into space, the life cycle of the AAUSAT- II is well defined. This section will briefly describe the different phases in the life cycle of AAUSAT-II. Development During the development stage, the different subsystems of the satellite are developed and tested separately. Integration In this stage, the different subsystems are interfaced (and tested) to each other one at a time. The result is the engineering model of the AAUSAT-II. Test During the test phase, the engineering model undergoes various tests to find possible design and/or implementation flaws, which can be corrected before the flight model of AAUSAT-II is built. Pre-launch The pre-launch stage is where the flight model undergoes the final tests and is shipped to the launch site. Launch AAUSAT-II is launched into a Low-Earth Orbit (LEO) as secondary or tertiary payload. Deployment When the rocket carrying AAUSAT-II has reached the designated orbit, the satellite is deployed via the Poly Pico satellite Orbital Deployer (P-POD) [24]. Power on After e.g. 5 minutes, the satellite is powered up by the Electrical Power Supply (EPS), and the antennas are deployed (should this be needed). Detumble The ACS system detumbles the satellite, so the angular rate is close to zero. Basic beacon When detumbling is complete, the EPS starts sending out a basic beacon, containing the satellite name and battery voltage.

17 Group 830a 2.3 Satellite structure 5 Establish two-way communication Once the ground station intercepts the basic beacon, twoway communication should be initiated to allow for download of advanced telemetry and upload of new parameters and/or software. Mission start Once two-way communication is achieved (when the satellite is over a suitable ground station), the science experiment and the active actuation test should start, and telemetry should be gathered. End of operation When the experiments are completed, and there is no point in continuing support of the satellite, end of operation should be declared, and the satellite should be shut down. Depending on the orbit altitude, the satellite will eventually burn up in the upper atmosphere. This report is carried out in the first phase, development. 2.3 Satellite structure The satellite consists of several different subsystems with each specific task. The subsystem abbreviations and the overall tasks of the particular system are listed below. ACS Attitude Control System. Control of the orientation of the satellite using electro magnetic coils and momentum wheels. ADCS Attitude Determination and Control System. Since ACS and ADS (below) are tightly coupled, the two subsystems are sometimes regarded as a single entity. ADS Attitude Determination System. Subsystem with the purpose of calculating the orientation of the satellite using rate gyros, solar panels and magnetometers. COM Communication. This subsystem consists of the radio and power-amplifiers needed to get two-way communication with earth. It also includes the antennas for the radios. EPS Electrical Power Supply. The system must charge the on board batteries with power generated by the solar panels. EPS is also responsible for sending out the basic beacon, which is a simple telemetry radio signal (for example using Morse code). GS Ground Station. This subsystem is not a part of the satellite itself, but is situated on earth as the main communication channel to the satellite. INSANE INternal Satellite Area NEtwork. This subsystem is the on board satellite communication channel between the different subsystems and consists of the physical communication layer (wires) and a protocol stack, which all subsystems must use to ensure compatibility. MECH Mechanical subsystem. The mechanical system must provide the satellite frame and casing, in which the other subsystems can be integrated.

18 6 Chapter 2 Mission analysis Attitude Control System for AAUSAT-II OBC On Board Computer. The on board computer is the main processing facility on board, and features most of the on board software like the flight planner and ADCS algorithms. P/L Payload. The payload consists of the DSRI-supplied Gamma Ray Burst (GRB) detector and controller, which is the key scientific experiment. CDH Command Data and Handling. This subsystem consists of the flight planner, which is the in-orbit scheduler for the various events which is to take place. CDH also includes the various telecommand and telemetry handles. The internal structure of the satellite is depicted on figure 2.1 on the next page, showing the different busses and communication channels on the satellite, namely the network bus named INSANE, and the power bus that delivers power to the systems which is the responsibility of the EPS. As can be seen on figure 2.1 on the facing page, the INSANE is used both as the satellite network between physical subsystems via a CAN bus and as the communication channel internally in the OBC via a mailbox system. The satellite link to earth is via a radio link between the radio unit and the GS. The EPS subsystem is directly connected to the solar cells and batteries, and all subsystems are connected to the EPS, which is responsible for monitoring the power usage. In the OBC subsystem two subsystems (SS) can be found. These represent a physical subsystem which will have a software system running on the OBC as well, for example the P/L, which needs data storage, and the ADCS system which might be too computational demanding to run on the decentralized physical subsystem itself.

19 Group 830a 2.3 Satellite structure 7 MECH OBC CDH SS SS INSANE (mailbox) INSANE (can) COM ADS ACS P/L EPS Power RADIO SOLAR PANEL BAT TERIES Radio link GS Figure 2.1: Overall satellite structure.

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21 CHAPTER Requirements This document states the requirements for the Attitude Determination and Control Systems for the AAUSAT-II satellite. 3.1 Mission The following two prioritized scientific missions for AAUSAT-II are: 1. Performance study of a single CdZnTe (CZT) detector 2. Gamma ray burst detection 3. Solar X-ray detection These missions prompt a need to adjust the attitude of the satellite which requires an attitude determination and control system. Requirements to the following subjects are set: Detumbling Pointing Fault detection and handling Autonomy 3.2 Definitions This section contains the prerequisites for the functional requirements Modes The term mode is used to indicate the requested state of the ADCS. The modes to be implemented are:

22 10 Chapter 3 Requirements Attitude Control System for AAUSAT-II Idle The ADCS is idle. Sensors and actuators use minimal power and the required computational demands are minimal. Detumbling The ADCS is performing angular rate reduction of the spacecraft. Pointing The ADCS is acquiring a requested attitude and maintaining this attitude. This mode has three submodes: Attitude control The ADCS is acquiring a requested attitude and maintaining this attitude. Direction control The ADCS is pointing the payload in a direction given by a target vector. Autonomous pointing The ADCS is pointing the payload in the direction of a target which is continuously calculated by the ADs States The term state is used to indicate the actual physical state of the spacecraft. The states referenced in the requirements to ADCS are: Detumbled state The satellite is detumbled and requirement [ADCS 1.2] in section applies. Stable pointing state The satellite is in detumbled state and a requested attitude has been acquired. Requirement [ADCS 2.2] in section applies Pointing The term pointing is defined to describe the act of acquiring and maintaining a given attitude referenced in the inertial coordinate system. This also includes direction control where the rotation around the payload axis is free. The term attitude error given as an angle is defined using the error quaternion in (3.1). q err q err[4] = q att q 1 = cos ref ( θerr 2 ) θ err = 2 cos 1 (q err[4] ) (3.1) The term pointing error is given as an angle between the payload axis and the vector indicating the requested direction in which to point.

23 Group 830a 3.3 Functional requirements Functional requirements This section states the functional requirements to the ADCS Detumbling ADCS 1 The ADCS has to be able to detumble the satellite in order to stabilize the attitude of the satellite. ADCS 1.1 The ADCS must be able to detumble the satellite from an angular velocity of 0.1 rad s down to rad within 5 orbits. s ADCS 1.2 In detumbled state the ADCS must keep the angular velocity below rad s axes. on all Pointing The GRB detection mission has no accuracy requirements for the attitude of the satellites other than is mst be able to point away from the Sun; but the X-ray detector alone determines how accurate the satellites attitude must be adjusted. The requirement for pointing accuracy from the X-ray mission is below 5, see [9]. A prerequisite for switching to pointing mode is that the satellite is in detumbled state and the ADCS is fully operational. ADCS 2 The ADCS must be able to turn the satellite to an attitude or direction given by the flight planner in pointing mode. ADCS 2.1 The ADCS must be able to maintain a stable pointing state continuously for 25% of an orbit. ADCS 2.2 The ADCS must be able to maintain a stable attitude of the satellite with an attitude error below 5 for 99% of the time in stable pointing state when in attitude control mode. ADCS 2.3 The ADCS must be able to maintain a stable direction of the pauload axis with a pointing error below 5 for 99% of the time in stable pointing state when in direction control mode. ADCS 2.4 The ADCS must be able to maintain a stable direction of the pauload axis with a pointing error below 5 for 99% of the time in stable pointing state when in autonomous pointing mode.

24 12 Chapter 3 Requirements Attitude Control System for AAUSAT-II Fault detection and handling ADCS 3 The ADCS must be able to detect sensor and actuator faults. ADCS 3.1 The ADS is in charge of monitoring the sensors and notifying the ADCS in case of a fault or failure. ADCS 3.2 The ACS is in charge of monitoring the actuators and notifying the ADCS in case of a fault or failure. ADCS 3.3 In case of failure in the redundant sensor system the ADS must still be able to determine the attitude of the satellite with an rough accuracy of 15. ADCS 3.4 The ACS must have at least one redundant actuator system. ADCS 3.5 In case of failure in the redundant actuator system the ACS must still be able to keep the satellite in detumbled state Autonomy ADCS 4 The ADCS must be able to autonomously point the payload in a direction given by a vector. ADCS 4.1 The ADCS must be able to autonomously point the payload at the Sun in pointing mode. 3.4 Budgets The budgets for the joint ADS and ACS is currently only a draft with a 200 g buffer for the mass, but the limits should not be exceeded Power The power budget of the ADCS system is 225 mw in active mode and 25 mw in idle mode. The expected average power generated by the EPS is 2.5 W Mass The mass budget for ADCS is 200 g. The maximum total mass of the satellite is 1000 g.

25 CHAPTER System configuration In this chapter the configuration of the sensors, actuators and other hardware of the ADCS is discussed. This is done by defining some basic design parameters such as the actuation strategy and which sensors to use. Following this, the physical configuration of sensors and actuators is set and considerations on electronic circuits and software design are made. 4.1 Actuator strategy To be able to control the attitude of the spacecraft at least one actuator system is needed. Actuators must be able to cause an angular acceleration of the spacecraft. According to equation (4.1) they must assert a torque on the spacecraft. N = ωi (4.1) As the torque is the derivative of the angular momentum, the actuator must change the angular momentum of the spacecraft, which according to Newtons law must be constant when the spacecraft is not affected by external forces. As a result there are only two ways to alter the attitude of the spacecraft. 1. By transferring angular momentum to an external object. 2. By transferring angular momentum to another part of the spacecraft. They can be achieved by a number of different methods. The most common ones are discussed below Chemical thrusters Chemical thrusters utilize a chemical reaction which accelerates a propellant and expels it from the spacecraft. In other words, momentum is transfered from the spacecraft to the propellant, thus chemical thrusters belong to the second group of actuators. They can be used to exert a torque on the spacecraft but they have a limited operational time which depends on the amount of fuel. As the chemical thrusters use propellant, they can not be used on a Cubesat, [24].

26 14 Chapter 4 System configuration Attitude Control System for AAUSAT-II Ion thrusters Ion thrusters expels an ionized propellant accelerated by an electric field. This makes the ion thrusters more fuel-efficient than the chemical thrusters, but with much lesser thrust. As with chemical thrusters the ion thruster belongs to the second group of actuators as the angular momentum of the satellite is transfered to the ionized propellant. Ion thrusters are a new and relatively unproven technology with very high complexity. The power consumption used for creating the electric field is very high and it is thus not feasible for use in a CubeSat Momentum wheels Momentum wheels consist of a motor and a flywheel. When the flywheel is accelerated by the motor it picks up angular momentum, which is transfered from the satellite frame (on which the motor is mounted). With three momentum wheels it is possible to transfer angular momentum from the satellite to the momentum wheels. Instead of using three momentum wheels it is also possible to use a single momentum wheel in a gyroscopic suspension. However there is a limit of how much angular momentum can be transferred to the momentum wheels as the motor has a saturation limit, and momentum wheels are thus often used in conjunction with another actuation system. The momentum wheels belong to the second group of actuators as the angular momentum is transfered to the momentum wheels. Momentum wheels are a viable actuation system for the satellite as it is possible to find motors of very small sizes and low power consumption Magnetorquers Magnetorquers belong to the first group of actuators as they work by generating a magnetic field, which through interaction with the Earth s magnetic field, transfers angular momentum to the Earth. Magnetorquers consists of a set of coils which generates a magnetic field when current runs through them. Three coils are needed to achieve control in all spatial directions. Their operation is limited since the torque produced depends on the cross product of Earth s magnetic field and the magnetic dipole generated by the coils. Thus the magnetorquers ability to control the spacecraft to a certain attitude is dependent on the position of the spacecraft relative to the Earth. It means that the precision, in which the attitude can be controlled, can vary substantially. Apart from varying accuracy of the magnetorquer attitude control, the coils are the ideal attitude actuator system for a CubeSat, as it has no mechanical parts and can be constructed arbitrarily small.

27 Group 830a 4.2 Sensor analysis Summary A magnetorquer is a mechanically simple actuator which has been proven to be very reliable for low Earth orbiting spacecraft like e.g. Ørsted and AAU CubeSat. Due to its capabilities to excert external torques on the spacecraft body and its availability the magnetorqer is chosen as an attitude control actuator for AAUSAT-II. The accuracy of magnetorquers is, however, limited so in order to achieve fine pointing control another actuator system is needed. For this purpose momentum wheels is seen to be the best choice as they do not involve propellant and it is possible to control the amount of angular momentum stored in the momentum wheels to a very high degree of accuracy. 4.2 Sensor analysis This document describes the possible sensors that can be used on the satellite. The section is based on information from [30, p.376] Sun sensor A sun sensor is a reference sensor which measures a direction in a known reference coordinate system. Sun sensors are visible light detectors which measure one or two angles between their mounting base and the incident sunlight. Sun sensors are popular, accurate and reliable but requires a clear field of view and are very expensive for the small scale of a CubeSat. Instead of using sun sensors, solar panels can be used. The angle of the Sun is calculated from measurements of the power generated by each of the panels. Solar panels are less precise than sun sensors as they are not designed to be used as sensors. However, as solar panels are the only mean for generating power on-board the satellite Star sensors Star sensors can be either scanners or trackers. Scanners work on spinning satellites where the attitude is determined by the light of stars passing through multiple slits in the field of view of the scanner. Star trackers recognize star patterns in the field of view of the sensor. Camera technologies as Charged-Coupled Devices (CCD), Active Pixel Sensors (APS) and CMOS could be considered for this task. The location of two or more stars is enough to determine the attitude of the satellite. This means that a star tracker alone can determine a three axis attitude when pointing towards the sky. For recognizing star patterns, an on-board star database is necessary. However, the problem with star trackers is they require the satellite to be stabilized to some extent before the tracker works. This stabilization may require extra sensors which increases the overall cost of the mission.

28 16 Chapter 4 System configuration Attitude Control System for AAUSAT-II Gyros A gyro is a sensor which measures the angular rate of the satellite or the angle of rotation from an initial reference, without any knowledge of external references. A gyro measures the angular velocity by means of the Coriolis effect which is defined by Newton s first law of motion: A body in motion continues to move at a constant speed along a straight line unless acted upon by an unbalanced force. The advantage using gyros is that they can provide high frequency angular rate information (up to hundreds of Hertz), which the other types of sensor may not be able to do. Also, gyros measure angular velocity directly, which eg. sun sensors and star sensors can not. When grouped together gyros can provide full 3-axis information Magnetometer Magnetometers are simple lightweight sensors that measure the direction and size of the magnetic field (eg. the magnetic field of the Earth). The precision of a magnetometer is not as high as that of a star sensor. Due to this, it is desirable to combine it with another type of sensor. Since the magnetic field of the Earth is limited to the vicinity of the Earth, the magnetometers can only be used in LEO. When the satellite uses magnetorquers to control the attitude, the magnetometer and the magnetorquer should be synchronized, so that the magnetorquer is not turned on when the magnetometer is measuring. Magnetometers can be used to get three axis attitude information GPS GPS is the most accurate positioning sensor. It communicates with satellites in GEO to calculate the current position and velocity of the sensor. The attitude of the satellite can be calculated using the difference between the measurements from two or more sensors. However the price of a GPS receiver, small enough to fit on a Cubesat, is very high Horizon sensor Horizon sensors detect the threshold between infrared light emitted from the atmosphere of the Earth and space by utilizing an infrared diode and a lens. There are two kinds of horizon sensors; scanners and horizon crossing indicators [21]. A horizon crossing sensor is fixed in the satellite structure and can provide valuable attitude information only when the sensors line of sight crosses the mentioned threshold. Due to the size, weight and complexity only static horizon crossing indicators are considered suitable for a Cubesat.

29 Group 830a 4.3 Configuration of momentum wheels Discussion of sensors Sensor comparison Sensor type Type Precision Horizon sensor Reference 0.1 deg Star sensor Reference 2 arcsec Sun sensor Reference 0.01 deg Gyro Inertial deg/hour Magnetometer Reference deg GPS Reference 1 cm 50 m Table 4.1: Comparison between different sensor types, [4] There are several aspects to take into account when choosing the sensors for the AAUSAT- II. Mass, volume and power usage are highly important factors, and also the precision of the sensors. The requirements stated in chapter 3 say that the satellite should be able to point within an accuracy of 5, so the combination of sensors chosen for the AAUSAT-II must, at least, be able to fulfill this demand. In table 4.1 different kinds of sensors and some of their characteristics are listed Choice of sensors The AAUSAT-II will be launched into LEO. Therefore the precise available models of the Earth magnetic field can be used. Magnetometers are light and do not require much power compared to eg. star sensors. Magnetometers are also to be used together with magnetorquers for detumbling the satellite as the control strategy requires measurements of the magnetic field. One of the goals of the AAUSAT-II mission is to test momentum wheel control, and see with what precision the satellite can be controlled. To be able to measure small changes of attitude, gyros or a star sensor have to be used. Since the momentum wheels will be tested on a zero-g parabolic flight star sensors are not considered, and gyros will be used instead. 4.3 Configuration of momentum wheels Designing an attitude control system using momentum wheels it is important to consider the placement and orientation of the wheels carefully before venturing into controller design. The physical constraints of the AAUSAT-II set limitations to the mass and volume that can be consumed by motors and flywheels. The laws of physics also play a role as at least three wheels are needed to do three-axis attitude control. Normally, a reaction wheel attitude control system would employ four wheels to be able to operate the motors at a bias speed while ensuring full three-axis control which requires the ability to generate angular momentum in any direction in the spacecraft frame. However, only three momentum wheels will fit inside AAUSAT-II which gives limitations to the three-axis control if the wheels are operated around a bias. These

30 18 Chapter 4 System configuration Attitude Control System for AAUSAT-II limitations are considered in the controller strategy discussion in section on page Orientation To achieve three-axis control the rotation axes of the momentum wheels must span all three dimensions of space. This is possible in many configurations, but the lack of knowledge of the mass distribution and hence the moments of inertia of the spacecraft, leaves only one obvious solution, orthogonal alignment. Figure 4.1 shows the configuration, where the wheels are aligned parallel to the axes of the spacecraft coordinate system. Figure 4.1: Momentum wheel orientation. The wheels are aligned along the spacecraft axes Operation When using momentum wheels to exert relatively small torques at high precision, as it is needed to control a spacecraft, the motors driving the wheels are operated at a bias. This means that the motors spin at half the maximum rate when the spacecraft is stabilized and torque is generated by either spinning the motors up or down between zero the maximum. The reasons for not changing the rotation direction of the motors are first of all, to avoid the non-linear Coulomb friction that acts when crossing zero (stopping and starting the motors). Secondly, according to Maxon Motor [23] extreme start/stop, left/right operation leads to a reduction in service life. The bias of the motors is referred to as S h bias = h mw 0.5 (4.2) Hardware configuration The configuration of the hardware of ADCS is depicted in figure 4.2.

31 Group 830a 4.4 Hardware configuration 19 Figure 4.2: Overall hardware architecture of the attitude determination and control system.

32 20 Chapter 4 System configuration Attitude Control System for AAUSAT-II In celebration of modularity and reusability the ADS and ACS parts of the complete ADCS are separated into two hardware units that are individually interchangeable. These hardware units work as interfaces to sensors and actuators respectively and are connected to the on board computer via CAN. The interface units communicates through INSANE 1 [19], thus it is very generic and easy to reuse in an eventual AAUSAT-III. 4.5 Software configuration The software of ADCS is distributed on multiple processing units: On board computer ADS hardware interface ACS hardware interface All control and determination algorithms will run as threads on the on board computer along with housekeeping software and supervisory control. These software units communicate with the programs running on the hardware interface units via the INSANE protocol. The software on the hardware interface units acts as device drivers communicating with sensors and actuators and relaying information between the physical systems and the software on the on board computer. 4.6 Perspectives The configuration of the actuation system may be subject to improvements if more detailed descriptions of the mechanical properties of the spacecraft are made available. This especially applies to the orientation of the momentum wheels as the choice of configuration allows the wheels to generate angular momentum in a certain subspace of the spacecraft coordinate system. By changing the orientations to span some non-orthogonal subspace it is possible to either, have the ability to generate more angular momentum, but in a narrower range of directions, or have a directional range at the cost of smaller angular momentum. The first option might be beneficial in the case that the difference in moments of inertia of the three axes of the spacecraft is big. By orienting the wheels, such that they can generate a larger angular momentum in a subspace containing the major axis of inertia, it is possible to achieve better control by being able to exert a greater torque around the major axis. This may minimize the difference in the angular acceleration which the controller can generate on the three axes. 1 INSANE is short for INternal Satellite Area NEtwork

33 CHAPTER Actuators In this chapter the design of the magnetorquers and momentum wheels for the AAUSAT-II is presented. The chapter also covers fault detection and isolation of faults in the actuators, which is a key topic in satellite design, due to the high degree of autonomy. 5.1 Magnetorquers design The magnetorquers are designed under the assumption that all three coils are the same. That concerns their physical characteristics such as structure, material and so on. The coils are to be bounded on three perpendicular plates, inside the satellite, so that the areas they encircle are as big as possible. The dimension of the coils is limited by the mechanical structure of the satellite. All the parameters are introduced in the figure 5.1. Figure 5.1: Coil. Mechanical drawing Mathematical description of a magnetorquer In the following the equations for one coil are described.

34 22 Chapter 5 Actuators Attitude Control System for AAUSAT-II Mass of one coil Defining: M c : mass of the coil V c : volume of the coil n: number of turns ρ: density of the material of the coil a w : wire cross sectional area C: circumference The mass of one coil can be described as: M c = V c ρ = nca w ρ (5.1) Power consumption Defining: P : power dissipation U c : voltage supplied to the coil I: current in the coil R: electric resistance of the coil The power dissipation can be described as: Where the resistance of the coil is given by: P = U c I = I 2 R (5.2) R = ncσ(t) a w (5.3) where: σ(t) = σ 0 (1 + α 0 T) α 0 : wire material resistivity with the temperature coefficient α 0 σ 0 (T): resistivity at the temperature 303 K σ(t): resistivity at the temperature T K

35 Group 830a 5.1 Magnetorquers design Maximum torque to detumble the satellite The maximum torque to detumble the satellite is found as: where: N det = I a o t o = 24.7 nnm N det : is the maximum torque required to detumble a o = 0.02 rad /orbit: maximum acceleration per orbit s I = kgm 2 : the moment of inertia (in the principle frame) t o = 5739 s: orbital period at the altitude of 550 km Total magnetic moment required To calculate the satellite magnetic torquing ability (magnetic moment) both torque to detumble the satellite and disturbance torque have to be taken into consideration and evaluated to have a maximum value. The total maximum torque due to the disturbances (section 6.5.5) is 36.2 nnm and the maximum torque to detumble the satellite is 24.7 nnm. Additionally a safety factor of 5 is used. Considering the moment of inertia and the torque to be scalars, the magnetic moment, that the magnetorquers are to be able to produce, can be calculated using the equation (6.37): where: m = 5 N s B = 5N det + N dist B = 0.01 Am 2 m: total magnetic moment required in Am 2. N dist : maximum torque produced by the disturbances (36.2 nnm)(6.5.5) N det : maximum torque needed to detumble(24.7 nnm) N s : minimum actuation torque that the coils have to be able to give in Nm. (N s = N dist +N det ) B: the smallest value for the magnetic field of the Earth, which is at the equator (B = nt) Design Each coil has to be able to generate the magnetic moment that can cancel the torque from the distubances and produce the one to turn the satellite. A preanalysis concerning the effect of choosing either aluminum or copper was done. It showed that aluminum is the most suitable material, minimizing the product of mass and power consumption. The reason for this is that, even though aluminum has a bad conductivity, its density is approx. three times smaller than the one for copper. These values are shown in table 5.1

36 24 Chapter 5 Actuators Attitude Control System for AAUSAT-II ρ [ kg ] m 3 σ [ Ωm] α 0 [ K 1 ] Al Cu Table 5.1: Parameters for aluminum and copper The necessary data needed is listed in the table 5.1 where: ρ: wire material density σ: wire material resistivity α 0 : temperature coefficient of resistivity Although a design with aluminum would be optimal it is not considered in the further discussion since either, manufacturers do not work with such material or the minimum diameter of wire provided is too large. It is designed under the assumption that the maximum voltage drop across the coil is 4.5 V and the number of windings is 500. All the calculations are made for the worst case, 70 C, considered the maximum temperature inside the satellite. Parameters of the coil, such as mass, maximum power usage and current flowing through the coil, can be calculated from the equations (5.1), (5.2) and (5.3). Maximum magnetic moment generated by each coil is: m = InA (5.4) Some of the possible sets of parameters using copper are represented in the table below. where, d: diameter of wire n: number of turns M: total mass of one coil I: current in coil R: electric resistance of one coil at 70 C V : voltage applied to produce the maximum magnetic dipole P : power consumption m: maximum magnetic dipole The wire of diameter 0.1 mm is chosen for the coil. The space required fits in the structure, power consumption and mass are within the budget and the magnetic dipole is almost twice of the value required. These reasons make this wire suitable for the magnetorquer requirements.

37 Group 830a 5.1 Magnetorquers design 25 d[ mm] n M[ g] I [ ma] R[ Ω] V[ V] P[ mw] m [ Am 2 ] Table 5.2: Design of coils Temperature[ C] R [Ω] I [ma] P [mw] m [A m 2 ] T max = T normal = T min = Table 5.3: Electrical properties of the coil

38 26 Chapter 5 Actuators Attitude Control System for AAUSAT-II Electrical properties of the coil The resistance, current and maximum magnetic dipole for minimum, normal and maximum temperature expected in the satellite are listed in table 5.3 The data were obtained using a voltage of 4.5 V to generate the maximum magnetic moment. In figure 5.2 the values of the maximum moment, that one coil is able to generate, are shown for different values of temperature m[am 2 ] T[ºK] Figure 5.2: Values for the maximum magnetic moment for the range of temperatures from 25 C to +70 C The worst case presented is at the highest temperature. This case was used previously for the design of the magnetorquers so that they are able to provide the required magnetic moment independently on the temperature Coil self inductance In order to make calculations easier, an equivalent square coil with the same area is considered. The side length of this equivalent coil is l = A = m 2 = 71 mm where A is the area of the rectangular coil designed previously. The magnetic field generated by a square coil is

39 Group 830a 5.1 Magnetorquers design 27 B mt = µ 0 2 2nI πl and the magnetic flux for each turn is Φ = B mt ds S where the integral is performed on the cross sectional area of the coil, A. Assuming a constant magnetic flux density B mt, gives Φ = B mt A then, the self inductance of the coil can be found as L = nφ I = 2 2µ 0 n 2 A πl = 20.1 mh where, B mt : magnetic field generated by the coil in Wb µ 0 : magnetic permeability for vacuum (µ 0 = 4π 10 6 Wb/(A m)) n: number of turns (500 turns) I: current in coil l: side of the equivalent square coil (71 mm) Φ: magnetic flux for each turn A: area enclosed by the coil ( m 2 ) Time constant τ The time constant of the coil is determined by its own resistance and self inductance. The values of τ for minimum, normal and maximum temperatures in the satellite are shown in table 5.4 Temperature [ C] Time constant τ [µs] T max = T normal = T min = Table 5.4: Coil time constant at different temperatures

40 28 Chapter 5 Actuators Attitude Control System for AAUSAT-II Magnetorquer Test Each of the coils must be tested separately and in an environment free of magnetic disturbances. The main test regards the functionality of the coils. Either a sensor or a multimeter must be used to measure the input current to the coil and the magnetometer will be used to verify the output (direction and intensity of the magnetic field). It is important to notice that a measurement of the Earth magnetic field must be subtracted from the one from the magnetometer in order to obtain the actual magnetic field generated by the coil. A possible alternative test could be to place the coil on a structure able to measure the torque acting on it. Then, applying a known current and magnetic field vector, the relation between current and torque can be found and verified. Further tests can be done, such as Measurement of the electric resistance Weighting the coil to confirm that the mass is below the budget Calculating the power consumption when applying maximum voltage (4.5 V) 5.2 Design of Momentum Wheels As decided in the section 4.1.5, that the ACS must feature momentum wheel control of the satellite. The requirements related to the momentum wheel actuation and due to the CubeSat size satellite need to be considered when designing momentum wheels for AAUSAT. A list of design criterias follows. Low power usage, due to the limitted power in the satellite. Tolerance to space environment, specially the vacuum problems. Low mass and small volume available. Safety issues. To design a suitable actuator it is needed to look into motor types and flywheel design Motor type selection To find a suitable motor for the space environment, was a Danish Maxon Motor dealer contacted. They have experience with actuators in space from the two mars rovers. Maxon motor agreed on sponsoring the motors, needed for the momentum wheels actuation, thus the only motors looked into is from the Maxon families.

41 Group 830a 5.2 Design of Momentum Wheels 29 Firstly the decision on which technology we need, fell on the brush less motor EC family, this was done due to the longer lifetime of the motors. But this would mean a higher power usage and a more complex driver circuit and sensory system. Instead the metal brush motor family RE was used. Because the driver circuit is less complex and the power usage is lower, but the lifetime is shorter. Since the satellite is expected to live 1 year or less, and the RE motors expected life time is more than one year. This solution should be suitable for this mission. The metal brushes do not generate dust like normal graphite brushes. Selection criterias The final actuators must be able to suppress the disturbances on the satellite, thus they have to have a maximum torque of at least Nm. See section for the calculation of the maximum disturbance. Another requirement is, that the momentum wheel must be able to store the angular momentum generated by the disturbances for at least half an orbit, without momentum unloading. This gives (5.5) since a orbit takes 5800 s at an altitude of 550 km. h mw,max = kg m2 Nm 5800/2 s 1 10 s (5.5) These requirements are used to pick a suitable motor, which is the RE13 or RE10. Since the RE13 uses more power than the RE10, the RE10 is chosen. Since none of the motors could store the h mw,max, it is necessary to put on extra inertia in the form of a flywheel. Additionally, the motor with the highest permissible angular velocity is choosen, since it gives the highest h mw,max with the lowest mass. Parameter Value Parameter Value Order number Stall torque Nm Nominal Voltage 3.0 V Max. permissible speed rad s Torque constant Nm Speed constant rad A s Rotor inertia kgm 2 Mechanical time constant s Terminal resistance 8 Ω Terminal inductance H Temperature range Motor weight kg Table 5.5: Maxon RE10 motor key informations from the data sheet. In the table 5.5 there is a short summary of the key informations about the Maxon RE10 motor. It is taken from the datasheet [22].

42 30 Chapter 5 Actuators Attitude Control System for AAUSAT-II Flywheel design The flywheel is designed such that the motor can store the angular momentum (5.5), thus a flywheel must have minimum moment of inertia calculated by (5.6). I > 2 h mw,max = = kgm 2 (5.6) ω max 1500 This matches the moment of inertia of a cylinder made of steel 1, with a Radius = 13 mm and a height = 6 mm. See figure 5.3 for the final mechanical design. The flywheel is screwed on to the shaft of the motor by using two screws 2, to retain symmetry and secure the shaft connection. See figure 5.3 for the mechanical drawings of the wheel. Figure 5.3: Mechanical drawings of the flywheel. The moment of inertia of the flywheel is calculated as kgm 2. The final moment of inertia and other motor parameters must be identified by tests, when the motor and flywheel are assembled. The following tests need to be made, to make sure that the actuator models are right when the final assembly is made: Determination of moment of inertia. Determination of mechanical time constant 1 The density of steel is ρ = kgm 3 2 The flywheel must also be glued on to the shaft, with a very small amount of epoxy adhesive, in the flight version of the satellite.

43 Group 830a 5.3 Fault Detection and Isolation system 31 Determination of friction coefficients Determination of current to torque constant. 5.3 Fault Detection and Isolation system Fault detection is carried out for different reasons. Firstly to extend lifetime of the actuator and satellite, but also to use it in fault tolerant control, to make the ADCS fault tolerant 3. A fault analysis is used to identify the different effects of a fault, thus giving the possibility to detect a fault when it occurs. The different faults are modeled, after the study of the effects, to determine how the faults is detected and isolated, thus making fault tolerant control a possibility Magnetorquers FDI Fault analysis Faults that occur in the operation of the magnetorquers can be due to the component faults or the change of the parameters, table 5.6. They can result in the reduced performance of the magnetorquer or lead to the destruction of some parts of the system. Thus it is important that fault handling is undertaken. In the following the aims are discussed and the ideas towards fault detection are presented. FAULTS Shortcut in winding (insulation breakdown) Winding breaks/disconnected Electrical fault or malfunction EFFECTS Local effects Higher level effects End effects Resistance drop, induction and time constant decrease. Current is zero for ever or a long time. Periodic shortcut, fluctuations in current. Maximum magnetic dipole reduced, more power usage, possible damage of the electronics. No magnetic dipole is generated, no power usage Possible damage of electronics Less control torque using the same power (loss of efficiency) No control torque Loss of efficiency and control ability Table 5.6: Magnetorquers fault analysis The electric resistance of the coil changes proportionally to its temperature as it was shown in section This is out of this projects scope.

44 32 Chapter 5 Actuators Attitude Control System for AAUSAT-II The effects due to the changes in temperature are shown in table 5.7. Temperature rises. EFFECTS Local effects Higher level effects End effects Resistance increases, time constant decreases. Temperature drops. Resistance decreases, time constant increases. Need more power to produce the same magnetic dipole, since V max = 4.5 V the maximum magnetic dipole producible is reduced. Need less power to produce the same dipole, the maximum magnetic dipole producible increases. Table 5.7: Temperature effects on the magnetorquers Less control torque, more power consumption, therefore loss of efficiency. Less power consumption. Model of faults and detection Once the list of faults have been established, they have to be detected and isolated. Some actions are required to reduce or eliminate the faults effects, if possible. The proposed way for detecting the fault, that is due to the shortcut in the winding and results in the sudden drop of the value of the resistance of the winding, is through the analysis of the rate of change of the resistance. It is expected that this kind of fault causes the peak in the chart of the derivative of the resistance, figure 5.4. By defining the threshold, the lower bound can be checked and the residual generated. The fault due to the break of the winding and the one due to electrical faults are detected and modeled in a similar way because an interruption in the current occurs in both of them. The difference between them is in the time that this interruption last. For the first case the interruption is permanent, the wire breaks and the current does not flow. For the second case it occurs from time to time, a fluctuation appears due to electrical malfunctions. The measurements of current and voltage in the coil are used. If the absence of current when a voltage is detected or a high value of the electric resistance (R = V ) appears, it means that I one of these faults has occurred. If these values are measured again and the fault still exists, it means that the one due to the break of winding is detected. If the fault disappears in posterior measurements then electrical malfunctions are detected. The second case is shown in figure 5.6, where applying a constant voltage, the current takes values of zero, resulting in peaks on the values of the resistance. Since this occurs eventually, the electrical fault is indicated in the residual.

45 Group 830a 5.3 Fault Detection and Isolation system 33 R dr/dt t threshold t Figure 5.4: Shortcut in windings. Fault detection. V I Division block R Check R static dr/dt lower r 1 bound Figure 5.5: Shortcut in windings. Model Fault detection in the momentum wheels Fault analysis This section discusses the different faults and effects, to provide the necessary information to generate the residuals, in order to detect and isolate the faults. The faults are divided into 3 groups. External faults, which is a direct result of another subsystems. These should be detected by the system that introduces them. But some are also detectable from the controller. Motor faults, all faults related to the motor and the mechanics in connection with the momentum wheels. All are detectable from the controller. Controller faults, which are related to the controller and controller hardware. These fault should be detectable from the controller. The external faults shown in table 5.8, where the faults due to power loss or brown out are described. f e1 in table 5.8, happens when the EPS turns off the power to the motors. This could happen due to a too high power usage. It is detectable since the motors stops and the angular velocities

46 34 Chapter 5 Actuators Attitude Control System for AAUSAT-II R I t V t Figure 5.6: Elecrical malfunction. Fault detection. t FAULTS f e1, Power loss f e2, Brown out EFFECTS Local effects Higher level effects End effects Momentum wheel Slower attitude control, Satellite spins up. spins down, until it due to less actuation. stops Momentum wheel Lower control momentum. Lower performance of looses power, thus the ACS. generating less momentum, slower reaction and lower max angular velocity. Table 5.8: Faults due to external subsystems. The table shows the effects of a Power loss or brown out situations.

47 Group 830a 5.3 Fault Detection and Isolation system 35 drops to zero. And the EPS system broadcasts that it has turned off the momentum wheels 4. f e2 is a result of the power level dropping. This results in a lower performance and less control momentum generated by the momentum wheel. The f e2 will not be detectable, since it probably would result in a brown out in the controller. Forcing the controller to reset, due to brown out detection electronics 5. It should be noted that a brown out does not give any permanent damage to the system, thus no action is needed on this fault, except to reset the controller, which should happen automatically due to the reset sequence of the controller hardware. In table 5.9 is a list of faults that relates to the motor both mechanical and electrical. FAULTS f m1, Loose shaft connection f m2, Shaft breakage f m3, Winding overheating Higher power usage, possible damage to motor due to vibrations. Possible damage to spacecraft due to the free flywheel. higher power usage and heating of the spacecraft. f m4, Insulation breakdown f m5, Shorted winding EFFECTS Local effects Higher level effects End effects Heat generation due to friction, higher chance for shaft breakage. Faster reaction time and less control momentum Resistance in winding increases, and the strength of the magnetic field in the stator decreases and higher possibility for insulation breakdown. Resistance decreases, power usage and heat generation increases resulting in faster insulation breakdown. Possible damage to driver electronics and motor Less control momentum. Loss of inertia, thus less control momentum Slower reaction and less control momentum Less control momentum and slower reaction time Loss of efficiency and control momentum. Possible damage to electronics due to smaller shorts in the winding. Higher power usage, loss of actuator. Table 5.9: Faults related to the motor, both mechanical and electrical. Some of the faults in table 5.9, can demage for the system, and therefore it is necessary to detect them and react on them before the damage becomes a problem. f m1 happens if the flywheel gets loose. This is difficult to detect because the effect could vary a lot, but it should be detectable as an increased friction 6. 4 The ACS system has 3 separate power channels, one for the micro controller and 1 for each actuator type. 5 A brown out in the controller should be logged thus detected. 6 This assumption should be verified by a test, when the actual fault detection system is implemented

48 36 Chapter 5 Actuators Attitude Control System for AAUSAT-II f m2 is the result of f m1, where the flywheel fly off. This faults is not detected since the result of a lost flywheel will possibly damage the satellite and therefore an automated detection is not needed. f m3 and f m4 are detectable by the controller and they are analyzed more. f m5 is detected in the EPS system, which detects a shortcut in the motors. And also detectable since the motor stops due to the power loss like the detection in f e1. In table 5.10 there is a list of fault on the momentum wheel in relation to the controller. FAULTS f c1,electrical malfunction EFFECTS Local effects Higher level effects End effects The momentum wheel spin at a fixed speed, and control of the momentum wheel is lost. Loss of 1 control axis in momentum wheel actuation system. Higher power usage, and slower attitude control. Table 5.10: Fault analysis showing the electrical malfunction fault in the controller or driver electronics and the effects of this fault. f c1 is detected by the controller, since the speed of motor is fixed or stopped and this possibly does not match the wanted speed. To summarize the fault analysis, there is three faults, f e1, f e2 and f m5, which do not need the model based fault detection. Due to power loss and brown out detection in the EPS and the direct speed measurements from the motors, which indicates the fault directly. The remaining faults, shown in table 5.9, are caused by two errors, abnormal power usage and angular velocity behavior. It is possible to detect these errors by investigating the system. By modelling the different faults and design a filter, which generates the residuals, it is possible to detect the different faults. Residual generation A model of the actuator is used to find an error vector e. It is filtered to generate different residuals. The method used is known as the geometric approach[5]. The fault detection system is shown on figure 5.7. The actuator output is matched to the output from the model to generate an error vector. The basic task is to find a residual vector r(k), such that r(k) is a function of all faults and r(k) 0,when f(k) 0 thus giving (5.7). r(k) = h(f m1 (k), f m2 (k),...,f c1 (k)) r(z) = H(z)f(z) (5.7) From the figure 5.7 and the model of the actuator, figure 5.8, it is possible to derive a relation between the error e and fault f acting on the actuator.

49 Group 830a 5.3 Fault Detection and Isolation system 37 Actuator y u + - e Filter/ residual generator r Actuator model ŷ Figure 5.7: Fault detection block diagram of the actuator system. Where u is the input, y is the actual measured output, ŷ is the expected output, e is the error and r is the generated residuals. 2 I_out 1 V_in Voltage saturation e e 1 den(s) Rotor impedance I Current saturation K N m mechanical motor constant 1 s Band Limited White Noise K Integrator inverse Inertia ω K ω 1 w_out 2 f1 Product Viscous friction 3 f2 Config.Km ω Electrical motor constant Figure 5.8: Model of momentum wheel actuator with disturbances and fault input.

50 38 Chapter 5 Actuators Attitude Control System for AAUSAT-II The error signal representing the difference between the outputs (5.8), e = y ŷ (5.8) is used together with the mathematical model of the actuator to determine the actual error signal as a function of the errors and disturbances. The model of the actuator is determined in section and the final model, including the fault inputs, is put on state space form (5.9) and (5.10). Where ω is the states in the 3 momentum wheels,f is the faults, which occurs, and k m,r,j and b are the momentum wheel parameters. y = ω = Rb + K2 m JR ω + k m RJ u + k m J f 1 1 J f 2 (5.9) [ ] [ ] [ ] ω I 0 = i km I ω + 1 I u (5.10) R R Tt is possible, by using (5.9), (5.10) and (5.8), to determine an expression for the error signal e(s). [ ] [ eω (s) 1 kmj 1 ][ ] J f1 e(s) = y(s) ŷ(s) = = e i (s) s + Rb+K2 m k2 m k m (5.11) JR JR JR f 2 By creating a filter which removes the dynamic part of (5.11) it is possible to measure the faults directly. 1 1 = G(s) (5.12) s + Rb+K2 m JR ( ) 1 1 G(s) = (5.13) s + Rb+K2 m JR With a samplings frequency of 10H this filter is expressed in the z-domain as G(z) = ê(z) e(z) = z 1 (5.14) A simulation of two faults, where the friction increases and the resistance in the windings increased to 10 Ω due to the change of the temperature, shows how the error signal ê(z) reacts to the faults. See figure 5.9. The two spikes on figure 5.9 are because of two input changes. The first one is a spin up of the motor to a bias point of 600 rad and the second one is a speed up to max velocity of 1200 rad. s s The two figures 5.9(a) and 5.9(b) show the two different error signals. It is imported to note that the current error e i reacts more to the friction error, which is applied at time=50 s. The velocity error e ω seems to react on both faults, thus both error are detectable. To isolate the faults, it is needed to use both error signals, because of the velocity error detecting both faults. e i detects only the friction. Thus it is possible to isolate the two fault types by comparing the errors to each other.

51 Group 830a 5.3 Fault Detection and Isolation system Time offset: Time offset: 0 (a) The e ω velocity error measured in ] and the time measured in [ s] [ rad s (b) The e i current error measured in [ ma] and the time measured in [ s] Figure 5.9: Error simulation of friction increase at 50 sec and a resistance change of +2Ω from 0 sec. Fault e ω e i u ω other f e1 0 EPS f m1 < f m3 < 10 < 0.01 f m4 > 10 < 0.01 f m5 0 0 EPS f c1 0 u 392.7u ± rad vs Table 5.11: The fault election system is based on a threshold criterias for each fault. The thresholds are put on the error, input and output signals to and from the system. EPS means that EPS will notify about this fault. The empty fields could be anything.

52 40 Chapter 5 Actuators Attitude Control System for AAUSAT-II To isolate the different fault a selection scheme is made. It is shown on table 5.11, where the fault name and criteria for the fault are shown. The measurements are filtered with a moving average filter averaging over 10 seconds to minimize the disturbances and to prevent a wrong error detection in a normal run. 5.4 Summary It is possible to design suitable actuators for AAUSAT-II within the given requirements. And a fault detection and isolation system has been designed to support future fault tolerant control of the spacecraft. Since the actuators, has not been delivered yet, no real test is made. Everything has been simulated and works. The next task for the actuator design and FDI system design, is to verify the designs against a real implementation of the actuators and controllers.

53 . CHAPTER Modelling This chapter describes the spacecraft model. It includes orbital, magnetic,disturbances, dynamic and kinematic,sensor and actuator modelling. An overview of the entire model is given, followed by detailed descriptions of each submodel. The Simulink implementations and verification of the individual models can be found in appendix B Before the actual ADCS system is built, it is necessary to perform simulations of the system in order to see if and how it works. This does not only relate to the sensors and the actuators of choice, but also for the algorithms used. If this is to be rated, it is necessary with a precise model of the environment including the natural disturbances. In [29] and namely in [4, p. 38] the disturbance torques can be seen as a function of satellite altitude. As the AAUSAT-II will be launched into a LEO, typically at an altitude of 550 km, it can be seen from [4, fig. 3.3] that the following disturbances should be taken into account when building an environment model: Radiation pressure Gravity torque Magnetic torque Aerodynamic torque 6.1 Coordinate system definitions To be able to describe the spacecraft orbit, environment and attitude a number of reference frames (coordinate systems) need to be defined. Below, a list of the reference frames used in this project along with a short description are listed. Inertial reference frame Used for simulation purposes as basis for physical laws. Earth fixed reference frame Used for simulation purposes as reference for environmental effects. Orbit reference frame Spacecraft centered frame, used to calculate offsets from the Earth to the spacecraft and vice versa

54 42 Chapter 6 Modelling Attitude Control System for AAUSAT-II Spacecraft reference frame Common reference frame for all on-board instrumentation and actuation systems. Controller reference frame Principal axes of spacecraft, used as operation reference frame of control system Inertial reference system To be able to use Newton s laws for modelling of a mechanical system an inertial coordinate system as the reference is needed. The choice of inertial reference system depends on the attitude reference system. In reality, no reference system can be a perfect inertial system. However some reference systems are better than others. Having a satellite orbiting the Earth, an Earth centered inertial (ECI) system could be used, [13]. This can be fixed with respect to the direction of vernal equinox or the direction of periapsis (perifocal reference frame). Both of these, however, are influenced by the Earth orbiting the Sun. That introduces an acceleration of the reference system. Hence, it is not a perfect inertial system. Placing the reference system in the center of the Sun can eliminate this acceleration. Of course, the Sun orbits the galaxy which also introduces an acceleration and so on and so forth. As the satellite is orbiting the Earth and the acceleration caused by Earth s rotation about the Sun is negligible, the Earth centered inertial (ECI) frame is chosen. The chosen ECI frame, which is shown in figure 6.1, is right orientated orthogonal and has the z-axis in the direction of the Earth s North Pole, the x-axis in the direction of vernal equinox (see section on page 54 or [29]), and the y-axis is the cross product between the z- and x-axis. Figure 6.1: ECI frame

55 Group 830a 6.1 Coordinate system definitions Earth fixed reference frame As some environmental modelling depends on the spacecraft position relative to a certain point on the surface of the Earth it is necessary to define an Earth Centered Earth Fixed (ECEF) reference frame, depicted on figure 6.2. ECEF is a right orthogonal coordinate system with the origin in the center of the Earth s. The z-axis is in the direction of the North Pole, the x-axis is in the direction of intersection between the equatorial plane (0 latitude) and the Greenwich meridian (0 longitude) and the y-axis is the cross product between the z- and x-axis. Figure 6.2: ECEF frame Orbit reference frame The orbit reference frame defines the coordinate system in which the spacecraft reference frames (ie. S and P frames) are given with respect to orbital elements. In this case a right orthogonal orbital frame, that follows the orbit of the spacecraft around the Earth, has been chosen. As shown in figure 6.3, the x-axis points in the nadir direction (towards the center of the Earth), the z-axis is in the direction of the spacecraft angular momentum vector (R V ) and the y-axis is the cross product between the x- and z axes (for a circular orbit this is in the direction of the spacecraft velocity vector) Spacecraft reference system The body-fixed spacecraft reference system is chosen such that certain features of the spacecraft are orientated along the axis of the spacecraft coordinate system. Since AAUSAT-II has an

56 44 Chapter 6 Modelling Attitude Control System for AAUSAT-II Figure 6.3: Orbit frame instrument pointing away from the satellite perpendicular to one of the cube sides, it would be natural to place the coordinate system in the geometrical center of the satellite. One axis should be in this direction and the other two should constitute a Cartesian coordinate system. Typically, these axes are perpendicular to two other sides of the cube Controller reference frame The attitude control system may benefit from using a reference frame which is slightly different from the spacecraft reference frame. It is still a body-fixed coordinate system but the axis are orientated according to the principal axis instead. The x-axis will be the major axis of inertia, y-axis is the minor axis of inertia and the third one is in the direction of the cross product vector of x- and y-axis. By using this coordinate system for dynamic calculations all the products of inertia are eliminated. 6.2 Model structure The model has been developed in Simulink with the goal of creating a modular model, where sub models can easily be reused. The top level model consists of five major blocks containing logically grouped sub blocks, as depicted in figure 6.4: Orbit and magnetic Simulates the spacecraft orbit and the magnetic field. Ephemerides Calculates positions of celestial objects based on time. Disturbances Calculates orbit and attitude disturbances for the spacecraft.

57 Group 830a 6.2 Model structure 45 Figure 6.4: Model overview

58 46 Chapter 6 Modelling Attitude Control System for AAUSAT-II Spacecraft dynamics and kinematics Simulates the spacecraft attitude. ADCS Simulates the Attitude Determination and Control System Ephemeris An ephemeris model calculates positions of celestial objects that are only dependent on time. In this case the ephemeris model uses the Julian date as input and outputs the position of the Moon and Sun in the ECI frame and the rotation from ECI to ECEF Disturbances The disturbance block calculates the torques and forces acting on the spacecraft. These disturbances are calculated from the aerodynamic drag, gravity gradient/perturbations and radiation pressure. To calculate these disturbances a number of parameters, shown below, are needed. Moon and Sun position Orbit position and velocity Spacecraft attitude Orientation of principal frame Spacecraft inertia and mass Center of mass position Orbit and magnetic field model The spacecraft orbital position and the magnetic field vector at the spacecraft position is calculated in this block. Concerning the orbit calculations, the following items are discussed: Simulation model The simulation model is based on Newton s acceleration model, which allows for simple addition of external disturbances and perturbations. Reference model Newton s model is verified using the Standard General Perturbations 4 (SGP4) model, which is the de-facto standard for orbital calculations, [6]. The model of the magnetic field of the Earth is based on the International Geomagnetic Reference Model (IGRF), [7]. The simulation implementation (which is based on the Rømer simulation implementation) of the IGRF is verified against data from the Øsrted mission.

59 Group 830a 6.3 Orbit and magnetic field modelling Spacecraft dynamics and kinematics The attitude and angular rate of the spacecraft is calculated in this block based on the initial conditions, the disturbances and actuator torques and the angular momentum. The inputs and parameters to the block are listed below. Initial attitude Initial angular rate Satellite inertia Disturbance torque Actuator torque Actuator angular momentum Orientation of principal frame ADCS The Attitude Determination and Control System block simulates the ADCS developed for the satellite. This includes sensor simulation, attitude determination system and filters/algorithms, controller and actuator simulation. To simulate the behavior of the spacecraft the ADCS block uses inputs from the orbital propagation, disturbances, etc. 6.3 Orbit and magnetic field modelling This section describes the orbit model used in the simulation and the reference model which used to verify the simulation implementation. The model of the magnetic field of the Earth can also be found in this section. The simulation implementation and verification of the orbit model can be found in appendix B.2 and for the magnetic field model in appendix B Orbit propagation models The primary orbit model used in the satellite simulation is a pure Newtonian propagation model based on the Newton s law of gravity

60 48 Chapter 6 Modelling Attitude Control System for AAUSAT-II R = GM R R 2 R R = GM R 3 R (6.1) This equation describes the gravitationally induced acceleration of a body with the mass m orbiting a celestial body of mass M. This acceleration is given by the product of the two masses and the gravity constant G divided by the distance between the center of mass of the two bodies in the opposite direction of the position vector R. R is the position of the orbiting body in reference to the celestial body. In order to compute the actual resulting acceleration, including other forces than the gravitation force, an externally introduced acceleration is added in equation (6.2). m R = a ext GMm R 2 R R R = F ext m GM R 3 R (6.2) This equation describes the orbit of a spacecraft orbiting a celestial body when including the external force F ext. This equation can be solved numerically by using two initial conditions: The initial position R 0 and the velocity V 0, which define the orbit when disregarding external forces. Apart from the Newtonian orbital model, the SGP4 model together with Two-Line Elements (TLE) obtained from [18] (which are obtained from the North American Aerospace Defense Command). A description of the SGP4 model and TLEs can be seen below. The implementation and verification of the Newtonian orbit propagator can be found in appendix B.2. SGP4 reference orbit model The SGP4 model was developed by NORAD and is used for calculating orbital descriptions of satellite movement, ie. position and velocity. SGP4 is based on the Keplerian orbit calculations but also takes a number of perturbations into account, eg. atmospheric drag and spherical harmonics. A detailed explanation of the propagation model and the algorithm can be found in [6]. On a daily basis, NORAD distributes actual data based on observations known as Two-Line Elements (TLE). The TLEs can be obtained for free (also on daily basis) on the CelesTrak homepage [18]. This makes the SGP4 model the most excact verification tool available.

61 Group 830a 6.3 Orbit and magnetic field modelling 49 Element Explanation 1 Line number A Satellite number B Classification (Unclassified) C Designator, last two digits of launch year D Designator, launch number of year E Designator, piece of the launch F Epoch year, last two digits of year [ year] G Julian date in current year [ JD] H First derivative of mean motion I Second derivative of mean motion J BSTAR drag term K Ephemeris type (0 for SGP and SGP4) L Element number M Checksum 2 Line number N Inclination [ deg] O Right Ascension of the Ascending Node [ deg] P Eccentricity (decimal point given implicit) [ eu] Q Argument of perigee [ deg] R Mean anomaly [ deg] S Mean motion [ orbits ] day T Revolution number at epoch Table 6.1: Two-Line Elements explanation. Units are given for the relevant elements. Two-Line Elements Data for the model are obtained as TLEs (sometimes called NORAD-lines), which explains some of the coefficients used in the SGP4 propagation model. A TLE consists of two lines of text: 1 AAAAAB CCDDDEEE FFGGG.GGGGGGGG +.HHHHHHHH +IIIII-I +JJJJJ-J K LLLLM 2 AAAAA NNN.NNNN OOO.OOOO PPPPPPP QQQ.QQQQ RRR.RRRR SS.SSSSSSSSTTTTTM A description of the various fields in the TLE are described in table 6.3.1, [18]. Some elements are exactly the same as those used in the Keplerian model, others are used for the higher order perturbation calculations as eg. BSTAR. Orbital elements The SGP4 model relies on the Keplerian orbital elements, which describe the orbital plane in which the satellite moves. They also describe the position of the satellite at a given point in time, and allow for propagation to calculate positions in the future.

62 50 Chapter 6 Modelling Attitude Control System for AAUSAT-II Figure 6.5: Orbital Elements In the following, the basic orbital elements are described. Epoch (t 0 ) The reference time for which the elements are specified, usually measured in the Julian Date format. Semi-major axis (a) Half of the major axis of the ellipse, describes the size of the orbit. In the TLEs the mean motion is used instead. Eccentricity (e) Defines the shape of the elliptical orbit, ie. the distance from center of ellipse to focus divided by (a) Inclination (i) The inclination describes the orientation of the orbit plane with regards to pitch or elevation, ie. the angle between orbit plane and Earth s equatorial plane. Right ascension of the ascending node (RAAN) (Ω) The second parameter which describes the orientation of the orbit plane. It can be regarded as the yaw of the orbit: The angle between vernal equinox and eastward to the point where the satellite ascends the equatorial plane (ie. a point on the line of nodes). Measured in center of the Earth. Argument of perigee (ω) The argument of perigee describes the position of the point where the satellite is closest to the surface of the Earth (the definition of perigee), ie. the angle from RAAN to perigee. Measured in center of the Earth, in the direction of the motion of the satellite.

63 Group 830a 6.3 Orbit and magnetic field modelling 51 Mean anomaly at epoch (M 0 ) The angle of the satellite position on the ellipse plane at epoch, measured from a at the given reference time. In this way, the mean anomaly describes the position of the satellite at time t 0. For a graphical representation of the mean anomaly, see figure 6.6. Mean motion (n) The average velocity of the satellite. In TLEs the mean motion is given in orbits per day. The mean motion can be used instead of the semi-major axis, as it is in the case of the TLE, since the following relation exists; n = µ where µ = GM a 3 earth is the gravitational constant for the Earth. The Julian date is the standard time format for spacecrafts and celestial objects. The Julian date counts the number of days since January 1st, 4713 B.C., at 12.00AM UTC. A Julian date is given as the day number and fraction of a day, eg which corresponds to June 16th, 2004 at 18:00 UTC. Since the orbit that the satellite is going to be inserted into is not known at the moment the TLE from the AAU CubeSat is used throughout the entire project where nothing else is specified. The TLE used is: U 03031G Apart from the basic orbital elements, a number of different anomalies exists, which all describes a point on the ellipse. Figure 6.6 shows the mean anomaly, M, the eccentric anomaly E and the true anomaly v. satellite M E v apogee perigee Figure 6.6: Mean-, eccentric- and true anomaly: M, E and v respectively. M is measured from perigee to the auxiliary circle, intersecting the satellite position.

64 52 Chapter 6 Modelling Attitude Control System for AAUSAT-II E is measured from the center of the orbit to the satellite position. v is measured from the perigee to the satellite position on the ellipse itself Magnetic field model This section explains the basics of the Earth s magnetic field, and how it is modelled. The implementation is made in Simulink and can be found in appendix B.6 on page 146. The main contribution to the magnetic field is from the Earth s crust and core. This field is approximately nt (nano Tesla) near the poles and about nt at the equator. The differential flow of ions and electrons inside the magnetosphere and in the ionosphere form current systems, which cause variations in the intensity of the Earth s magnetic field. These external currents in the ionized upper atmosphere and magnetosphere vary on a much shorter time scale than the internal main field and may create magnetic fields as large as 1 of the main 10 field. Since the ionosphere only stretches out to about 400 km, it will not have any effect on the satellite, since the altitude of the AAUSAT-II is expected to be around 550 km. The standard model for the internal main field is the International Geomagnetic Reference Model (IGRF), [7], which is the empirical representation of the Earth s main magnetic field. The field is predicted within about 100 nt. The IGRF model estimates the magnetic field strength and direction at a given date, latitude, longitude and altitude, which are provided by the orbit propagator. The field strength and direction are given in the control frame, centered in the center of the mass of the spacecraft. The IGRF model uses spherical harmonics expansion to describe the magnetic field. The IGRF coefficients are updated every 5 years. The implemented simulation model of the IGRF consists of 144 coefficients. The following section describes the equations used in the IGRF simulation model. The IGRF model The magnetic field vector B can be expressed as a sum of spherical harmonics B n,m, as given in the following Equation [25]. B = n=1 m=0 n B n,m (6.3) The contribution of spherical harmonics are of degree n and order m, and are given by equation (6.5) B n,m = Kn,man+2 R n+m+1 { gn,mcm+hn,msm R [(s λ A n,m+1 + (n + m + 1)A n,m )ˆr A n,m+1 ê 3 ] (6.4) ma n,m [(g n,m C m 1 + h n,m S m 1 )ê 1 + (h n,m C m 1 g n,m S m 1 )ê 2 ]} where the different variables are given by

65 Group 830a 6.4 Ephemeris model 53 a Mean radius of the Earth, r earth 6371 km. a is used to ease the readability. R Magnitude of the position vector from the geometric center of the Earth. This is provided from the orbit model providing the position vector of the satellite. ˆr Unit vector in direction of R found using the orbit model. ê Mutually perpendicular unit vectors ê 1, ê 2 and ê 3 parallel to the x, y and z axes of the ECEF frame. n Degree of contributing spherical harmonics. m Order of contributing spherical harmonics. K n,m Coefficients that relate Schmidt functions to associated Legendre functions. A n,m Derived Legendre polynomials of degree n and orders m. s λ Sine of geographic latitude λ (latitude is normally La) g n,m, h n,m Schmidt-normalized Gauss coefficients of degree n and order m. The Gauss coefficients used for the magnetic field model are obtained from DSRI for the epoch 2000, [8] The implementation of the IGRF model is available in appendix B.1. As can be seen in table B.1 and table B.2 on page 140, there is little gain when the order of the model is higher than seven. However a seventh order model also fulfills the precision demand stated in chapter 3. The precision of the IGRF model should also be adjusted with regards with the chosen magnetometer: The magnetometer to be used (see [14]) has a resolution of maximum 85 µg which can be utilized by using an IGRF order of 7, since 85 µg = 85 nt (the order was obtained from table B.1). 6.4 Ephemeris model In order to get precise gravitational disturbance models, the position of the Sun and the Moon must be calculated at a given time. Furthermore, the position of the Sun must be known on-board, so that the sun sensor measurements can be compared to the position vector of the Sun. Also, the rotation of the Earth must be known, so that the ECI-frame can be referenced correctly to the ECEF-frame. Using the ECEF-coordinates the magnetic field vector can be found.

66 54 Chapter 6 Modelling Attitude Control System for AAUSAT-II Earth rotation (ECI to ECEF) The rotation of the Earth is based on a sidereal day, which is defined as the time that the vernal equinox takes to rotate 360. The mean sidereal day is 23h 56m s. This yields an angular velocity of 5 rad ω = s Furthermore, the Earth s attitude must be known at a fixed point in time. The fixed point can be regarded as the exact coincidence between the ECI and ECEF frame, which according to The 1993 Astronomical Almanac [27], took place on December , 17h 18m s. This translates to a Julian date of f s = This yields the formula for calculating the ECI to ECEF rotation, ie. the vernal equinox to Greenwich line relation, which is known as ψ T r = 2π ω ( N r = mod Tr (t fs ) s ) ψ = ω ( ) t N r T r (6.5) where the function mod x (y) is modulus x of y. T r is the revolution time and N r is the number of revolutions since time s. Time t and s are measured in seconds, hence the conversion to seconds per day. Using equation (6.5) the ECI to ECEF quaternion, E q can be formed as I E I q = 0 0 sin( ψ 2 ) cos( ψ 2 ) (6.6) The above equations do not take neither nutation (the nodding of the Earth) nor precession (the spin-axis circular motion) into account. The precession is negligible as it varies very slowly (one precession revolution takes approx years), whereas the nutation has a period of 18.6 years. However, the nutation varies only between approx. ±15 arc second, so it is negligible as well. [29] Sun position The model presented in this section is based on [29]. The position of the Sun in the ECI-frame depends only on time and a number of constant parameters and can be described by a Keplerian orbit as the Earth s orbit around the Sun. This orbit can also be described the other way, ie. the Sun orbiting the Earth. This is not true but leads to the same solution (the Sun vector). The

67 Group 830a 6.4 Ephemeris model 55 orbital elements used in the Sun vector calculation are given for the opposite case explained above. The mean anomaly of the Sun at a fixed date, M se, is known for January at noon UTC (epoch), and is M se = Also, the mean motion of the Sun is known, using d for days, is n s = /d The mean anomaly at given time can now be calculated as M s = M se + n s JD E Where JD E is the Julian Date since the defined epoch. The ecliptic longitude is [29, p. 141] Lo s = M s sin(m s ) sin(2m s ) The ecliptic is the plane in which the Sun orbits the Earth. This plane is tilted by ǫ = regarding the ECI-frame. Using the ecliptic longitude and the tilt of the spin axis of the Earth, the sun vector is cos(lo s ) I R S = cos(ǫ) sin(lo s ) (6.7) sin(ǫ) sin(lo s ) The position of the Sun depends only on time. The sun vector has only the direction to the Sun. The distance to the Sun is calculated as [29] D s = (1 e 2 ) 1 + e cos(m s ) Where e is the eccentricity of the Earth s orbit around the Sun, which is e = Sunlight-eclipse determination Since one of the inertial reference frames used for attitude determination relies on a vector to the Sun, it is important to determine whether the satellite can see the Sun or not. This means finding out whether the satellite is in sunlight or eclipse. An equation for determining this is derived in this section.

68 56 Chapter 6 Modelling Attitude Control System for AAUSAT-II If the satellite is between the Sun and the Earth it will always be exposed to the Sun. If the satellite is farther away from the Sun than the Earth there is a possibility for being in the eclipse. This is checked by calculating the perpendicular distance from the center of the Earth to the line of sight from the spacecraft towards the Sun, and comparing this distance to the radius of the Earth. This is illustrated in figure 6.7 and expressed as equation (6.8). I R SC ( I R S I R SC ) ( I R I R S I R SC 2 S I R SC ) + I R SC > r earth (6.8) Figure 6.7: Sunlight/eclipse determination strategy. Where R SC and R S are the positions of the spacecraft and Sun respectively relative to the Earth and r earth is the Earth radius. If equation (6.8) is satisfied or if the satellite is nearest the Sun, ie. I R S I R SC < I R S (6.9) then the satellite is in sunlight. The points where the satellite passes from sunlight to eclipse and vice versa are not treated. It is expected to take very short time due to the high speed of satellites in LEO.

69 Group 830a 6.5 Disturbance modelling Moon position As the orbit of the Moon is highly non-linear, and the development of a precise orbit model is out of the scope of this project, the algorithms are omitted, [29, p. 141]. However, the simulation requires the position of the Moon as input since the gravity disturbances depend on it. The simulation implementation of the Moon orbit is used as a black box, using the Spacecraft Control Toolbox (SCT) from Princeton Satellite Systems. Only wrappers for the MoonV2 algorithm has been made, in order to allow continuous updates of the position. [27] 6.5 Disturbance modelling This section describes the modelling of the environmental disturbances, which are the external forces and torques acting on the satellite Solar radiation disturbance A spacecraft moving in the solar system experiences perturbations of it s trajectory due to the incidence of solar radiation upon its illuminated surfaces. This results in a force and torque about the center of mass of the satellite. Below 800 km altitude, acceleration from drag is greater than the one from solar radiation pressure; above 800 km, acceleration from solar radiation pressure is greater, [30, p. 146]. There are four major radiation sources, listed in order of significance: The direct radiation from the Sun, the albedo (reflected radiation from the illuminated Earth hemisphere), the Earth infra red radiation and finally the directed IR radiation emitted from the spacecraft [20]. Direct solar radiation is the dominant source and is generally the only one considered. The force produced by the solar wind is also normally negligible relative to the solar radiation pressure, [26]. Solar radiation analysis The three major factors that influence the magnitude of the solar radiation torque, according to [26], are: Intensity and spectral distribution of the incident radiation. Geometry of the surface and its optical properties. Orientation of the Sun vector relative to the spacecraft. The equation that calculates the solar radiation pressure F R is, [26] F R = P C a A (6.10)

70 58 Chapter 6 Modelling Attitude Control System for AAUSAT-II where P = F S c (6.11) F S is the mean solar energy flux at the spacecraft c is the velocity of light C a is the absorption coefficient (C a = 1 for absorption,c a = 2 for specular reflection and C a = 1.4 for diffuse reflection). [30, p. 146] [26, p. 65] A is the cross sectional area of the satellite perpendicular to the sun line P is the momentum flux from the Sun ( kg/ms 2 ), see [26, p. 65] and [30, p.145] Cross sectional area analysis. Maximum and minimum perturbation Assuming that all values in equations (6.10) and (6.11) are constants except A and the absorption coefficient for specular reflection (C a = 2), the minimum and maximum values of the perturbation can be obtained analyzing the maximum and minimum values of A: The minimum area corresponds to one side of the cube structure of the satellite fig 6.8. Since a is the edge of the cubic structure A = a 2 hence F Rmin = C a P a 2 = kg m/s 2 N solarmin = 1.76 nn m The maximum cross sectional area is indicated in figure 6.8. where hence F Rmax = C a P 3 A = a a 2 = kg m/s 2 N solarmax = 3.05 nn m Torques where found considering the worst case of CoM (center of mass) placed 2 cm from the geometrical center.

71 Group 830a 6.5 Disturbance modelling 59 sunline sunline (a) (b) Figure 6.8: Geometric description of the minimum and maximum cross sectional area exposed to the Sun Radiation disturbances model The primary influence on the radiation disturbances, which results in a torque on the satellite, is the orientation of the sun vector relative to the spacecraft. The other radiation disturbances caused by the Moon and the Earth albedo are not considered in this model. The effect of eclipse is also neglected. The variations of the force, caused by solar radiation in the inertial frame, is implemented as I F rad = A P C a I R s I R sc I R s I R sc (6.12) where 6 kg P momentum flux from the Sun, P = m s 2 C a absorption coefficient, C a = 1.4 eu for diffuse reflection I R s vector of the Sun in ECI frame I R sc vector from the center of the Earth to the spacecraft in ECI frame In equation (6.12) the cross sectional area which is exposed to the sun line, A, is obtained and implemented in Simulink. The details can be found in appendix B.3. The radiation torque is calculated in the spacecraft frame with the following 4 inputs: I F rad vector of the disturbances force caused by solar radiation in ECI frame I q quaternion from the spacecraft frame to ECI frame SC SCq quaternion from ECI frame to the spacecraft frame I SC R com vector from the CoM to the geometrical center of the satellite

72 60 Chapter 6 Modelling Attitude Control System for AAUSAT-II The vector of the disturbances force in the ECI frame, I F rad, is rotated into the spacecraft frame by the I S q and S I q quaternions, denoted as S F rad. The radiation disturbance torque in the spacecraft frame S N rad is calculated through S N rad = S F rad S R com (6.13) The implementation and verification of the model can be found in appendix B.3 on page Atmospheric drag disturbance In this section the influence of the atmospheric drag is addressed. In the orbit the satellite experiences collisions with gas molecules. This elastic impact without reflection removes energy from the spacecraft orbit. The influence of this disturbance is highly significant in a LEO where the atmospheric density near the Earth is the highest. Atmospheric disturbance model The force, df drag, acting on a small surface element of the satellite, da, can be modeled by df drag = 1 2 C Dρv 2 ( ˆN ˆV ) ˆV da (6.14) where ˆN is the normal vector to the area element da, v is the orbit velocity, and ˆV is the unit vector in the direction of the translational velocity. The dot product ˆN ˆV projects the area element on a surface perpendicular to the velocity (if the value is below zero it is set equal to zero). The parameter ρ is the atmospheric density and the tabular values for it are available in [30]. At an altitude of 550 km the maximum pressure is ρ 550km = kg m 3. The last parameter, C D, which is the drag coefficient is approximated by C D 2.2 eu according to [30, p. 145]. The total aerodynamic drag acting on the satellite is calculated by integrating equation (6.14) over the entire exposed area. Since the satellite is a cube it has a simple and symmetric structure. This simplifies the equation (6.14) to where A is the total exposed area. f drag = 1 2 C DAρV 2 ˆV (6.15) The minimum exposed area of the satellite is when only one side of the satellite is exposed to the wind. In this case A min = 0.01 m 2. The maximum area where three sides are equally exposed, which gives A max = 3A min. Substituting the values of A min and A max in the equation (6.15) gives f dragmin = 585 nn and f dragmax = 1014 nn.

73 Group 830a 6.5 Disturbance modelling 61 The effect of the atmospheric drag on the dynamic of the spacecraft, the aerodynamic torque N drag acting on the satellite due to the force df drag, can be written as N drag = r s df drag (6.16) where r s is a vector from the center of mass to the surface element da. By disregarding the difference of the velocity of the surface elements due to satellite rotation and utilizing the simple structure of the satellite, equation (6.16) can be reduced to N drag = C P f drag (6.17) where C P is a vector from the center of mass to the center of pressure. It is noticeable that the center of pressure is the geometric center of the exposed cross sectional area. Since the satellite has to satisfy the CubeSat requirements, the center of mass should be within the radius of 2 cm from the geometric center, [28]. At an altitude of 550 km this displacement causes a maximum torque of N dragmax = f dragmax 2 cm = 20.3 nnm (6.18) The acceleration of the satellite can be calculated from equation (6.15) as where m is the mass of the satellite. V = 1 2 ρc DA m V 2 ˆV (6.19) These values can then be used to give a rough estimation of the lifetime of the satellite. For a near circular orbit the change in the altitude change per orbit can be approximated as a rev = 2π C DA m ρa2 (6.20) which in orbit altitude of 550 km gives a rev = 10.6 m/rev.. A circular orbit at an altitude of 550 km has approximately 15 orbits per day which yields a day = m/day. The model of this disturbance is implemented in Simulink and verified in appendix B Gravitational disturbances This section describes the gravitational disturbance and its model. The model includes the gravitational forces of the Earth, the Sun and the Moon. Gravitational disturbances are also caused by the non-uniform mass distribution of the Earth, known as zonal harmonics. These disturbances cause translational and rotational acceleration of the satellite. The last gravitational disturbance to be modeled is the gravity gradient, which is related to the mass distribution of the satellite.

74 62 Chapter 6 Modelling Attitude Control System for AAUSAT-II Sun and Moon The gravity contribution from the Sun and the Moon is calculated under the assumption that both the Sun and the Moon are point masses. A vector representing the distance from spacecraft to Sun/Moon is calculated as I R ssc = I R s I R sc, where I R s is the vector from the center of the Earth to the Sun and I R sc is from the center of the Earth to the spacecraft. It is used to calculate the force acting on the spacecraft: I F G = M s M sc I R ssc IˆR ssc A similar equation is used for the Moon. Earth zonal harmonics Information about the zonal harmonics was obtained from [29, p. 776]. The variations of the Earth gravitational field because of zonal harmonics are implemented as a S-function in Simulink (see appendix B) that implements the following equation β( R sc, θ) = µ o R sc 4 n=2 ( ( R ) 0 R sc )n J n P n0 cos(θ) (6.21) where θ is the coelevation measured from the rotational axis of the Earth. Equation (6.21) can be approximated to where U G M e [U J2 + U J3 + U J4 ] r U J2 = ( R 0 I R sc )2 J (3cos2 (θ) 1) U J3 = ( R 0 I R sc )3 J (cos3 θ 3 5 cosθ) U J4 = ( R 0 I R sc )4 J (cos4 θ 6 7 cos2 θ ) (6.22) From the gravitational potential the force is found as M sc U, where U is

75 Group 830a 6.5 Disturbance modelling 63 U = , GM GM ( ( 1 2 R 2 0 J 2 (3 cos(θ) 2 1) R sc R 3 0 J 3 (cos(θ) cos(θ)) R sc 3 R 4 0 J 4 (cos(θ) cos(θ) ) ) ( / Rsc 2 + GM R2 0 J 2 (3 cos(θ) 2 1) R sc 4 R sc 3 R 3 0 J 3 (cos(θ) cos(θ)) R sc R2 0 J 2 cos(θ) sin(θ) R sc R 4 0 J 4 (cos(θ) cos(θ) ) ) / Rsc, R sc 5 R 3 0 J 3 ( 3 cos(θ) 2 sin(θ) sin(θ)) R sc 3 R 4 0 J 4 ( 4 cos(θ) 3 sin(θ) + 12 ) cos(θ) sin(θ)) 7 /( Rsc sin(φ)) 2 R sc 4 Since this gives the gradient in a plane that is rotated around the φ angle, this is inserted in the gradient before its converted back into cartesian coordinates. Gravity gradient The gravity gradient is calculated in the principal frame and depends on the following four parameters: P I Vector containing the diagonal values form the moments of inertia tensor I R sc Vector from center of earth to spacecraft I Sq quaternion from Spacecraft to ECI frame Sq quaternion from the principal frame to spacecraft fixed P These parameters are then used to calculate the gravity gradient, N GG, as N GG = 3µ 0 P [ PˆR Rs 3 sc ( P I PˆR sc )] Worst case gravity gradient In order to find the maximum value of N GG (which will deliver the worst case gravity gradient), the most unprofitable mass distribution must be taken into account. This mass distribution can

76 64 Chapter 6 Modelling Attitude Control System for AAUSAT-II be calculated when the entire mass of the satellite is located in the two cross corners of the cube. The center of mass is in this case displaced by 2cm of the geometrical center, which is shown in figure 6.9. The x 3 vector designates the mass displacement from the geometrical center. m 1 y x 1 x 3 x z x 2 m 2 Figure 6.9: Geometric description of the mass distribution. x 1 = ; x 2 = ; x 3 = Knowing that the mass of the satellite is equal to 1 kg the values of m 1 and m 2 can be found by using the equation for calculating the center of the mass as 0 = m 1x 1 + m 2 x 2 m 1 + m 2 m 1 + m 2 = 1 kg And choose the displacement to be one the m 1 side yields m 1 = kg m 2 = kg The inertia tensor defined for the SC frame with the origin in the center of the mass of the satellite [16] can now be found as I worst = I xx I xy I xz I xy I yy I yz I xz I yz I zz = where the scalar elements are given by

77 Group 830a 6.5 Disturbance modelling 65 I xx = I yy = I zz = I xy = I xz = I yz = 2 (y 2 + z 2 )ρdv = (x 2 yi + x2 zi )m i V V V V V (x 2 + z 2 )ρdv = (x 2 + y 2 )ρdv = xyρdv = xzρdv = yzρdv = i=1 2 (x 2 xi + x2 zi )m i i=1 2 (x 2 xi + x2 yi )m i i=1 2 (x xi x yi )m i i=1 2 (x xi x zi )m i i=1 V i=1 2 (x yi x zi )m i (6.23) Thus, the gravity gradient N GG = Nm with a gravity direction of R s = [ ] T becomes N GG = N GG = Nm Magnetic residual disturbance Magnetic residual field torques are generated by interactions between the spacecraft residual magnetic dipole and the Earth s magnetic field. This spacecraft residual magnetic dipole is caused by current running through the spacecraft wiring harness. The residual dipole exhibits transient and periodic fluctuations due to power switching between different subsystems. These effects can be minimized by proper placement of the wiring harness, but since this will be addressed in the satellite integration phase, a estimate has been made on the magnetic residual. A dipole magnet approximation with a field strength of 0.1 ma m 2 has been selected Residual dipole torques decrease with the inverse cube of the distance from the Earth s primary dipole. This cyclic disturbance is influenced primarily by the orbit altitude, the residual spacecraft dipole and the orbit inclination.

78 66 Chapter 6 Modelling Attitude Control System for AAUSAT-II Magnetic residual model The equation that describes the residual disturbance is [26] N mres = D B where N mres is the torque due to the spacecraft residual magnetic dipole, D is the residual dipole in [A m 2 ] and B is the Earth s magnetic field in [T]. An approximation for the magnetic field B can be calculated as B = 2M r 3 where M is the magnetic moment of the Earth ( T m 3 ) and r is the radius from the dipole center of the Earth to spacecraft in m, which can be found as r = r earth + Al = 6378 km km = 6928 km where Al is the altitude of the satellite, for which 550 km will be used, and r earth = 6378 km is the radius of the Earth. This yields equation (6.24) for the force caused by magnetic residual N mres = D B = D 2M r (6.24) Inserting the values in equation (6.24) yields a force of N mres = 0.1 ma T m 3 = N m The simulation implementation of the magnetic residual model can be found in appendix B.6 on page Total disturbance torque Once the disturbances are analyzed and modeled, the worst case disturbance torque is found adding them together. It is done under the assumption that all the torques are acting in the same direction which is very unrealistic. This value, together with other requirements, will be helpful in the design of the actuators. The values of these torques and the total one are shown in table 6.2 hence the maximum total torque is N dist = 36.2 nn m The most predominant torque is created by the aerodynamic force (56%). At higher altitude the predominant ones would be the ones with magnetic field interactions.

79 Group 830a 6.6 Spacecraft dynamics and kinematics 67 Disturbance Maximum torque [nn m] Solar radiation 3.05 Atmospheric drag 20.3 Gravity gradient 8.02 Magnetic residual 4.79 TOTAL 36.2 Table 6.2: Maximum values for the disturbance torques 6.6 Spacecraft dynamics and kinematics The dynamics of a satellite is a description of how forces (torques) acting on the satellite (internal or external) are affecting the rotational acceleration of the satellite. The rotational acceleration is then transformed into a rotational rate and an attitude of the satellite which is described by the kinematics. Usually, when describing the dynamics of a mechanical system, it is desirable to use a simple model structure, and the simplest three dimensional mechanical structure is a rigid body. As the structure of the AAUSAT-II is to be with no moving bodies, except momentum wheels, it is chosen to consider (approximate) the satellite as a rigid body Rigid body dynamics This section lists the basic equations for rigid body dynamics and their general properties are discussed. The derivations of the equations can be found in both [29] and [12]. Mathematical modelling The basic dynamic equations for a rigid body is based on Newtons laws of conservation of momentum and Eulers equation for rotational dynamics ḣ = ω h = I ω h = k (6.25) h is the angular momentum of the satellite, ω is the instantaneous rotational rate and I is the moment of inertia tensor. The change in angular momentum is proportional by the inertia matrix to the change in rotational rate, which depends on the rotational rate of the body. This causes a free floating satellite to wobble in a motion known as nutation, [29, chap. 5]. As the satellite contains momentum wheels it is not a rigid body, thus the equations of motion have to be adjusted accordingly. Then total angular momentum of the spacecraft can be described by equation

80 68 Chapter 6 Modelling Attitude Control System for AAUSAT-II h = Iω + h mw where h mw is the angular momentum of the momentum wheels. ω = I 1 (h h mw ) (6.26) Inserting eq. (6.26) into eq. (6.25) yields eq. (6.27), which describes the dynamics as a quasi rigid body. The first equation is more suitable for numerical integration as it does not contain both the momentum wheels angular momentum and the time derivative in the same equation, ḣ = ( I 1 (h h mw )) h = I ω + ḣmw d dt (h h mw) = ḣmw ω (Iω + h mw ) = I ω (6.27) Adding control and disturbance torques is straight forward, resulting in equation (6.28) ḣ = N ext (I 1 (h h mw )) h = I ω + ḣmw d dt (h h mw) = N ext ḣmw ω (Iω + h mw ) = I ω (6.28) The principal frame To simplify the dynamic equations it is important to choose the right reference frame for calculating the dynamics. Calculating the dynamics in an inertial frame or any other non spacecraft fixed frame causes the moment of inertia to be a true tensor with time varying components. If a body fixed frame is chosen instead, the inertia tensor becomes constant, which simplifies the equations considerably. The equations can even be simplified further if the frame is chosen to coincide with the principal axis, which are the axis of greatest inertia, smallest inertia and the cross product between the two. This results in an inertia tensor with the structure as in equation (6.29) where the products of inertia are zero, ie. I xx 0 0 I = 0 I yy 0 (6.29) 0 0 I zz As the mechanical structure of the satellite is not determined at this point the inertia tensor is unknown. Because of this, the inertia tensor from AAU CubeSat is used, which is defined as I = (6.30)

81 Group 830a 6.6 Spacecraft dynamics and kinematics 69 This result is seen in equation (6.28) can be expressed as ḣ x N ext,x (h y h mw,y ) h z Iyy 1 + (h z h mw,z ) h y Izz 1 ḣ y = N ext,y (h z h mw,z ) h x Izz 1 + (h x h mw,x ) h z I 1 xx (6.31) ḣ z N ext,z (h x h mw,x )h y Ixx 1 + (h y h mw,y )h x Iyy 1 When choosing a spacecraft fixed reference frame it is important to consider that the angular velocity (momentum) of the spacecraft is to be measured as seen from the inertial frame and transformed to the principal frame Spacecraft kinematics The spacecraft kinematics can be described in several ways (see [29] and [30]) of which the Euler-Rodriques quaternion has the best properties for computational purposes. To describe the kinematics by quaternions, a time derivative of the quaternions must be used. Appendix A contains the most basic quaternion algebra. Using this algebra, the representation of a time derivative quaterion (see appendix A.2) is q = 1 2 Ωq = 1 2 [ ] ωq4 + q 1:3 ω ω q 1:3 (6.32) where Ω is the 4 4 skew-symmetric cross product matrix of ω, which can be written as (6.33). 0 ω 3 ω 2 ω 1 Ω = ω 3 0 ω 1 ω 2 ω 2 ω 1 0 ω 3 (6.33) ω 1 ω 2 ω 3 0 This representation will be used in the system equation (ie. dynamic and kinematic equations) of the satellite System equations Combining the spacecraft dynamics and kinematics results in a set of system equations, (6.34), describing the attitude of the spacecraft. [ḣ ] [ N = ext ( I 1 (h h mw ) ) ] h 1 q 2 }{{} } {{ Ωq } ẋ f(x,u,w) [ ] [ ] [ ] [ ] ω I 1 0 = 3 4 h I 1 + h q q }{{} mw }{{} } {{ }}{{} } {{ } u y H x J (6.34) (6.35)

82 70 Chapter 6 Modelling Attitude Control System for AAUSAT-II The implementation of dynamic and kinematic of the satellite in Simulink can be found in appendix B Modelling of actuators To dimension the actuators of the satellite, it is necessary to model the actuators and establish the constraints for the actuators by simulation Modelling of magnetorquers Magnetorquers, as actuation devices, use magnetic coils or electromagnets to generate magnetic dipole moments. Magnetorquers can compensate for the spacecraft s residual magnetic fields or attitude drift from minor disturbance torques. They can also desaturate momentum-exchange systems but usually require much more time than thrusters. A magnetorquer produces torque proportional (and perpendicular) to the Earth s varying magnetic field. [30, p.369] Model By considering a single plane wire loop enclosing an area, A mt, through which a current, I mt, is flowing, the magnetic moment, m mt, is given by m mt = I mt A mtˆn mt where ˆn mt is a unit vector normal to the plane of the loop. The positive sense of the magnetic moment is determined by the right-hand rule. For a coil of n mt turns: m mt = n mt I mt A mtˆn mt (6.36) The magnetic dipole moment depends on the material enclosed by the current-carrying coil and is given by d mt = µ r m mt where µ r is the relative permeability of the core material (permeability of free space µ 0 = 4π 10 7 [N/A 2 ]) Thus, for a coil or an electromagnet enclosing a plane area, A mt, the magnetic dipole moment is given by d mt = µ r (n mt I mt )A mtˆn mt To generate the required dipole moment, parameters such as core material, coil configuration and the current level, must be appropriately selected. The selection is constrained by mission

83 Group 830a 6.7 Modelling of actuators 71 requirement and is influenced by considerations such as mass, power consumption and heat generated limitations. [26]. Magnetorquers use the Earth s magnetic field, B, and electrical current through the torquer to create a magnetic dipole (d mt ) that results in torque (N mag ) on the spacecraft. [30, p. 370] N mag = d mt B = µ r m mt B (6.37) where B is the magnetic flux vector of the Earth and N mag is the control torque generated by magnetorquers acting on the satellite [30, p. 370]. In B.8 the model of the magnetorquer is implemented in Simulink and verified Momentum wheels To simplify the actuation hardware and electronics we use regular DC motors. This constrain on the design lowers the lifetime of the momentum wheels. However as the expected lifetime of the satellite is lower than one year a regular DC motor which can last up to 20, 000 hours (approximately 2 years) [23]. Electrical model From the DC motor s electrical equivalent diagram shown on figure 6.10, it is possible to derive the relation between the voltage and current driving the motor, and the angular velocity and torque generated. + R L u i + e m Figure 6.10: The electrical equivalent diagram of a DC motor, where u is the armature voltage, i is the current through the motor, R is the resistance in the coils, L is the inductance of the coils and e m is the electromotive force. The electromotive force is the force generated by rotating a conductor in a magnetic field and is approximated by a linear combination of the angular velocity of the rotor and a constant K m, thus giving the following equation. e m = K m ω m (6.38) The torque N m generated by the motor is proportional to the current i and the motor constant K m. It gives the relation between current and torque N m = K m i (6.39)

84 72 Chapter 6 Modelling Attitude Control System for AAUSAT-II From figure 6.10 and eq. (6.38) it is possible to derive the following relation between the armature voltage and angular velocity of the motor. u = Ri + L di dt + K mω m (6.40) Mechanical model The mechanical part of the model is derived from the free body diagram of the rotor and the load, shown on figure N m J r + J l b (J r + J l ) d dt ω bω Figure 6.11: Free body diagram of the motor and load From the free body diagram on figure 6.11 and using Newtons 2 nd law, it is possible to derive (6.41) (J r + J l ) dω m dt = N m N f (6.41) where J r + J l is the combined moment of inertia of the rotor and the load, ω m is the angular velocity of the rotor, N m is the torque generated by the motor and N f is the torque generated due to friction in the motor. In the model only the viscous friction is taken into account. It applies where b f is the viscous friction coefficient. N f = b f ω m (6.42) By combining (6.39), (6.40) and (6.41) it is possible to build up the model for a DC motor. It is implemented in Simulink and the implementation is verified in appendix B.9.

85 Group 830a 6.8 Summary Summary In this chapter, a number of models for both the spacecraft and the environment has been presented. All models have been implemented in Simulink. The Simulink block diagrams of the models can be found in appendix B. The following models have been modelled and implemented: Orbit position and velocity Magnetic field strength and direction Ephemeris models for Earth (ECI to ECEF quaternion) Sun Moon External disturbances Solar radiation Atmospheric drag Gravity gradient Magnetic residual Spacecraft kinematic model Spacecraft dynamic model Furthermore, specific to the attitude control system, models of two kinds of actuators has been presented, namely: Magnetorquer Momentum wheel 6.9 Perspectives Even though the system has been designed and the model has been implemented in Simulink many improvements can still be made. Magnetic and Orbit modelling The IGRF simulation model of the magnetic field uses 144 coefficients. However it is possible to exchange the model to one that uses 16,000 parameters to describe the geometric field and thus making it more accurate. The ephemeris modelling The model would be more precise if the nutation and precession of the Earth was considered. Atmospheric drag As it is now the atmospheric density in the model is just a constant, but the model could be improved by modelling the atmospheric density as it depends on the satellites altitude.

86 74 Chapter 6 Modelling Attitude Control System for AAUSAT-II Spacecraft dynamics modelling The model of the spacecraft can be extended by including non-rigid effects. An accurate estimate of the satellites inertia tensor is also necessary to be able to simulate the spacecraft dynamics. Actuators modelling Neither disturbance nor noise were included in the models of actuators. Therefore, the models of magnetorquer and momentum wheels should be verified against actual actutors.

87 CHAPTER Controllers This chapter describes the different control strategies of the attitude control system. Detumble and pointing control algorithms are designed, implemented and tested in a simulation environment. Supervisory control is considered and the states of the control supervisor are described. 7.1 Control strategies There are a number of different ways of stabilizing and controlling the attitude of a satellite using the actuators available on AAUSAT-II: magnetorquers and momentum wheels. Table 7.1 lists some of the features of these actuators. 3-axis control High precision Slew maneuvers Long term stabilization Momentum unloading Saturation Position dependance Magnetorquers Momentum wheels Table 7.1: Features of magnetorquers and momentum wheels Detumbling Detumbling of the satellite after orbital insertion is the first priority of the attitude control system, hence it should be done by a very reliable system. As detumbling requires unloading of angular momentum from the satellite body the best actuators for detumbling are the magnetorquers. Combined with a three-axis magnetometer it is possible to reduce the angular rate of the satellite using only electromagnetic actuation. The method employed is refered to as B-dot which is described in section 7.3 on page 83.

88 76 Chapter 7 Controllers Attitude Control System for AAUSAT-II Attitude control For controlling the attitude of the spacecraft both momentum wheels and magnetorquers are used. It is possible to use only magnetorquers for attitude control but the physical nature of this actuation system makes it a time variant system as it depends on the geomagnetic field of the Earth which is periodically time variant. At any given point in time it is only possible to control two axes using magnetorquers but over time all three axes are controllable. For this reason, momentum wheel control is employed to do fine adjusting of the attitude and maintaining attitude by suppressing disturbances. Momentum wheel control If only momentum wheels are used for attitude control the wheels will be used for both major attitude corrections and disturbance rejection. In order for the spacecraft body to be stabilized the total angular momentum of the spacecraft must be equal to the angular momentum of the momentum wheels: stable attitude h tot : h tot = h mw (7.1) This means that they must be able to store angular momentum that is transfered to the spacecraft body by the disturbance torques and onto the wheels by the attitude controller. It also means that at any given attitude the momentum wheels must be able to generate an angular momentum in the direction of the total angular momentum of the spacecraft system (including the momentum wheels). Using only momentum wheel actuation and disregarding disturbances the total angular momentum of the spacecraft is constant in the inertial system as there are no externally applied torques to transfer angular momentum to the spacecraft. The result is that any change in the attitude of the spacecraft in the inertial reference system corresponds to a change of the direction of the total angular momentum in the spacecraft reference system. Figure 7.1 illustrates the direction of the total angular momentum of the spacecraft in two different attitudes. The configuration of the momentum wheels employed in AAUSAT-II is such that the wheels can only generate angular momentum in the first octant of the spacecraft coordinate system because the motors only spin one way. This physical contraint makes it impossible to satisfy (7.1) given the attitude in figure 7.1(b) which means that the momentum wheels cannot stabilize the spacecraft in attitudes where S h tot is outside the first octant of the spacecraft coordinate system. From a stable situation where the momentum wheels are at bias and the satellite is detumbled the total angular momentum of the system equals angular momentum of the wheels which is (7.2). S h tot = S h mw = h 1 bias 1 (7.2) 3 1

89 Group 830a 7.1 Control strategies 77 (a) Total angular momentum is in first octant of spacecraft coordinate system. (b) Total angular momentum is not in first octant of spacecraft coordinate system. Figure 7.1: Total angular momentum of satellite including momentum wheels in two situations. Given the bias point in (7.2) a rotation about an arbitrary axis can only be guaranteed to succed up to an angel of 45 without changing I h tot by means of the magnetorquers because S h tot would be required to enter another octant outside the range of angular momentum of the momentum wheels. The physical symptom is that the momentum wheels reach their saturation limit. Due to this constraint the momentum wheels will be used for small-angle corrections and stabilization. If the attitude change is greater than 45 it must be computed if the attitude change is possible in one step, else it will be done stepwise managed by the control supervisor. Between the attitude correction steps the momentum wheels are desaturated using the magnetorquers. Momentum wheel stabilization Though any given attitude maneuver may not always be possible in one step using momentum wheels, it is always possible to stabilize the satellite when the momentum wheels are at bias and the size of the angular momentum of the satellite body is less than or equal to the half the maximum angular momentum of one momentum wheel 1. This follows from the fact that for the satellite to be stable (7.1) must be true and in general for a rigid body with no external torques (7.3) holds true. S h tot = k and h tot = S h mw + S h sat S h mw + S h sat = k (7.3) 1 When the momentum wheels are biased at half their capacity.

90 78 Chapter 7 Controllers Attitude Control System for AAUSAT-II where S h mw = S h bias + S h mw (7.4) The momentum wheel angular momentum offset S h mw R 3 has a maximum size of half the maximum angular momentum of one momentum wheel for the worst case direction 2. To meet the detumble requirement of (7.1) then (7.5) must be true: S h mw = S h sat (7.5) This is obtainable as long as the size of the angular momentum of the satellite rigid body S h sat is less than the maximum change of angular momentum of the momentum wheels from their bias: S h sat < S h mwmax (7.6) The attitude controller using momentum wheels is described in section 7.4 on page Desaturation of momentum wheels When performing attitude corrections and disturbance rejection the momentum wheels will eventually reach saturation and angular momentum must be unloaded from the spacecraft system using the magnetorquers. This desaturation is carried out in continuous mode while maintaining attitude stabilization which is described in section 7.5 on page Control supervisor When operating in different modes, different control tactics are employed to handle the specific requirements for each mode. The control situations to handle are described in the following. Detumbling Reduce the angular rate of the spacecraft after orbital insertion. Whenever the ADCS s angular rate is above a certain level the ACS should be in detumbling mode. Attitude control When in attitude control mode the supervisor is in charge of sending the reference quaternion to the attitude controller. When receiving the reference from the flight planner it is necessary to 2 Aligned with one of the momentumwheels.

91 Group 830a 7.2 Control supervisor 79 check if the controller is able to obtain the desired attitude in one maneuver as the momentum wheel attitude controller has a limitied range as described in section To determine if the the attitude maneauver must be devided into steps, the direction of the total angular momentum measured in the spacecraft frame at the end of the maneuver, S h totend, needs to be determined using (7.7). [ S h totend 0 ] [ S ] h = q tot ref q ref 0 = q ref [ S I S ω + S h mw 0 ] q ref (7.7) If S h totend is in R + the maneuver can be completed in one step else the maneuver needs to be devided into several steps. The strategy behind the attitude maneuver division algorithm is to find a rotation that rotates S h totend to R +. This rotation is then devided into steps such that S h totend before the last rotation is in R +. It is then as stated above possible using a small angle maneuver to obtain the desired attitude. The rotation between S h totend and S h tot is calculated using (7.8) and (7.9). e = S ĥ tot S ĥ totend (7.8) ( θ = cos 1 S ĥ tot Sĥ ) totend (7.9) It is guarantied that any rotation of maximum 45 can be done in one step when the satellite is detumbled and the momentum wheels are at bias. Adding a 10 margin taking into account disturbances and a small offset in momentum wheel bias the rotation angle needed to move S h totend to R + is calculated as θ h = θ 35. Using the θ h and e to construct a new attitude reference quaternion using (7.10) it is tested if the new rotation is possible. If not, the angle is devided by two until a possible rotation has been found. where q is the current attitude of the satellite. [ ( e sin θh2 )] q h = cos ( ) θ h2 q ref = qq h (7.10) The method described is alternatly described using the following algorithm. 1. Find S h tot 2. Rotate S h tot to S h totend 3. If S h totend is in R + goto Find rotation between S h tot and S h totend

92 80 Chapter 7 Controllers Attitude Control System for AAUSAT-II 5. Subtract 35 from θ 6. Calculate new q ref 7. Rotate S h tot to S h totend 8. If S h totend is in R + goto Find rotation between S h tot and S h totend 10. θ h = θ goto Set attitude reference to q ref. When the new attitude has been reached and the momentum wheels desaturated the algorithm is run again with q ref set as the desired attitude. Attitude control When in attitude control mode the ACS is given a constant attitude quaternion from the flight planner and the ACS is required to keep this attitude. Direction control In stead of sending an attitude quaternion to the ACS it is also possible to request the ACS to point the bore axis which in this case is the same as the payload axis, S P of the payload in a certain direction. This vector is then converted by the supervisor to an equivalent quaternion. This can be done by first rotating I P r to the spacecraft frame by S B r = S q I S P I r q (7.11) I The axis to rotate about can now be calculated as e = and the angle to be turned around that axis is S P S P r (7.12) θ = cos 1 ( SˆP SˆP r ) (7.13) which can be used to compose the error quaternion to be feed to the ACS: q control = q att q err (7.14) Autonomous pointing Since the payload is requested to point at the Sun for extended periods of time the flight planner can tell the ADS to send the position of the Sun at a sertain frequency to the ACS and by that calculate the required attitude continuously by the same method as used in direction control. Momentum wheel desaturation When the angular momentum of the momentum wheels have moved away from the bias by a certain amount the supervisor has to turn the desaturation controller on.

93 Group 830a 7.2 Control supervisor 81 Mode Initiation conditions Termination conditions Idle External command External command Detumbling External command External command Poiting External command External command Desaturation If the absolute bias error h err of at least one momentum wheel is more than h on. If the absolute bias error h err of all momentum wheels is less than h off. Table 7.2: Rules for switching between control modes Mode switching The basic conditions for switching between control modes are listed in table 7.2. State description Main states The main states of the control system are depicted as a state diagram in figure 7.2. The state transitions are triggered by events from the flight planner (FP.event) or by a time triggered event (T) garded by some conditions. Conditions to transistions are shown as [condition]. Actions during transitions are marked with /action. The flight planner can tell the ACS to enter either detumbling, pointing or idle mode. When receiving the FP.point event a transition to Pointing is made only if the spacecraft is detumbled which is indicated by a variable. Figure 7.2: Main states and transitions of the attitude control system. Desaturation states When operating the attitude controller the desaturation controller is operated in parallel as shown in the state diagram in figure 7.3. When the desaturator is in the

94 82 Chapter 7 Controllers Attitude Control System for AAUSAT-II waiting state it is monitoring the angular momentum of the momentum wheels and if they come above a sertain distance from the momentum bias the desaturation system changes mode from passive to active desaturation. When all the wheels are desautrated to their nominal angular momentum the desaturator changes from active to passive again. Figure 7.3: Desaturation states. Event logging To get feedback from the ACS it is needed to lock different parameter and mesurements. This is done by the House keeping system on the OBC. Which logs relevant messages on the INSANE. In the following table 7.3 is a list of different Telemetry messages, that ACS generates. Message ID Description Value frequency ACS.TM.1 Angular velocity of momentum wheels ω mw 20 Hz ACS.TM.2 Angular momentum h mw 10 Hz ACS.TM.3 Control torque N mw 1 Hz ACS.TM.4 Control dipole moment m mt 1 Hz ACS.TM.5 Fault message f i Table 7.3: Telemetry package list including ID, description, value name and frequency of the telemetry message. Inaddition to data logging, gives this the supervisor possibility to make fault tolerant control.

95 Group 830a 7.3 Detumbling controller based on B-dot Detumbling controller based on B-dot The first and most important attitude control task to be executed, after orbital insertion of the satellite is stabilizing its angular rate, i.e. detumbling. This procedure should be done by a robust and failsafe system which does not depend on very complex systems being operational like e.g. attitude estimation filters. A very simple solution to detumbling using magnetic actuation is the Ḃ (B-dot) algorithm. The principle of a Ḃ-controller is to minimize the derivative of the magnetic field vector measured by a magnetometer. As the spacecraft orbits the Earth, the magnetic field vector in the spacecraft reference frame changes, depending on the position of the spacecraft. However, the dominant rate of change of direction direction of the field vector is caused by the tumbling of the satellite, as it may tumble with angular rates much larger than the orbital rate. Minimizing the change in the measured field vector by means of actuation, causes the spacecraft to approach an angular rate close to the orbital rate, which is achieved by forcing the derivative of the measured B-field, Ḃ, to zero B-dot control law The control law for Ḃ can be written as m mt = CḂ (7.15) where m mt is the magnetic dipole moment vector to be generated by the magnetic actuators in the three axes of the spacecraft. C is a controller gain and Ḃ is the time derivative of the magnetic field vector. The controller gain is negative, in order to actuate opposite the rotation, thus taking kinetic energy out of the system. The reason that Ḃ can be used directly without any cross product is that the changes in the B- field that the controller seeks to minimize, are caused by a rotation of the spacecraft and hence the derivative of the B-field is perpendicular to the field vector. This means that the control law gives an output to the actuators, which is a dipole moment perpendicular to the B-field. Ḃ can be written as Ḃ B ω sc (7.16) given the assumption that the direction and magnitude of the B-field with respect to the orbit fixed coordinate system, O B, is constant. This assumption leads to the conclusion that the rate of change of the B-field in the spacecraft reference frame is mainly due to the rotation of the spacecraft.

96 84 Chapter 7 Controllers Attitude Control System for AAUSAT-II Estimating B-dot One problem with the Ḃ-algorithm is that S Ḃ cannot be directly measured by the magnetometer and differentiating its output may give peaks of unwanted noise. Continuous time estimation A continuous time filter that estimates the rate of change of the B-vector can be realized. Simply multiplying its output by the controller gain C gives the controller output to the magnetorquers as a dipole moment reference. See figure 7.4. Figure 7.4: Filter for estimating the time derivative of the B-field. The transfer function of the filter, disregarding C, is given in (7.17). H cont (s) = ˆḂ B = ω cs s + ω c (7.17) Discrete time estimation The controller will be implemented on a computer and therefore needs to be discrete. Using pole-zero matching and gain matching the transfer function of the Ḃ-estimator can be written in the z-domain as shown in (7.18). p cont = ω c p disc = e ωcts z cont = 0 z disc = 1 H disc (z) = K z 1 z e ωcts (7.18) The gain is matched in center of the bandwidth, that is at ω 0 = ωc and the gain correction K is 2 computed as follows: K = K cont = H cont(s) K disc H disc (z) s=jω 0,z=e jω 0 Ts (7.19)

97 Group 830a 7.3 Detumbling controller based on B-dot Periodic measurement and actuation The control law described in section is stated without any constraints to the measured B-field or the output to the magnetic actuators. However, usage of the actuators while trying to measure the B-field with the magnetometer, causes a disturbance to the measurements, that introduces a feedback in the control loop, which cannot easily be estimated. In order to avoid this potential problem the actuators and the sensor are not used simultaneously but a periodic time-sharing policy is adopted. The period of the control/measurement cycle is T cycle = T sensor + T actuator. During the period T sensor the sensor readings are fed to the discrete Ḃ-estimation filter which settles to an estimate of the rate of change of the B-field. During the rest of the time of the cycle period, T actuator, the output from the controller is held at a constant value, yielding a constant magnetic dipole moment from the actuators. All readings from the magnetometer are discarded in the actuation period and the input to the estimation filter is held at zero. Figure 7.5 illustrates the principle. Figure 7.5: Periodic measurement and actuation Stability of the B-dot controller In order to analyze stability of the detumbling controller the ideal 3 continuous time Ḃ-controller is first considered. Lyapunov stability The stability using the control law in (7.15) can be proven by the Lyapunov direct method, described in chapter 5 in [31]. Since the stability criterium for detumbling implies that the rotational kinetic energy of the satellite should converge to zero 4, a Lyapunov candidate function is (7.20). E kin = 1 2 ωt sc I scω sc (7.20) In order to ensure energy dissipation and thus stability the derivative of the kinetic energy must be negative definite. Neglecting external disturbances the change in kinetic energy is only due 3 By ideal it is meant that the assumption that the exact time derivative of the B-field can be measured. 4 This is an assumption as the orbital rate of the satellite is not considered.

98 86 Chapter 7 Controllers Attitude Control System for AAUSAT-II to the torque exerted by the magnetorquers. Hence, the change in energy is Ė kin = ω T sc N mt (7.21) Including the control law into (7.21) and using (7.16) a simple expression for the change in kinetic energy can be achieved: Ė kin = ω T sc ( CḂ B) = Cω T sc (Ḃ B) = Cω T sc (S(Ḃ)B) = Cω T sc (S(B)Ḃ) = C(S(B)Ḃ) T ω sc = CḂ T (S(B) T ω sc ) = CḂ T (S(B)ω sc ) = CḂ T (B ω sc ) = CḂ T Ḃ = C Ḃ 2 (7.22) Equation (7.22) 5 describes the change of rotational kinetic energy of the spacecraft when applying the Ḃ control law. This equation is negative definite, thus proving that energy is dissipated from the system during detumbling. The control gain C determines the rate of energy dissipation and can be selected according to the detumbling requirements and electrical power constraints etc. The result of the Lyapunov analysis also shows that energy dissipation in the detumbling phase is proportional to Ḃ 2 which means that angular rates are reduced rapidly after initiating B-dot control and slowly converging over time. Stability including the B-dot estimation filter The implemented controller is not an ideal Ḃ controller and the Lyapunov stability analysis suggested in [31] must be used with some modifications/extensions. The result in (7.22) can be adopted to include the estimated derivative of the B-field as follows: Ė kin = C ˆḂ T Ḃ (7.23) In order for (7.23) to stay negative definite the vector dot-product ˆḂ T Ḃ must be negative at all times, which can only be ensured if the absolute angle between ˆḂ and Ḃ is less that 90. If 5 S(B) is the skew symmetric cross product matrix that is used to reduce cross products to matrix multiplications. The properties of skew symmetric matrices have been used to manipulate the equation.

99 Group 830a 7.3 Detumbling controller based on B-dot 87 the angle is more than 90 then ˆḂ T Ḃ becomes negative and (7.23) becomes positive and the kinetic energy rises. In this case the Lyapunov analysis does not prove stability. Figure 7.6 illustrates the problem. Figure 7.6: The B-dot vector and the estimated B-dot vector used in the Lyapunov stability analysis. The absolute angle between thees vectors must be less than 90 in order for the stability analysis to be valid. By rewriting (7.17) the relationship between Ḃ and ˆḂ is found: ˆḂ Ḃ 1 s = ω c s s + ω c (7.24) ˆḂ Ḃ = ω c s + ω c (7.25) This transfer function has a phase in the interval [0 ; 90 [ and a phase of 45 at the bandwidth frequency. The interpretation of this is that when the rate of change of Ḃ is large the angle between Ḃ and ˆḂ increases and stability becomes marginal; i.e. the phase margin decreases. Phase of estimated B-dot The phase of (7.25) is dependent on the rate of change of Ḃ, that is the size of B. Equation (7.26) expresses B with the assumption that there are no spinning momentum wheels in the spacecraft. Also, to simplify the equations it is assumed that the geomagnetic field is constant in the inertial coordinate system. Ḃ = B ω sc B = Ḃ ω sc + B ω sc ω sc = I 1 (N ctrl ω sc Iω sc ) B = Ḃ ω sc + B (I 1 (N ctrl ω sc Iω sc )) (7.26) It is clear that the phase increases with the actuation torque, which is proportional to the controller gain C and the size of Ḃ and inverse proportional to the inertia of the satellite. Also,

100 88 Chapter 7 Controllers Attitude Control System for AAUSAT-II there will be a contribution to the rate of Ḃ from the local variations in the B-field, caused by the change of position of the spacecraft (i.e. its position in orbit). Measurement noise and pure time delay from the magnetometer may add even more phase to the estimated Ḃ. A simulation has been made with some pre-selected values of ω c and C, which are based on the suggested values from [11] in which a similar control problem is handled. Setting ω c = 0.7 and C = the control law including the Ḃ yields the behavior illustrated in the simulation results of figure 7.7 and 7.8. The satellite is detumbling from an initial angular velocity of [ ] rad s. Figure 7.7 shows the norm of the angular velocity of the satellite during the simulation. The angular velocity norm is monotonously decreasing as predicted by the Lyapunov analysis, which seems reasonable when observing the phase of the estimated Ḃ in figure 7.8. Figure 7.7: Result of a simulation with the continuous time B-dot controller enabled after t=1000 seconds. The graph shows the norm of the angular velocity of the satellite while detumbling. Figure 7.8 shows that the phase of the estimated derivative of the B-field is approximately in the interval [0 ; 20 ], which gives a margin of 70 to the dangerous 90, that may cause instability. Stability using periodic measurement and actuation As described in section the derivative of the B-field cannot be estimated during the actuation period due to magnetic disturbances from the magnetorquers. This means that the ˆḂ estimate is constant for the period of actuation. As torque is applied when actuating, the actual Ḃ will change, creating a difference between Ḃ and ˆḂ, which is equivalent to a change in the angle between these. This angle must be added to the phase of the ˆḂ-estimation filter when considering stability. According to figure 7.8 the maximum additional angle allowed is approximately 70. The figure shows that the period of Ḃ is at least 20 s. Hence, the maximum time allowed for the controller to use a constant estimate of Ḃ is s = 3.8 s, which potentially (worst case) adds 70 to

101 Group 830a 7.3 Detumbling controller based on B-dot 89 Figure 7.8: Result of a simulation with the continuous time B-dot controller enabled after t=1000 seconds. First graph is the real derivatives of the B-field. Second graph is the estimated derivative. Third graph is the angle between B-dot and the estimated B-dot. Last graph is the dot-product of the two first. The peaks in the graphs are caused by bugs in the Rømer IGRF simulation software.

102 90 Chapter 7 Controllers Attitude Control System for AAUSAT-II the angle between Ḃ and ˆḂ. There are no requirements to the time of measuring the B-field between actuation regarding stability of the system. However, the measurement time should be long enough for the filter to settle to an acceptable estimate of Ḃ, otherwise the error introduced here also contributes to the phase of ˆḂ Implementation The controller is implemented in Simulink to test it together with the AAUSAT-II simulation library. Estimation filter Equation (7.27) is the discrete time filter for estimation of Ḃ calculated with a sample time of T s = 0.1 s. The discrete filter is equivalent to the continuous time filter with ω c = 0.7. ˆḂ(z) Ḃ(z) = z 1 (7.27) z 1 Controller gain The controller gain C in the control law is set to C = which has been found to be optimum through simulations. Measurement and actuation following values are used: For the periodic time shared measurement and actuation the T cycle = 5.5 s T sensor = 2.0 s T actuator = 3.5 s Magnetorquers Three identical magnetorquers are used. Their parameters are depicted in table 7.4. The maximum current at i = 6.8 ma causes a saturation limiting the dipole moment to a maximum of 17 mam 2. A coil [m 2 ] n i max [ma] r[ω] V max [V] m max [mam 2 ] p max [mw] Table 7.4: Magnetorquer parameters.

103 Group 830a 7.3 Detumbling controller based on B-dot Test The detumble controller has been verified in Simulink. Figure 7.9 shows the angular velocity of the satellite when detumbling from an initial angular velocity of ω sc = [ ]. Figure 7.9 shows the absolute value of the angular velocity. After approximately 5000 s the angular rates are down to rad 1. Figure 7.9: Result of a simulation with the discrete time B-dot controller including periodic measurement and actuation. Figure 7.10: Result of a simulation with the discrete time B-dot controller including periodic measurement and actuation. The graph shows the norm of the angular velocity of the satellite while detumbling Conclusion The B-dot detumbling controller algorithm has been implemented and tested in Simulink. Simulation results show that the controller is capable of reducing the angular rate of the satellite to below the requirement.

104 92 Chapter 7 Controllers Attitude Control System for AAUSAT-II 7.4 Momentum wheel attitude controller Requirements The momentum wheel attitude controller has to be able to point the satellite in inertial space using momentum wheel actuation. The controller must be able to point the satellite with a maximum error of 5. The pointing error is also affected by the precision of the determined attitude, therefore the target of the controller is to be able to point the satellite within 1 of the determined attitude. Thus assuming that the attitude determination is accurate within 4. When the satellite is stabilized it is a system requirement that the angular rate is kept below rad s. There are no mission requirements setting the maximum time to change the satellites attitude, but a goal is to be able to turn the satellite within 50 s. Furthermore, after reaching the desired attitude, the satellite has to be completely detumbled within 100 s. Since the system is relatively slow and there is a limited number of computational resources available to the ACS, the goal is that the controller frequency is at maximum 1 Hz. Due to the very low power available on the satellite the controller should also use as little power as possible Controller structure The control structure for a constant gain feedback controller can be seen in figure q ref q err K SC q ω Figure 7.11: Controller structure In order to include a quaternion reference the rotational difference between the current and the desired attitude has to be calculated which is done according to (7.28). q err = q att q 1 ref (7.28) It is possible to achieve two attitudes because the controller only controls q ref[1:3] so the fourth parameter is free. To avoid this, the error calculation is amended to (7.29), which equals the Gibb s parameter of the attitude difference. q G[1:3] = q err[1:3] q err[4] (7.29)

105 Group 830a 7.4 Momentum wheel attitude controller 93 However, the Gibb s parameter does have a singularity, as it is not possible to represent an attitude using three parameters without having singularities. The singularity emerges when the attitude error is a 180 rotation, which causes q err[4] = cos ( θ 2) = 0 and therefore qg[1:3] = Lyapunov stability analysis of constant gain controller To determine if the controlled system exhibits asymptotic stability, the Lyapunov stability criteria can be used. For asymptotic stability a function of the system, V (x), must be found such that it is positive definite and its derivative is negative definite, see (7.30). if V (x) : R n R such that : 1 V (x) is positive definite 2 d V (x) is negative definite dt then ẋ = ζ(x) is asymptotically stable (7.30) Often an energy function of the system is used and V (x) or its derivative is brought on the form shown in (7.31). V (x) = x T Qx (7.31) Where x is the state vector and Q is a constant matrix for LTI-systems. The advantage of this form is that if Q is positive definite then V (x) is positive definite and vice versa. For a simple mechanical system such as a rigid body it is natural to try the kinetic energy function E = 1 2 ωt Iω + (1 q 4 ) (7.32) with I being the rigid body inertia matrix, ω the angular rate of the spacecraft and q 4 the real component of the attitude quaternion. For a momentum-wheel controlled spacecraft (7.32) does not hold true, as the kinetic energy contribution from the momentum wheels is not included. However, this contribution is not necessary to consider when performing a Lyapunov stability analysis, as the requirement of V (x) is that it s a function of the system. Also an addition of 1 q 4 such that the equation includes the attitude quaternion. Differentiating (7.32) yields (7.33). Ė = 1 2 ( ω T Iω + ω T I ω ) q 4 = 1 2 ( ω T Iω + ω T I T ω ) ωt q 1:3 = ω T Iω ωt q 1:3 (7.33)

106 94 Chapter 7 Controllers Attitude Control System for AAUSAT-II Inserting the dynamics of the satellite, (7.34), into (7.33) yields the Lyapunov function, (7.35) used to test the stability of the controlled system. ω = I 1 [ N mw ω (Iω + h mw )] (7.34) Ė = [ I 1 ( N mw ω (Iω + h mw )) ]T Iω ωt q 1:3 (7.35) Where N mw and h mw are respectively the torque produced by and the angular momentum of the momentum wheels. Reducing (7.35) yields (7.36). Ė = [ N mw ω (Iω + h mw )] T (I 1 ) T Iω ωt q 1:3 = [ N mw ω (Iω + h mw )] T ω ωt q 1:3 = N T mwω [ω (Iω)] T ω [ω h mw ] T ω ωt q 1:3 = N T mwω [S(ω) (Iω)] T ω [S(ω)h mw ] T ω ωt q 1:3 = N T mw ω (Iω)T S(ω) T ω h T mw S(ω)T ω ωt q 1:3 = N T mw ω + (Iω)T S(ω)ω + h T mw } {{ } S(ω)ω + 1 } {{ } 2 ωt q 1:3 0 0 = N T mw ω ωt q 1:3 (7.36) Using (7.36), any controller for momentum wheels can be tested for stability by inserting the appropriate control law for N mw. Failing to prove stability with this Lyapunov candidate does however not imply instability. The stability criteria for a simple proportional gain feedback controller is to be examined, and the control law, (7.37), is inserted into (7.36) and brought to the preferred form seen in (7.31) yielding (7.38). N mw = Kx (7.37) Here K is the proportional gain feedback matrix and x is the state vector [ ω q 1:3 ] T. Ė = [ Kx] T ω ωt q 1:3 [ = x T K T ω + x T ] x = x T ( [K T [ ] ]) x (7.38)

107 Group 830a 7.4 Momentum wheel attitude controller 95 Having brought the Lyapunov function on the preferred form it can be deduced that for the spacecraft attitude control to be stable Q = [ K T [1:3,1:3] K T [3:6,3:6] ] (7.39) must be negative definite or Q must be positive definite. This is not possible for any choice of K and using this lyapunov candidate it is therfore not possible to prove the stability of a constant gain feedback controller using momentum wheels. Another lyapunov candidate has been sought but is has this far not been possible to find one that proves stability for attitude control. It is however possible to prove stability for detumbling using only x = [ω] T as the state vector. Utilizing the same derivation as (7.36) the Lyapunov function (7.40) E = 1 2 ωt Iω (7.40) (7.41) Ė = ω T K T [1:3,1:3]ω (7.42) The controlled system is therefore stable in ω if K T [1:3,1:3] is negative definite Linearization of system equations To be able to design a controller for the satellite using linear control theory a linearized model of the system has to be developed. In this section a small signal linearized model of the spacecraft dynamics and kinematics is computed for controller design use. For controller design it is more convenient to represent the equations in section using the state vector x = [ ω h mw q ]T, as the states that are to be controlled are q and ω. The second state is the angular momentum of the momentum wheels, h mw, and it has to be included to avoid having two dependent inputs, resulting in the alternative system representation ω I 1 [N ext N mw ω (Iω + h mw )] ẋ = h mw = N mw (7.43) 1 q Ωq 2 where N ext is the sum of external torques acting on the satellite including torques produced by the magnetorquers.

108 96 Chapter 7 Controllers Attitude Control System for AAUSAT-II Taylor linearization of spacecraft dynamics Splitting (7.43) into a linear and a non-linear part yields (7.44). ẋ = [ ω h mw ] [ ] I 1 [ ω (Iω) ω h = mw ] 0 } {{ } Non linear [ I 1 + I 1 ][ ] N mw N ext } 1 0 {{ } Linear (7.44) As the linearization is done with respect to an operating point, x 0 = [ ] T ω 0 h mw0, where the states are in equilibrium, the Taylor linearization of the non-linear part can be simplified to (7.45). x 0 + x x 0 }{{} 0 f(x 0 ) + f(x) (x x 0 ) x x=x0 + x f(x 0 ) + f(x) x f(x) x } {{ } 0 x x=x0 x x=x0 0 x f(x) x } {{ } 0 x (7.45) x=x0 Calculating the Jacobian of the non-linear part of the system yields (7.46). J {f(x), x} = = = = x=x0 [ I 1 ( S(ω)Iω ω ( I 1 ( I 1 S(ω) ω ) ( + S(ω)hmw I 1 S(ω)Iω ω h mw 0 0 ) + S(ω)hmw h mw ] x=x 0 (7.46) ) Iω 0 + S(ω 0 ) Iω ω S(hmw)ω ω I 1 S(ω 0 ) ω=ω0 ω=ω0 x=x0 0 0 ) ω + S(ω 0 )I ω S(h mw0 ) ω I 1 S(ω 0 ) ω=ω0 ω=ω0 ω=ω0 0 0 ] (7.47) S(Iω 0)ω [ I 1 {S(Iω 0 ) S(ω 0 )I + S(h mw0 )} I 1 S(ω 0 ) 0 0 Substituting the linearized equation into (7.44) results in (7.48), which is the linearized spacecraft dynamics given in the operating point [ ω 0 h mw0 ] T. [ ω h mw ] [ ][ ] I 1 {S(Iω = 0 ) S(ω 0 )I + S(h mw0 )} I 1 S(ω 0 ) ω 0 0 h mw [ ][ ] I 1 I 1 N + mw 1 0 N ext (7.48) (7.49)

109 Group 830a 7.4 Momentum wheel attitude controller 97 Linearization of spacecraft kinematics The quaternion at the time t+ t where t is a small interval of time can be described as (7.50) with q 0 being the quaternion operating point and q describing the small attitude change. q = q(t + t) = q(t)q( t) = q 0 q q = q 0 q (7.50) The kinematic equation for the spacecraft is given in (7.51), where q ω = [ ω 0 ] T. q = 1 2 Ωq = 1 2 qq ω (7.51) Expressing ω as an addition of an operating point, ω 0, and a small deviation, ω enables q ω to be written as (7.52). [ ] ω0 + ω q ω = 0 = q ω0 + q ω (7.52) Taking the derivative of (7.50) and inserting (7.51) yields (7.53) using (q 1 q 2 ) q ω = q ω. = q 2 q 1 and q = q 0 q + q 0 q = 1 [ q 2 0 qq ω + (q 0 q ω0 ) q ] = 1 [ q 2 0 qq ω + q ω 0 q 0 q] = 1 [ qqω q ] 2 ω0 q (7.53) Inserting (7.52) into (7.53) yields (7.54). q = 1 [ qqω0 q ] 2 ω0 q qq ω (7.54) As q for small angles can be approximated to (7.55) equation (7.54) can be simplified to (7.56). e 1 sin ( ) θ 2 lim q = e 2 sin ( ) 0 θ 2 θ 0 e 3 sin ( ) θ 2 cos ( = 0 ) 0 (7.55) θ 1 2

110 98 Chapter 7 Controllers Attitude Control System for AAUSAT-II q = 1 2 [ qqω0 q ω0 q ] q ω = 1 2 [ qqω0 q ] ω0 q + 1 [ ] ω 2 0 (7.56) [ ] S(ω0 ) Furthermore qq ω0 q ω0 q = 2 q as can be seen from (7.57), which inserted into 0 (7.56) yields (7.60). [ ][ ] S ( q1:3 ) + 1 q ω0 q = 3 3 q 4 q 1:3 ω0 q T 1:3 q 4 0 [ ] S ( q1:3 ) ω = 0 + q 4 ω 0 q T 1:3ω 0 [ ] S (ω0 ) q = 1:3 + q 4 ω 0 q T (7.57) 1:3ω 0 [ ] ] S (ω0 ) ω qq ω0 = 0 [ q1:3 ω T 0 q 4 [ ] S (ω0 ) q = 1:3 + q 4 ω 0 q T 1:3 ω (7.58) 0 [ ] S (ω0 ) q qq ω0 q ω0 q = 1:3 + q 4 ω 0 S (ω 0 ) q 1:3 q 4 ω 0 q T 1:3 ω 0 + q T 1:3 ω 0 [ ] S(ω0 ) = 2 q (7.59) 0 q 1:3 = S(ω 0 ) q 1:3 + 1 ω (7.60) 2 Linearized system equations Combining the linearized equations of dynamics and kinematics results in a system, (7.61), with nine state variables instead of the original ten. This is due to the fact that the small signal change of the real part of the attitude quaternion is zero. ω I 1 {S(Iω 0 ) S(ω 0 )I + S(h mw0 )} I 1 S(ω 0 ) ω h mw = h mw 1 q 1 1: S(ω 0 ) q 1:3 + I 1 I [ N mw N ext ] (7.61)

111 Group 830a 7.4 Momentum wheel attitude controller Controllability To be able to design a controller for the system, the system must be controllable. When only momentum wheel are used for actuation it can be seen that the linearized system equations (7.61) are not controllable, as the angular momentum of the momentum wheels and the angular rate of the spacecraft can not be controlled independently. Furthermore it can be confirmed mathematically. If the controllability matrix defined by (7.62) is non-singular the system is controllable. With the system defined as ẋ = F x + Gu and n = rank(f ). C = [ G F G F n 1 G ] (7.62) Before calculating the controllability matrix for the linearized system an operating point is chosen. The operating point for the momentum wheels is chosen at their bias, which is found to 5 kgm2 be in section As the controller is to point the satellite in inertial fixed space, s the operating point for ω is zero. Choosing the operating point for ω to be zero results in the controllability matrix to be singular, with a rank of 6 and thus uncontrollable. For some systems it is possible to move the operation point slightly from the desired value, and thus achieve a controllable system. As can be seen on figure 7.12 this is not the case for this system, as small deviations in the operating point for ω results in nearly singular controllability matrices. 9 x 106 Condition number of controllability matrix Condition number Angular velocity norm Figure 7.12: Condition number of controllability matrix for varying ω 0. The minimum value of the condition number is A solution to this problem is to divide the system into a controllable and a non-controllable part.

112 100 Chapter 7 Controllers Attitude Control System for AAUSAT-II As the rank of the controllability matrix for ω 0 = [ ]T is 6, the system contains six states which can be controlled independently. To find these six states the system can[ be brought ] on a xuc controllability staircase form, using a similarity transform matrix T for which = T 1 x. x c Where x c represents the controllable states and x uc represents the uncontrollable states. The transformation of the system described by (7.63) to (7.65) divides it into a controllable part independent of the uncontrollable states. [ ] F F T = uc 0 = T 1 F T (7.63) F 21 F c G T = [ ] G uc G c = T 1 G (7.64) ] [ ] T = xuc x = T 1 x (7.65) [ xuc x c T xc The controllable system is then defined as (7.66) and a constant gain controller can be designed using any desired method. ẋ c = F c x c + G c u (7.66) Having designed the control feedback matrix for the controllable system, it is necessary to transform the feedback matrix to control the original system. Using (7.65) the control matrix K for the attitude control can be written as (7.67). u = K c x c = K c T xc x = Kx K = K c T xc (7.67) Discretization As the controller is to be implemented on a computer it is necessary to compute a discrete equivalent of the system. As the controlled signal is sampled using zero order hold the discrete equivalent matrices can be computed by (7.68). Φ = e F T Γ = Where T is the sample time. T 0 e Fη Gdη (7.68)

113 Group 830a 7.4 Momentum wheel attitude controller Pole placement method The pole placement method of designing a controller is based on placing the poles such that the desired system response is obtained. Discretizing the system, ẋ = F x + Gu equations using (7.68), with the minimum sample time T = 1 s, given by the controller requirement specification, yields the discrete equivalent, (7.69) to the system. ẋ = Φx + Γu (7.69) Calculated earlier in section the rank of the controllability matrix is 6, meaning that it is possible to find a transformation T such that six of the states of the transformed system are controllable independent of the uncontrollable states. Deriving the transformation matrix, which divides the system into a controllable and an uncontrollable part, can be done using a physical understanding of the system. As the angular momentum of the satellite is constant, any change in angular momentum of the momentum wheels causes the angular velocity of the satellite to change, proportional to the satellites inertia tensor, in the opposite direction. Therefore it is not possible to drive both states to zero from an arbitrary initial condition. As the objective of the controller is to control the angular velocity and the attitude of the spacecraft the angular momentum of the momentum wheels is considered as a free state. Therefore the transformation matrix, T, can be derived using (7.65) resulting in (7.70). Using (7.70) the controllable system (7.71) can be found. h m w ω ω = T 1 h m w q 1:3 q 1: T 1 = T = (7.70) ẋ c = Φ c x c + Γ c u (7.71) From the eigenvalues of Φ c it can be seen that the system is marginally stable as four poles lie in z = 1 and the last two in z = ± ı Using the K c feedback matrix these poles can be moved to any arbitrary position as can be deduced from (7.72).

114 102 Chapter 7 Controllers Attitude Control System for AAUSAT-II ẋ c = (Φ c Γ c K)x c Poles = eig(φ c Γ c K c ) (7.72) Having chosen the desired poles solving for K c in (7.72) can be done using a robust pole assignment method [17] as implemented in Matlab s place function. Choosing the poles There are no finite method of choosing the poles for the system. One method for coming up with a good first guess, which can then be iteratively improved, is to analyze the system as a second order system. This method involves defining the rise time, settling time and damping ratio from the requirements. Poles for a second order system are then chosen to meet these requirements. To make the system mimic a second order system, the poles are placed such that two of the poles coincide with those found for the second order system. And the other poles are placed further left such that they have significantly less influence on the system behavior. The attitude controller requirements states that the settling time must be maximum 100 s and the attitude must be reach within approximately 50 s. As described in section on page 76 the saturation of the momentum wheels limits the controller to small angle maneuvers after which the momentum wheels has to be desaturated. The settling time is related to the rise time by (7.73). t r = 1.8 ω n ω n = 4.6 ζt s tr t sζ 2.56 (7.73) As the damping ratio, ζ, maximally can be 1, the rise time is maximally 40 s. Choosing a damping ratio of ζ = which roughly corresponding to an overshoot of 1% and a settling time of 100 s, yields a boundary for the placement of the second order poles shown in the polezero map in figure As a first attempt the complex conjugate pole pair is moved just inside the design boundary and the remaining poles are placed much closer to zero. This yields the two second order poles z = ± ı , two poles are placed in z = 0.5 and two in z = 0.51 as shown on the pole-zero map ζ = 1 is not practically possible

115 Group 830a 7.4 Momentum wheel attitude controller 103 Pole Zero Map Imaginary Axis Real Axis Figure 7.13: Pole-zero map of the uncontrolled system, with design requirement boundaries. Pole Zero Map Imaginary Axis Real Axis Figure 7.14: Pole-zero map of the first controlled system with design requirement boundaries.

116 104 Chapter 7 Controllers Attitude Control System for AAUSAT-II Calculating the poles of the entire system is done using (7.74). Where T xc is determined from (7.65) yielding (7.75). Poles = eig(φ ΓK c T xc ) (7.74) [ ] T xc = (7.75) Comparing the poles of the entire system with those of the controllable system it is seen that three poles at z = 1 has been added. These poles originate from the pure integration of N mw to h mw thus the only state that is marginal stable is the uncontrolled state h mw. Simulating the system response To verify that the system responds as desired, the control loop has been simulated using the designed controller. The simulation is done without disturbances such that the pure system response is not affected. The spacecraft is initially in a stable state with the angular velocity equaling zero and the momentum wheels at bias. The initial attitude is q init = [ ] T and the attitude reference input is chosen as a sequence of rotations about the x-axis as shown in table 7.5. Time( s) Angle( rad) quaternion [ ] T [ ] π T [ ] 300 π T [ ] 500 π T [ ] T Table 7.5: Input sequence used for simulation of system response The resulting attitude and angular velocity is shown in figure 7.15, where it can be seen that spacecraft attitude follows the attitude nicely while stabilizing the satellite between each attitude maneuver. In the closeup of the second attitude maneuver, shown in figure 7.16, it can be seen that the high frequency poles of the system causes the controller to be aggressive. Thus overcompensating for the systems cross coupling resulting in a rotation about the other two axes before the attitude is stabilized. Measuring the performance of the system it can be seen that the rise time, detumble time, error angle and the angular velocity error fulfills the requirements as listed in table 7.6. To remedy the agressiveness of the controller the poles should be moved closer to their original position thus minimizing the control effort. The four recessive poles are therefore moved to the right resulting in the poles z = { ± ı , 6.1, 6.1, 6, 6}.

117 Group 830a 7.4 Momentum wheel attitude controller 105 Attitude of spacecraft Attitude quaternion q1 q2 q3 q Time [s] Anguar velocity [rad/s] Angular velocity of spacecraft w1 w2 w Time [s] Figure 7.15: Angular velocity and attitude responce. Requirement Goal value Rise time max 40 s 35 s Detumble time max 100 s 0(no disturbances) Error angle max (mean = ) Error angular velocity max rad s rad s Table 7.6: Performance versus requirements ( mean = rad s )

118 106 Chapter 7 Controllers Attitude Control System for AAUSAT-II Attitude of spacecraft 0.2 Attitude quaternion Time [s] Angular velocity of spacecraft Anguar velocity [rad/s] Time [s] Figure 7.16: Close up of angular velocity and attitude responce.

119 Group 830a 7.4 Momentum wheel attitude controller 107 Performing the test described above with the new controller, yields the response shown in figures 7.17 and 7.18, which shows that the new controller is less aggressive, but still conforms to the demands as can be seen from table 7.7. Attitude of spacecraft Attitude quaternion q1 q2 q3 q Time [s] Anguar velocity [rad/s] Angular velocity of spacecraft w1 w2 w Time [s] Figure 7.17: Angular velocity and attitude responce. Requirement Goal value Rise time max 50 s 40 s Detumble time max 100 s 4s(no disturbances) Error angle max (mean = ) Error angular velocity max rad s The resulting K feedback matrice is (7.76) rad s Table 7.7: Performance versus requirements ( mean = rad s ) K = (7.76)

120 108 Chapter 7 Controllers Attitude Control System for AAUSAT-II Attitude of spacecraft 0.2 Attitude quaternion Time [s] Angular velocity of spacecraft Anguar velocity [rad/s] Time [s] Figure 7.18: Close up of angular velocity and attitude responce.

121 Group 830a 7.4 Momentum wheel attitude controller 109 Stability Using the results from section the controller is stable in ω if K [1:3,1:3] is negative definite. Using the controller matrice developed in this section, (7.76), yields the following eigenvalues of the symmetric part of the matrix [ ]. K [1:3,1:3] is therefore negative definite and the controlled system is stable in ω. Stability reagarding attitude control has not been proved but simulations indicate that the controlled system in fact is stable LQR-controller Another approach for designing an attitude controller for the satellite is by the use of optimal control theory. Opposed to the classical way of designing a controller by pole placement the design in optimal control is done by optimizing the controller performance according to some performance index N I = H (x(k), u(k)) (7.77) k=0 where k is the sample number, x is the state vector and u is the input signal. Thus the goal is to find an input sequence u(k) which minimizes I. The function H is the weighting function which is used to weigh the size of the input signals versus the size of the states. In the following the linearized system described by x(k + 1) = Φx(k) + Γu(k) (7.78) is considered and is used to deduce the general time-varying optimal control law. This deduction is mainly build on [10] and [3]. For this controller the H function is selected to be quadratic in both x and u thus (7.77) can be rewritten into N ( I = x T (k)q 1 x(k) + u T (k)q 2 u(k) ) (7.79) k=0 where Q 1 and Q 2 are square matrices 7. Q 1 is a matrix that punishes large states and is required to be positive definite and Q 2 is used to punish large control signals and is required to be positive semi definite. In order to find a minimum for (7.79) constrained by (7.78) the performance index can be rewritten into I = N ( x T (k)q 1 x(k) + u T (k)q 2 u(k) + λ T (k + 1)( x(k + 1) + Φx(k) + Γu(k)) ) k=0 (7.80) 7 Thus the name: Linear Quadratic Regulator.

122 110 Chapter 7 Controllers Attitude Control System for AAUSAT-II by the use of Lagrange multipliers. The necessary conditions for I to reach an extremum is then I 0 = x(k) = 2xT (k)q 1 λ T (k) + λ T (k + 1)Φ (7.81) 0 = 0 = I u(k) = 2uT (k)q 2 + λ T (k + 1)Γ (7.82) I = x(k + 1) + Φx(k) + Γu(k) λ(k + 1) (7.83) The optimal solution defined for x(k), u(k) and λ(k) is given by the coupled differential equations (7.81), (7.82) and (7.83) providing that the initial or final conditions are known. In the case of x(k) it should be known from the beginning. According to (7.79) u(k) should be zero in order to minimize the performance index. Using this in conjunction with (7.82) and (7.81) the following boundary condition can be found 0 = 2u T (k)q 2 + λ T (k + 1)Γ u(n)=0 λ(n + 1) = 0 (7.84) 0 = 2x T (N)Q 1 λ T (N) + λ T (N + 1)Φ λ(n) = 2Q 1 x(n) (7.85) This results in the two-point boundary-value problem given by (7.83) and (7.81) with (7.82) inserted and the final condition of λ given by (7.85). A method for solving this two-point boundary-value problem (described in [3]) is to assume that λ(k) = S(k)x(k) (7.86) is a solution to the problem and using this transformation to transform the problem into a onepoint boundary-value problem in S. By applying (7.86) to (7.82) and isolating u this gives 0 = 2Q 2 u(k) + Γ T λ(k + 1) 0 = 2Q 2 u(k) + Γ T S(k + 1)x(k + 1) 0 = 2Q 2 u(k) + Γ T S(k + 1)(Φx(k) + Γu(k)) u(k) = ( 2Q 2 + Γ T S(k + 1)Γ ) 1 Γ T S(k + 1)Φx(k) (7.87) If the suggested solution (7.86) is substituted into (7.81) for λ and x(k + 1) is substituted according to (7.83) and the input u(k) substituted with (7.87) this yields 0 = 2x T (k)q 1 λ T (k) + λ T (k + 1)Φ 0 = S(k)x(k) + 2Q 1 x(k) + Φ T S(k + 1)x(k + 1) 0 = S(k)x(k) + 2Q 1 x(k) + Φ T S(k + 1)(Φx(k) + Γu(k)) 0 = S(k)x(k) + 2Q 1 x(k) + Φ T S(k + 1)(Φx(k) Γ ( 2Q 2 + Γ T S(k + 1)Γ ) 1 Γ T S(k + 1)Φx(k)) (7.88)

123 Group 830a 7.4 Momentum wheel attitude controller 111 In (7.88) it can be seen that all the terms are multiplied by the state, so by rewriting the equation it gives 0 = ( S(k) + 2Q 1 +Φ T S(k + 1)(Φ Γ ( 2Q 2 + Γ T S(k + 1)Γ ) ) 1 Γ T S(k + 1)Φ) x(k) (7.89) Since (7.89) has to hold true for every possible value of x then the backward differential equation (7.90) must be a solution to S(k) S(k) = 2Q 1 + Φ T S(k + 1)(Φ Γ ( 2Q 2 + Γ T S(k + 1)Γ ) 1 Γ T S(k + 1)Φ) (7.90) The scalars in front of the weighting matrices in (7.90) can now be included in the weighing matrices thus yielding the standard discrete Riccati equation S(k) = Q 1 + Φ T S(k + 1)Φ Φ T S(k + 1)Γ(Q 2 + Γ T S(k + 1)Γ) 1 Γ T S(k + 1)Φ (7.91) where the boundary condition can be found by relating (7.85) to (7.86) S(N) = Q 1 (7.92) This single boundary problem given by (7.91) and (7.92) has to be calculated backward since the requirement is given at the endpoint. In order to find the optimal input signal (7.87) can be used to solve for u thus giving u(k) = L(k)x(k) (7.93) with L(k) = ( 2Q 2 + Γ T S(k + 1)Γ ) 1 Γ T S(k + 1)Φ (7.94) thus L(k) becomes the desired optimal time-varying feedback gain matrix. Design Since the time-varying gain solution is computational heavy it is desirable to find a constant gain matrix. A good candidate for such a constant optimal gain matrix is the steady state gain matrix L. This can be seen as the value that L(k) and thus S(k) converges to as N. Since (7.91) has to be solved recursively a simple computer implementation is to calculate it numerically. A general algorithm for doing this is as follows

124 112 Chapter 7 Controllers Attitude Control System for AAUSAT-II 1. S = Q 1 2. REPEAT 3. L 0 = L 4. L = ( Q 2 + Γ T S 0 Γ ) 1 Γ T S 0 Φ 5. S = Q 1 + Φ T SΦ Φ T SΓ(Q 2 + Γ T SΓ) 1 Γ T SΦ = Q 1 + Φ T S(Φ ΓL) 6. UNTIL L 0 L L < ǫ where ǫ is a break condition that is used to indicate how big the changes in S are allowed to be before terminating and saying that L = L. Since the two weight matrixes Q 1 and Q 2 are found by hand-tuning a simplification of the weight matrix selection is required. One possible way to simplify the selection process is by making the two weighting matrixes diagonal with the following sizes 8 Q 1 (i, i) = 1 x 2 i,max Q 2 (j, j) = 1 u 2 j,max (7.95) and introducing a scalar weighting factor σ between states and inputs, thus (7.79) can be rewritten to I = ( N n x 2 i (k) + σ x 2 i,max k=0 i=1 m u 2 j (k) ) u 2 j=1 j,max (7.96) The maximum values for the state vector are: ω max = rad and q s max = 1. And the kg m2 maximum value for the torque generated by the momentum wheel is h wmax = 267, thus s 2 giving the following weight matrices 9 : ω max ω max ω max Q 1 = q max q max q max Q 2 h wmax h wmax h wmax (7.97) 8 This also makes the performance index unit less. 9 The momentum of the momentum wheels are not considered here since it is not desirable to control their momentum.

125 Group 830a 7.4 Momentum wheel attitude controller 113 By using these weighting matrices as an initial guess the L matrix has been calculated to be L 1 = T (7.98) An attitude change test has been conducted with this matrix and the result can be seen in figure From the resulting rise and settling times listed in table it can be seen, that the rise and settling time requirements are not met. By looking at the initial guess it can be seen, that the two primary limiting factors are the initial weight of the angular velocity and the weight of the input. Since the input is limited in the momentum wheel driver a first try would be to lower the weight of the angular velocity. L 2 is calculated with a 10 th of the weight and as it is seen in table 7.8 there is a good improvement of the settling time. By decreasing the weight a bit further to a 15 th of the initial guess a satisfactory result is found. This design results in the following feed back matrix L 3 = T (7.99) The rise and settling time for L 3 is shown in table 7.8 and comparing it to the previous two it can be seen that the lower settling time has the price of a higher control effort and thus a higher energy usage. Control matrix Rise time (s) Settling time (s) Energy usage L L L Table 7.8: Controller performance. 10 A measure of the control effort can be calculated as the total required torque, i.e.: 3 i=1 0 N mw i.

126 114 Chapter 7 Controllers Attitude Control System for AAUSAT-II 1 Quaternion rotation Time [s] Angular velocity 0.01 Angular velocity [rad/s] Time [s] 7 x 10 5 Angular momentum of momentum wheels 6 Angular momentum [kg m 2 /s] Time [s] Figure 7.19: Simulation results with L 1 and attitude steps varying from π 16 to π 4.

127 Group 830a 7.4 Momentum wheel attitude controller Quaternion rotation Time [s] Angular velocity Angular velocity [rad/s] Time [s] 9 x 10 5 Angular momentum of momentum wheels 8 Angular momentum [kg m 2 /s] Time [s] Figure 7.20: Simulation results with L 3 and attitude steps varying from π 16 to π 4.

128 116 Chapter 7 Controllers Attitude Control System for AAUSAT-II Stability Using the results from section the controller is stable in ω if L 3,[1:3,1:3] is negative definite. Using the control matrice developed in this section, L 3, yields the following eigenvalues of the symmetric part of the matrix [ ] L 3,[1:3,1:3] is therefore negative definite and the controlled system is stable in ω. Stability reagarding attitude control has not been proved but simulations indicate that the controlled system is stable Test The test enviroment used to test the two controllers agains each other is the total simulation including all disturbances as described in 6 on page 41. The limitations on the momentum wheel assembly used in the tests are listed in table kgm2 Maximum angular momentum s Minimum angular momentum 0 5 kgm2 Angular momentum bias 7 10 s Maximum torque ± Nm Table 7.9: Parameters used in the test. The tests have been conducted by requirering the controller to rotate the satellite around a given axis with the following steps in the angle: π, 8 π, 3π and π with an interval of 300 s. The two main comparable parameters are the settling time and the energy usage during the test. The results are summarized in table 7.10 and a plot of how the two controllers behaved during test 3 can be seen in figure 7.21 and 7.22 for the K- and LQR-controller respectively. Test number In order to test the two controllers against each other a number of tests have been conducted in order to conclude which controller is performing the best according to the controller requirements. K settlinergtling time ergy usage K en- LQR set- LQR en- Rotation vektor time [s] usage [s] [ ] T [ ] 4 T [ ] 4 T Table 7.10: Results of the comparative test Conclusion From table 7.10 it can be seen that there is a general tendensy to that the K-controller uses more energy compared to the LQR-controller. On the other hand the LQR-controller is set-

129 Group 830a 7.4 Momentum wheel attitude controller Quaternion rotation Tid [s] 0.04 Angular velocity 0.02 Angular velocity [rad/s] Tid [s] 1.4 x 10 4 Angular momentum of momentum wheels 1.2 Angular momentum [kg m 2 /s] Tid [s] Figure 7.21: Simulation results with the K-controller.

130 118 Chapter 7 Controllers Attitude Control System for AAUSAT-II 1.2 Quaternion rotation Tid [s] 0.02 Angular velocity Angular velocity [rad/s] Tid [s] 1.2 x 10 4 Angular momentum of momentum wheels 1 Angular momentum [kg m 2 /s] Tid [s] Figure 7.22: Simulation results with the LQR-controller.

131 Group 830a 7.5 Desaturation of momentum wheels 119 tling slower that the K-controller. Compared with the requirements for the attitude controller in on page 92 that states that the controller should have a settling time below 100 s and use as little energy as possible it can be concluded, that the LQR-controller is preferable. Another advantage in using the LQR-controller is that it is fairly easy to alter its preformance, should it be required. 7.5 Desaturation of momentum wheels When operating the attitude control system using momentum wheels the motors will eventually reach their saturation limits. The reason is that the controller seeks to suppress external disturbances by transferring angular momentum generated by disturbance torques onto the momentum wheels. The result is that the wheels spin either faster or slower until they spin at maximum velocity or stop. Angular momentum can be unloaded from the system again by applying a controlled external torque that transfers angular momentum to another body in the inertial system 11, which in this case is the Earth. The only practical way of unloading angular momentum from the satellite in this case is using the magnetorquers capability of exchanging angular momentum with the Earth by exerting a torque on the geomagnetic field Requirements The requirements to desaturation cannot be derived directly from the mission requirements as they depend on the performance of the attitude controller. Simulations of the LQR-controller have shown that the momentum wheels saturate during attitude maneuvers in the order of π 4 rad. Between such maneuvers the wheels must desaturate all wheels before initiating the next attitude maneuver. Given the requirement from section 7.4.1, stating that the attitude control system must be able to turn the satellite to any attitude within one orbit, it is clear that the desaturation controller has to desaturate the momentum wheels four times in that period of time. One orbit is appr s so desaturation must be done in maximum 1500 s. This requirement is set to 750 s in order to have a margin of factor 2. The momentum wheels are said to be desaturated when their deviation from bias is below ±10% Desaturation strategy Desaturating the momentum wheels using only electromagnetic actuation can be done in different ways. Some possibilities are: 11 The momentum in an inertial system is constant which means that the only way of changing the momentum of an object is by exchanging momentum with other objects in the inertial system. A spacecraft can achieve this by e.g. ejecting mass at high speed (like a rocket) or by exerting a force on a celestial body like the Earth by means magnetic interaction.

132 120 Chapter 7 Controllers Attitude Control System for AAUSAT-II Periodic brute force desaturation by using the Ḃ-algorithm after having spun the motors back to their operating bias periodically. Stepwise periodic A variant of the first option is still to use the Ḃ-algorithm periodically, but without spinning the motors to bias at once. Instead the motors are graduately and stepwise spun towards the bias while detumbling the spacecraft between the steps in the desaturation. Continuous desaturation using magnetorquers while maintaining attitude stability can be done via a momentum wheel bias-error minimizing controller, that continuously unloads momentum. This is obtained through the attitude controller itself by controlling the magnetorquers so they slowly turn the spacecraft in such a way, that the pointing controller automatically spins the momentum wheels towards their bias, in order to maintain a stable attitude. The desaturation strategy employed for AAUSAT-II is the continuous mode option, which allows for longer durations of accurate pointing, providing more quality time for the mission Desaturation control law A control law for continuous desaturation of the momentum wheels using only electromagnetic actuation is derived by considerations of preservation of momentum. If the angular momentum of a momentum wheel is too high, then angular momentum is externally applied to the spacecraft body in the opposite direction via magnetorquers. By transferring this angular momentum to the momentum wheel the angular momentum of the wheel decreases. The control law for transferring momentum from the spacecraft body to the momentum wheels is either one of the attitude controller described in section on page 109 or 7.4 on page 92. Equation (7.100) is the desaturation control law. m mt = C d (h err B) (7.100) Where h err is error in angular momentum of the momentum wheels given as the difference between current angular momentum and the angular momentum when the wheels are at bias. See (7.101). h err = h wheel h wheel bias (7.101) The torque generated by the desaturation control law is given in (7.102). N mt = m mt B (7.102) Figure 7.23 illustrates the directions of the bias error vector, the magnetic field vector, the magnetorquer dipole vector and the magnetorquer torque vector.

133 Group 830a 7.5 Desaturation of momentum wheels 121 Figure 7.23: The directions of the error vector, the magnetic field vector, the magnetorquer dipole vector and the magnetorquer torque vector when applying the desaturation control law. The pointing controller acts opposite N mt, thus minimizing the components of the bias error that are not parallel to the B-field. The magnetorquers only produce torque perpendicular to the B-field meaning that bias error components aligned with the B-field will not be affected Periodic measurement and actuation As described in section on page 85 it is not practical to generate magnetic dipole with the magnetorquers while using the magnetometer due to the magnetic disturbances introduced. The same periodic measurement and actuation strategy used for detumbling is adopted for the desaturation controller Stability The desaturation control law is not intended to stabilize the system and it will actually render the system unstable if no other stabilizing controller is active. In order to secure stability the attitude controller must be able to compensate for the torque applied Test The desaturation controller has been tested in Simulink along with the LQR-controller using the following parameters for desaturation:

134 122 Chapter 7 Controllers Attitude Control System for AAUSAT-II The results are shown in figure T cycle = 1.0 s T sensor = 0.5 s T actuator = 0.5 s C = kgm2 H bias = [ ] 10 s 5 kgm2 H err on = [ ] 10 s 5 kgm2 H err off = [ ] 10 s During the simulation the attitude reference to the LQR-controller is changed 180 in four steps while having the desaturation controller is in wait state. When the angular momentum of at least one of the momentum wheels exceeds H bias + H err on the desaturation controller becomes active and starts disturbing the attitude controller, resulting in momentum unloading. When the angular momentum of all momentum wheels is below H bias + H err off the desaturation controller returns to standby. The test results confirm this behavior and figure shows that the angular momentum of the 5 kgm momentum wheels oscillates around zero with an amplitude of approximately s 2 While desaturating the momentum wheels the LQR-controller is still capable of maintaining a stable attitude with fluctuations in angular velocity of the spacecraft body in the magnitude of 5 rad 10 s. 7.6 Perspectives In this chapter a complete set of controllers for the AAUSAT-II has been designed such that it is possible to detumble and control the attitude of the satellite and desaturate the momentum wheels. However the attiude controller is only able to do small angle maneuvers thus it is necessary for large angle maneuvers to turn the satellite using a series of small angle maneuvers and desaturation of the momentumwheels. As an alternative the magnetorquer attitude controller developed for AAUSAT-I, could be used for the initial rotation of the satellite until the target is within the range of the momentum wheel attitude controller. Controller robustness The controllers has been designed under a number of assumptions, regarding the physical parameters of the satellite. Specifically that the inertia tensor and actuator properties estimates are perfect. A logical next step is to analyse the stability and performance dependency of the controllers with respect to pertubations in the estimated moments of inertia and orientation of

135 Group 830a 7.6 Perspectives 123 (a) Angular momentum of momentum wheels. (b) Angular momentum deviation from bias. (c) Angular velocity of satellite. (d) Magnetic dipole moment of magnetorquers. Figure 7.24: Simulation results from test of desaturation controller.

136 124 Chapter 7 Controllers Attitude Control System for AAUSAT-II the principal axis. The effects of missalligned momentum wheels, pertubantions in the flywheel inertia and actuator performance also needs to be considered. For the B-dot controller the characterisitics of the magnetometor have not been considered, thus they can add to the phase between the actual Ḃ and the estimated ˆḂ resultning in a decreased stability margin. Constant B-field contributions such as the residual magnetic dipole of the satellite has no consequence for the B-dot controller, and it is therefore mostly sensor missallignment that can cause problems. However the desaturation controller does rely on the B-vector directly and it is influenced by both sensor missallignment and residual magnetic contributions. This does not have an effect on the stability of the attitude controller as long as the attitude controller is able to absorb the maximum torque created by the desaturation controller. However it can decrease the performance of the desaturation making it necessary to increase the accepted bias error of the momentum wheels. Fault handling The controllers designed are not fault tolerant, but using the fault detection principles described in chapter 5 a fault can be detected such that appropriate actions can be taken. If the magnetorquer attitude controller develeoped for AAUSAT-I is implemented in parallel to the controllers developed in this chapter a simple fault handling scheme could be devised selecting the controller based on which actuation system is working nominally. Integration with ADS As the attitude determination system has been developed parallel to the attitude control system the two systems needs to be synchronised. The attitude determination system needs to sample the sensors approximatly ten times faster than the rate of change of the sensor readings. A problem may arise as the ADS has a limited amount of computational resources available which limits the frequency of the sampling. Specifically, the gyro measurements are of concern as their rise time depends on the torque applied to the spacecraft. Therefore, the controllers needs to ensure that they do not exite the satellite in such a way that the angular velcity changes to fast. The torque limitation of the momentum wheels used in this chapter forces the ADS to sample to gyros with 5 Hz. If this frequency is not obtainable by the ADS, the control torque can be limited either by lowering the saturation limit on the momentum wheels or, for the LQR controller, weighing the input higher. This will decreaase the controllers performance, such that attitude maneuvers and desaturation will take more time and the ability of the controller to supress disturbance torques will be lessend.

137 . CHAPTER Test This chapter describes the test strategies employed to verify the functionality and performance of the attitude control system. No physical tests have p.t. been carried out as the ACS subsystem has not yet been implemented in hardware but potential tests scenarios are discussed. 8.1 Introduction Testing the performance of an attitude control system for a satellite is not a trivial task as it requires an environment which is practically inaccessible, space. The characteristics of space which are absolutely essential to testing the ACS are first of all the absence of constraint forces caused by gravity. The SatLab at Aalborg University has at its disposition a gimbal system capable of emulating a state of weightlessness in the sense that constraint torques are minimized by suspending the satellite in its center of mass while allowing free rotation around all axes. This system, however, is designed for micro satellites weighing a hundred times more than a CubeSat which renders it useless in this context due to the mounting constraints and magnitude of disturbance torques. 8.2 Feel free - feel zero-g The obviously optimal test environment for the attitude control system of a CubeSat is to subject it to a completely free fall in vacuum thus removing all constraint forces. The closest available environment with properties resembling a free fall is inside an aeroplane during a zero-g parabola maneuver. Through ESA s Outreach programme the AAUSAT-II attitude control system team has been accepted on the 7 th Student Parabolic Flight Campaign (SPFC 7) in Bordeaux, France, On the parabolic flight an experiment will be subject to micro gravity for durations of up to twenty five seconds and in this period an experiment can be free floating for shorter durations. No constraint forces other than the atmospheric disturbances act on a free floating body in the micro gravity phases of the flights which makes it possible to test the attitude control system in space-like conditions. In the course of a week, two flights will be carried out providing a total of sixty parabolas each lasting approximately twenty seconds.

138 126 Chapter 8 Test Attitude Control System for AAUSAT-II 8.3 Test strategies Since the ACS system is accepted on the 7 th SPFC the following tests are divided in two categories, one for zero-g testing, and other tests, which can be carried out on ground. It is needed to test the performance of different controllers and the actuators. This is done through various test cases which are defined in section and Zero-G tests The test case in this section is designed for Zero-G test, on a parabolic flight see section 8.2 on page 125 for information of the flight. Since each test lasts 20 seconds, they have to be well prepared and structured for the test to perform correctly. This is done through preprogrammed test cases for each parabola. A list of tests are shown in table 8.1. Test id Test case No. Description ACS.tSPFC.1 Inertia test 1 4 Spin up of each momentum wheel one at the time. ACS.tSPFC.2 Inertia test 2 3 Spin up of all momentum wheels at once. ACS.tSPFC.3 Brake test 1 4 Spin down of one momentum wheel at a time. ACS.tSPFC.4 Brake test 2 4 Spin down of all momentum wheels. ACS.tSPFC.5 Detumble 1 5 Momentum wheel detumble using a classical controller. ACS.tSPFC.6 Detumble 2 5 Momentum wheel detumble using an LQR controller. ACS.tSPFC.7 Attitude change axes attitude change. ACS.tSPFC.8 Attitude change axes attitude change. ACS.tSPFC.9 Attitude change axes attitude change and roll. Table 8.1: List of different test cases for ACS during the SPFC, including the Test id,test case name, No. (number of runs) and a short description of the test case. All of the test cases listed in 8.1, will be carried out on the ESA Outreach SPFC in late june. And the test result will be posted on the AAUSAT-II web site [1].

139 Group 830a 8.3 Test strategies Other tests In addition to the SPFC test cases, some test cases which can be made on ground has been created and listed in table 8.2. Test id Test case Description ACS.tGND.1 Basic self test A functional test of the ACS ACS.tGND.2 Driver test A test of current limitations and power usage. ACS.tGND.3 Sensor test A test of sensor measurements. ACS.tGND.4 Momentum wheels A test of the momentum wheels, to determine the parameters and real power usage for the actuator. ACS.tGND.5 Magnetorquers Test of the magnetic field generated by the magnetorquers. And a determination of parameters and power usage. ACS.tGND.6 Controller test Performance test of controller, to check that they fullfill the requirements. Table 8.2: List of different test cases for ACS during ground testing, including Test id, Test case name and a short description of the test case. The test cases in table 8.2 has not been carried out since the system isn t implemented and the actuators are not available, but these tests and results will be posted on the AAUSAT web site [1].

140

141 CHAPTER Conclusion The chapter summarizes the objectives and results of the design of the attitude control system for AAUSAT-II. 9.1 Project objectives The attitude control system for AAUSATI-II is being developed in close cooperation with the attitude determination system team. Simulation In this context a satellite simulation library has been developed. The library contains a full set of tools for simulating the spacecraft orbit and attitude including disturbance and magnetic field simulations. The ADCS is integrated in the simulation library including sensor and actuator simulations and an implementation of the attitude determination and control algorithms. Furthermore a set of auxiliary tools has been developed such as a 3-D orbit and attitude visualization engine, solar panel power output simulator, vector and quaternion libraries. Actuators The first step in designing the attitude control system has been selecting the actuators to be used. Magnetorquers and momentum wheels are chosen resulting in an over actuated system facilitating three axis control of the spacecraft. The magnetorquers have been designed and dimensioned to minimize power, volume and mass budgets while retaining adequate range of output torque. The momentum wheels are driven by Maxon DC motors that have been chosen due their low mass, high power efficiency, durability and availability. Momentum wheels have been designed especially for these motors to satisfy the requirements to torque and angular momentum. The actuators are modelled as part of the simulation library and fault detection analysis and design has been made. Controllers The control strategy is composed of momentum wheel attitude control and magnetorquer desaturation and detumbling. A supervisory controller is developed to manage the attitude controllers to allow autonomous behavior such as momentum wheel bias control and target tracking. For

142 130 Chapter 9 Conclusion Attitude Control System for AAUSAT-II detumbling a B-dot controller has been designed that uses only unconditioned readings from the magnetometer to determine the control signal to the magnetorquers. Two tactics have been employed to tune a state space attitude controller using momentum wheels: classical poleplacement and optimal control. This controller stabilizes the spacecraft at a given attitude and performs attitude maneuvers either to acquire a fixed attitude or to point the payload in a given direction calculated by the supervisory controller. Simulations of the detumble controller shows that it is capable of detumbling the spacecraft to angular rates below rad. According to requirement [ADCS 1.2] the rates should be below s rad which cannot be guaranteed using only the B-dot controller. However, the attitude s controller using momentum wheel have shown to be able to stabilize the spacecraft to angular rates down to rad which satisfies the detumbling requirement. The attitude error of the s attitude controller is less than which satisfies requirement [ADCS 2.2]. 9.2 Study objectives Control and supervision has been designed for an autonomous system. The mechanical system and its environment has been modeled and simulated using simulink and magnetorquer and momentum wheel actuators has been chosen to fulfill the requirements. A number of different control strategies and modes have been explored and compared resulting in the development of a number of controller and design principles for an autonomous satellite attitude control system capable of operating under a wide range of situations. During the project the foundation for a complete attitude control system has been made resulting in a good vantage point for further exploration of pico-satellite attitude controllers. 9.3 Perspectives The work done in this project comprises strategic and algorithmic design and verification of algorithms through simulation. In order to finalize the complete operational attitude control system for AAUSAT-II it is necessary to design the electrical and mechanical hardware and the software that implements the algorithms developed. The attitude control algorithms have been tuned using the simulation but important parameters including the moments of inertia of the satellite used in the simulation are guessed from the AAU CubeSat and may be very different in AAUSAT-II. One possible way of enhancing the performance of the controllers is to get a good estimate of the moments of inertia and use the values for tuning the controllers. Using system identification methods moments of inertia may be estimated during the tests in zero-g on the parabolic flight campaign.

143 Bibliography [1] Aausat-ii officiel web site, 2004, [2] S. L. Altmann, Rotations, Quaternions, and Double Groups. Oxford University Press, [3] J.. Y. H. Arthur E. Bryson, Applied Optimal Control. Hemisphere Publishing Corporation, [4] T. Bak, Spacecraft attitude determination - a magnotometer approach, Ph.D. dissertation, Aalborg University, [5] M. Blanke, Fault Tolerant Control - an Engineering Approach. Department of Control Engineering, [6] N. A. A. D. Command, Spacetrack report no. 3, NORAD, Tech. Rep., December 1998, pdf-file: [7] DGRF/IGRF, Geomagnetic Field Model , N. MODELWeb, Ed. IGRF, 2000, [8] DSRI, Models of the Geomagnetic Field. Danish Space Research Institute, 2000, [9] DSRI Payload requirements, Web-page, 2004, index.php?page=payload.html. [10] G. F. Franklin, J. D. Powell, and M. Workman, Digital Control of Dynamic Systems. Addison-Wesley, [11] T. Graversen, M. Fredriksen, and S. Vedstesen, Attitude control system for aau cubesat, Aalborg University, Tech. Rep., 2001, [12] C. D. Hal, Spacecraft attitude dynamics and control, Virginia Tech, online PDF, 2003, cdhall/courses/aoe4140/rigid.pdf. [13] C. D. Hall, Reference Frames for Spacecraft Dynamics and Control. Virginia Tech - AOE, 2004, cdhall/courses/aoe4140/refframes.pdf.

144 132 Chapter 9 BIBLIOGRAPHY Attitude Control System for AAUSAT-II [14] Honeywell, Hmc axis magnetic sensor, Online PDF, May 2004, [15] R. Izadi-Zamanabadi and D. Bhanderi, Aausat-ii requirements, Un-published document, 2004, aausat-ii CVS: aausatii/systems/urd/aau/aau_requirements.pdf. [16] J. J.Craig, Introduction to Robotics, Mechanics and Control. Addison-Wesley Publishing Company, [17] J. Kautsky and N. Nichols, Robust pole assignment in linear state feedback, International Journal of Control, vol. 41, pp , [18] T. S. Kelso, Celestrak, Web-page, June 2004, date: [19] J. Kjær, K. Laursen, and D. Pedersen, Internal Satellite Area Network, AAUSAT-II OBC, May 2004, not published - can be found in the AAUSAT-II repository. [20] H. Klinkrad and B. Fritsche., Orbit and attitude perturbations due to aerodynamics and radiation pressure. [21] W. J. Larson and J. R. Wertz, Space mission analysis and design, 3rd ed., J. R. Wertz, Ed. Kluwer Academic Publishers, [22] M. motor, Re w datasheet, 2004, [23] M. Motor, Technology-Short and to the point, 2004, ftp://ftp.maxonmotor.com/public/download/catalog_2004/pdf/04_technik_kurzund_buendig_dc_24_25_e. [24] C. Poly, Official website of the international cubesat project, Web-Page, 2004, [25] C. M. Roithmayr, Contributions of spherical harmonics to magnetic and gravitational fields, p. 15, december [26] J. R.Wertz, Ed., Spacecraft Attitude Determination and Control. D.Reidel Publishing Company, [27] P. S. Systems, Spacecraft control toolbox, MatLab Toolbox, [28] A. C. Team, AAU CubeSat, 2003, date: [29] J. R. Wertz, Spacecraft Attitude Determination and Control, J. R. Wertz, Ed. Kluwer Academic Publishers, [30] J. R. Wertz and W. J. Larson, Space mission analysis and design, 3rd ed., J. R. Wertz, Ed. Kluwer Academic Publishers, [31] R. Wisniewski, Satellite attitude control using only electromagnetic actuation, Ph.D. dissertation, Aalborg Universitet, 1996.

145 Appendix

146

147 APPENDIX A Quaternion This appendix describes the fundamental laws of calculus (axioms) behind quaternion, which is used throughout the project. The main reason for using quaternion is that no representation of rotations can be singular, as in the case with Euler angles and direction cosines matrices. Also, quaternion takes up very little space, and multiple rotations are easily computed (lightweight computation). This appendix is based on information found in [29] and [2]. A.1 Axioms and definitions A quaternion is represented by the four parameters (q 1, q 2, q 3, q 4 ) which is normally expressed by a single letter, q, which is given by q = q 4 + iq 1 + jq 2 + kq 3 (A.1) where the vectors i, j and k are hyper imaginary terms, which have the relationships i 2 = j 2 = k 2 = 1 (A.2) ij = ji = k (A.3) jk = kj = i (A.4) ki = ik = j (A.5) The values of the different components are defined as ( ) Θ q 1 e 1 sin 2 ( ) Θ q 2 e 2 sin 2 ( ) Θ q 3 e 3 sin 2 ( ) Θ q 4 cos 2 (A.6) (A.7) (A.8) (A.9) where e is the vector which is to be rotated around with angle Θ.

148 136 Chapter A Quaternion Attitude Control System for AAUSAT-II To a certain extent quaternion can be regarded as extensions to complex numbers. For example, addition and subtraction use the same rules as complex numbers and the associative law is true. The commutative law is not valid because of equation (A.4) to (A.5). Multiplication of two quaternion, q A and q B, is defined as q C = q A q B (A.10) = (iq A1 + jq A2 + kq A3 + q A4 )(iq B1 + jq B2 + kq B3 + q B4 ) (A.11) = i(q A1 q B4 + q A2 q B3 q A3 q B2 + q A4 q B2 ) +j( q A1 q B3 + q A2 q B4 + q A3 q B1 + q A4 q B2 ) +k(q A1 q B2 q A2 q B1 + q A3 q B4 + q A4 q B3 ) +( q A1 q B1 q A2 q B2 q A3 q B3 + q A4 q B4 ) (A.12) Equation (A.12) can be described as the matrix multiplication found in equation (A.13), which can easily be implemented in software q C1 q B4 q B3 q B2 q B1 q A1 q C2 q C3 = q B3 q B4 q B1 q B2 q B2 q B1 q B4 q B3 q A2 q A3 q C4 q B1 q B2 q B3 q B4 q A4 (A.13) The norm of a quaternion is like standard complex numbers defined as q q q q q 2 4 (A.14) One of the properties of the quaternion algebra is that an inverse quaternion, q 1 A, such that q A q B = q C q B = q 1 A q C (A.15) The conjugate of a quaternion, q, is defined as q = q 4 iq 1 jq 2 kq 3 (A.16) Furthermore, if the norm (see equation (A.14)) of the quaternion equals to 1 then the conjugate equals the inverse, ie. q = 1 : q 1 = q (A.17)

149 Group 830a A.2 The time derivative of a quaternion 137 A vector v in R 3 can be composed as a quaternion with the real part is zero q V = iv 1 + jv 2 + kv 3 (A.18) A hyper imaginary quaternion as described in equation (A.18) can be rotated by another quaternion q by using the following multiplication, where the rotation result is given by the hyper imaginary quaternion q V q V = q q V q (A.19) Mathematically it can be shown that q V becomes hyper imaginary which can then be converted back to R 3 as in equation (A.18). A.2 The time derivative of a quaternion The time derivative of a quaternion can be found by using a quaternion which describes the rotation to time t, and another quaternion which describes the rotation in the time span t. The multiplication of these two yields the rotation to the time t + t. If the two quaternion are functions of time, they can be expressed as q(t + t) = q(t)q( t) (A.20) For small angles, the sine function can be approximated as lim sin(θ) = Θ Θ 0 (A.21) Using this approximation equation (A.6) to (A.9) can be written as q( t) Θ(t) e 1 2 Θ(t) e 2 2 Θ(t) e = [ ] Θ 2 1:3 1 (A.22) By using the multiplication matrix given in equation (A.13), equation (A.22) can be rewritten as q(t + t) [ ] I Ω t 2 q(t) (A.23)

150 138 Chapter A Quaternion Attitude Control System for AAUSAT-II Using equation (A.20) and (A.23), the following expression of the time derivative of a quaternion yields dq dt q(t + t) q(t) = lim t 0 [ t ] I4 4 + Ω t 2 q(t) q(t) = lim t 0 t Ω t 2 = lim t 0 t q(t) = 1 Ωq (A.24) 2 where Ω is a matrix which defines the velocity of the quaternions in the reference coordinate system and is given by 0 ω 3 ω 2 ω 1 Ω ω 3 0 ω 1 ω 2 ω 2 ω 1 0 ω 3 ω 1 ω 2 ω 3 0 (A.25) Finally, equation (A.24) can be rewritten to dq dt = 1 2 [ ] ωq4 + q 1:3 ω ω q 1:3 (A.26)

151 APPENDIX B.... Implementation of simulation In this appendix the central parts of the Simulink simulation framework can be found. Please note that the simulation framework does not contain the ADCS, but only the simulation of the environment, sensors and actuators. B.1 IGRF model To ease the implementation of the on-board IGRF model it is chosen to implement it using an S-function in Simulink and programmed in C. The input to the model is the position of the satellite in Cartesian coordinates, from which the magnetic field strength vector is calculated in the ECEF frame. Since the Rømer satellite project already has an implementation of the IGRFmodel in C, it is decided to use this model. The only change is made to the IGRF coefficients, which are updated so that they are expected to apply until The Rømer Simulink IGRF model has two parameters, the year and the order of the simulation. B.1.1 Verification To verify that the model is accurate with the Ørsted data has been used. In figure B.1 the test setup in Simulink is displayed. In tables B.1 and B.2 the maximum error of the magnetic field vector, the root mean square of all the errors and the phase between the models. Figure B.1: Figure of Simulink verification of the IGRF model

152 140 Chapter B Implementation of simulation Attitude Control System for AAUSAT-II Magnetic field vector error between the IGRF model and Ørsted data Model order Error RMS [nt] Error MAX [nt] Table B.1: Magnetic field vector error between the IGRF model and Ørsted data Phase between the IGRF model and Ørsted data Model order Error RMS [deg] Error MAX [deg] Table B.2: Phase between the IGRF model and Ørsted data B.2 Orbit model The orbit model is implemented in Simulink as a block diagram shown in figure B.2. Figure B.2: Implementation of Newtonian orbit model in Simulink. The inputs, R_sci(I) and V_sci(i), set the initial values of the two numeric integrators in the diagram. The initial values are obtained from a TLE. The third input is the external force generated by disturbances which is used to calculate the resulting acceleration by adding it to the acceleration introduced by gravity. This acceleration is integrated two times to yield the position vector R from which the gravitational acceleration is calculated.

153 Group 830a B.2 Orbit model 141 B.2.1 Verification Equation ( 6.2 on page 48) is verified using the SGP4 as a reference and a TLE from AAU CubeSat. Using a zero-disturbance input to the Newtonian propagator together with the SGP4-based initial position and velocity, the Newtonian- and the SGP4-model are simulated in Simulink. The length of the simulation is set to 7 days in orbit, which relates to approx. 100 orbits. The result of the verification is shown in figure B.3. The TLE, used for initialization, is from CubeSat (identification 27846, 2004 Julian day ). The numerical method for solving was the Runge-Kutta method. Distance [m] 1 x SGP4 Newton error (x) Distance [m] Distance [m] Time [days] 2 x SGP4 Newton error (y) Time [days] 2 x SGP4 Newton error (z) Time [days] Figure B.3: SGP4 and Newtonian orbit simulation of position, error graph of SGP4 - Newton for the three cartesian axes. Furthermore, the RMS and maximum error was calculated, which can be found in table B.3.

154 142 Chapter B Implementation of simulation Attitude Control System for AAUSAT-II Axis X Y Z Mean RMS m m m m Max. error m m m m Table B.3: RMS and maximum error of SGP4 versus Newtonian orbit model. B.2.2 Verification As can be seen from figure B.3, the Newtonian orbit propagator diverges as time passes. However, even after a week in orbit, the error is at maximum 139 km, which is expected to be good enough for simulation purposes. With regards to the on board orbit model, it is expected that a new TLE will be uploaded once a week to correct the drift. Please note, that the Newtonian orbit model is not used for on board purposes. B.3 Solar radiation Figure B.4 shows the implementation of the solar radiation disturbance model in Simulink. Figure B.4: Simulink model of the radiation disturbances. The block diagram uses another block, exposed area, which calculates the area which is exposed to the Sun. B.3.1 Verification The Simulink model of the radiation disturbances is proved to be correctly implemented by running it with several specific values in two concrete situations. Assuming the absorption coefficient C a = 1.4, the minimum and maximum values of the force and the torque acting on the satellite due to the influence of the Sun are calculated as: F Rmin = C a Pa 2 = kg m s 2

155 Group 830a B.4 Atmospheric drag 143 N solarmin = nnm F Rmax = C a P3 1 3 a 2 = kg m s 2 N solarmax = nnm The radiation model is verified by comparing the results obtained from calculation and simulation, they are found to be equal, which concludes that the solar radiation disturbance model can be used in the environment simulation. B.4 Atmospheric drag A schematic of the atmospheric drag model can be seen in figure B.5. The three inputs of the model are: The velocity of the satellite given in the ECI frame. The attitude of the satellite The location of the center of mass in the SC frame The outputs generated by the model are: The torque generated in the SC frame. The force produced in the ECI frame. 3 R_com(SC) 1 q(i >SC) quaternion exposed area vector Exposed area Product 1 N_atm(SC) 2 V_sc(I) MATLAB Function V 2 Square u v To the power q q Vq V Vector rotation by q 9.25e 13 In to unit vector Out Product1 Product2 Product3 Product4 1/2 Gain 2 F_atm(I) Density 2.2 C_D Figure B.5: Schematic of the aerodynamic drag model implementation in Simulink.

156 144 Chapter B Implementation of simulation Attitude Control System for AAUSAT-II Verification Since there is not measured data available it is not possible to verify the model correctly using real data. It is important to notice that the model can be verified. A test in a wind tunnel would be a suitable choice but its realization is not possible since the testing facilities are not available. B.5 Gravitational disturbances The disturbances introduced by gravitational forces is implemented in Simulink. The disturbance model takes the following input: Position of spacecraft Position of the sun Position of the moon Mass of the spacecraft Inertial tensor matrix Orientation of spacecraft Orientation of the principal frame The two outputs are the force and torque disturbances on the spacecraft. B.5.1 Gravitational disturbance forces The modeled forces acting upon the satellite are originating from the Moon and the Sun and the zonal harmonics of the Earth. The model implemented in Simulink is depicted in figure B.6. B.5.2 Gravitational disturbance torque The modeled disturbance which generates torque is the gravity gradient of the spacecraft. The Simulink model is depicted in figure B.7. B.5.3 Verification The point mass disturbances have been compared with calculations with respect to objects of the same mass and distance from each other. The zonal harmonics have not been extensively tested since no actual data could be found to compare it to.

157 Group 830a B.5 Gravitational disturbances 145 Figure B.6: Gravitational disturbance force Simulink implementation. Figure B.7: Gravitational disturbance torque implementation in Simulink.

158 146 Chapter B Implementation of simulation Attitude Control System for AAUSAT-II The gravity gradient torque has been tested to yield the same results as the worst case calculation in section on page 61. B.6 Magnetic residual The effect of the residual magnetic dipole generated by the electronics in the satellite has been modeled in Simulink. A schematic of the model is depicted in figure B.8 The four inputs to the model are: Magnetic dipole moment The spacecraft attitude The Earth rotation The magnetic field vector in the ECEF frame The single output is the torque generated in the SC frame. 1 Magnetic electronics dipole, d in(sc) 2 Attitude q(i >SC) MATLAB Function cross product 1 N_mres(SC) 3 Earth rotation q(i >E) q(i >SC) q(i >E) B(E) B(SC) 4 Magnetic field B(E) Magnetic field E >SC1 Figure B.8: Schematic of the magnetic residual model implementation in Simulink. B.6.1 Verification It has not been possible to verify the model since no data is available for the specific case treated in this project. But by following the structure given in [26], the model is assumed to be correct as the implementation is very simple.

159 Group 830a B.7 Spacecraft dynamics and kinematics 147 B.7 Spacecraft dynamics and kinematics The dynamic and kinematic model of the spacecraft has been implemented in Simulink as an S-function programmed in C. As the dynamics and kinematics are calculated in the principal frame, all inputs has to be converted to this frame. The advantage of using S-functions is that the equations of motion can be implemented directly as ordinary differential equations, which solved by Matlab using numerical integration methods. Furthermore the C implementation of S-functions causes the code to be compiled to native machine code and therefore execute faster, which is important for the simulation to run smoothly. The resulting Simulink diagram can be seen in figure B.9, and the S-function source code can be found on the enclosed CD-ROM. 1 Initial Values W_i(SC) q q Vq V Vector rotation by q 3 Principal frame q(sc >P) q q_opposit Reverse quaternion rotation 5 Actuator N_act(sc) 6 Actuator H_mw(SC) 7 Disturbance N_dist(SC) q q Vq V Vector rotation by q1 q q Vq V Vector rotation by q2 q q Vq V Vector rotation by q3 scdynamics sc dynamics q q Vq V Vector rotation by q4 q1 q_res q2 Rotation of q1 by q1 2 Angular rate W(SC) 1 Attitude q(i >SC) 4 Spacecraft Inertia I(P) 2 Initial Values q(i >SC) q1 q_res q2 Rotation of q1 by q2 Angular velocity estimated quaternion Initial quaternion w to q Figure B.9: The spacecraft dynamics and kinematics implementation in Simulink. B.8 Magnetorquers Two main blocks are developed in Simulink, the first one is a model of the magnetorquer driver, shown in figure B.10, and the second one is the model of the magnetorquer, represented in figure B.11. They are both used three times, once for each coil. The input to the magnetorquer driver is the magnetic field of the Earth and the torque required from the control system. Both are given in the spacecraft reference frame. The output of this block is the necessary current to be applied to the specific coil in order to obtain the torque required.

160 148 Chapter B Implementation of simulation Attitude Control System for AAUSAT-II 1 N_mt(SC) Dot Product N N Product1 K Gain 1 I out 2 B (SC) MATLAB Function MATLAB Fcn MATLAB Function V u v Math Function 1 Constant Figure B.10: Model of the magnetorquer driver This output from the magnetorquer driver is used as input to the magnetorquer model block. The magnetorquer model outputs the torque applied to the spacecraft. 1 K I Gain 1 N N MATLAB Function MATLAB Fcn Product N_mt(sc) 2 B(SC) Figure B.11: Model of the magnetorquer B.8.1 Verification To verify the model the required value of the torque, given as the input, is compared with the value that is obtained as the output, figure B.12. Granted that the direction of the torque is perpendicular to the direction of the magnetic field of the Earth, the compared values are the same. B.9 Momentum wheels The model for a DC motor is shown on figure B.13. In the implemented Simulink model a voltage and current saturation are considered in this way the limitations in the dynamics of the motor due to the driver are taken into account.

161 Group 830a B.9 Momentum wheels 149 1e 06 1e 06 0 Display2 1e 06 1e 06 0 Display4 (1 1 0) N_mt Constant I out I in (0 0 1) Constant3 B(SC) Magnetorquer drivers Saturation N_mt(SC) B(SC) Magnetorquers 1 N_act(SC) 0 0 4e 05 Display3 Figure B.12: Verification of the magnetorquers model 2 N_out 3 1 V_in Voltage saturation e 1 den(s) Rotor impedance I K Current N mechanical m saturation motor constant 1 s Integrator K inverse Inertia ω H_out 1 w_out K Viscous friction Config.Km Electrical motor constant Figure B.13: Simulink Block diagram of a DC motor, including voltage and current saturation limits.

162 150 Chapter B Implementation of simulation Attitude Control System for AAUSAT-II Verification The Simulink implementation is verified against the parameters from the datasheet for the Maxon RE10 DC motor[22]. The parameters used are shown in table B.4. Rotor inertia, J r kgm 2 Motor constant, K m Nm A Rotor resistance R 8 Ω m Rotor inductance L H 9 kgm2 Friction Coefficient b f 7 10 s Table B.4: Parameters used for verification of the DC motor model. Since the actual motors has not been delivered yet, a simple speed voltage relation rad sv and the mechanical time constant t m = 7e 3 s, taken from the motor datasheet [22], are used to verify the implementation. Previous experience with maxon motors reveals that the datasheet parameters are very close to the actual parameters. But it is still important to verify the model using the actual motors RE W impulsresponse 3V 1000 ω ss = angular velocity ω % of ω ss t m s time in seconds Figure B.14: Verification of motor model using parameters from the datasheet. In figure B.14, the impulse response of the motor is, with an input step of 3 V at t = 0, is shown. The ω ss = = rad s is reached in steady state, and the mechanical time constant is matched. Therefore the implementation has been verified, but the real motor parameters needs verification.

163 APPENDIX C Interface Control Document The Attitude Determination and Control System (ADCS), is responsible for the control of the satellites attitude and the determination of the orbit, attitude and references to different objects like the Sun, Moon and Earth. To make the ADCS fit in the satellite, it is needed to identify and specify all interfaces (IF) regarding the ADCS internally and interfaces to other subsystems. This is done throughout this Interface Control Document. OBC: FP ACS ADS INSANE PSU ACS Actuators ADS Sensors Figure C.1: Block diagram of the ADS and ACS subsystems including external subsystems needed for operation. A part of the overall satellite block diagram is shown in figure C.1. It shows which subsystems the ADCS interfaces to. C.1 System modes This section contain a description of the different modes of the spacecraft. Then the common modes of ACS and ADS are described. The definitions of the distinct modes of ACS and ADS are described in the end of this section. The following list contains the different system modes the ADCS uses. Off The off mode is when the subsystem is powered off. Ready Off This mode prepares the subsystem to be switched off, if the subsystem could benefit from a gracefull shutdown. It is not required for a subsystem to implement this mode, as

164 152 Chapter C Interface Control Document Attitude Control System for AAUSAT-II it is only a service. Maintenance This mode is used to upload new software or data to the subsystem and verifying the software. Operating This is the standard mode, where the subsystem is operating and performing it s defined task(s). While in this mode, it must be possible to control the different parts of the subsystem. Debug This is the same mode as Operating, but includes a verbose debugging mode for extended gathering of telemetry. The descriptions of the internal operating modes of ACS and ADS are in the following sections. C.1.1 ACS operating modes The different operating modes of the ACS are Idle The ACS is idle and waiting for a command from the flight planner. Detumbling the satellite by the means of magnetorquers, magnetometers and the Ḃ-algorithm. This mode must be very robust and simple. Pointing The ACS is acquiring a requested attitude and maintaining this attitude. This mode has three sub-modes: Attitude control The ACS is acquiring a requested attitude and maintaining this attitude. Direction control The ACS is pointing the payload in a direction given by a vector. Autonomous pointing The ACS is pointing the payload in the direction of a target, which is continuously calculated by the ADS. C.1.2 ADS operating modes The different operating modes of ADS are Detumbling In the detumbling mode, the ADS waits for commands from the FP or ACS to make specific measurements. This mode is used, when the satellite is detumbled. Pointing When the ADS is in this mode, it will continuously determine the attitude of the spacecraft by using all available sensors, an Extended Kalman Filter and the q-method. This is the primary operating mode of the ADS. The ADS will by default boot in detumbling mode and shift to the pointing mode when commanded by the FP.

165 Group 830a C.2 External Interfaces 153 C.2 External Interfaces A definition of all the external interfaces to and from ACS and ADS is stated, in this section. C.2.1 ACS external IF Power interface The ACS needs 3 power channels from EPS. The different power channels are described below. 3.3 V channel to supply the necessary voltage for MCU and tachometers on the motors. 3.3 V channel to supply the necessary voltage for the momentum wheels. 5 V channel to supply the necessary voltage for the magnetorquers. The expected power usage is shown in the table C.1, where a list of power usages for all parts of the ACS is shown. Operation mode Power usage mw Component supply voltage V average minimum peak Detumbling Micro controller Magnetorquers Pointing Micro controller Magnetorquers Momentum wheels Off Table C.1: Power budget for ACS, approximated number with a small margin of 5 10% Mechanical interfaces The interface to the MECH subsystem is as follows. Magnetorquers 3 magnetorquers are needed, each one is mounted on the inner side of the satellite, such that the magnetorquers is mounted perpendicular to each other. The coils have the following dimensions: height= mm, outer width= mm and depth= 2 mm. Each of the coils can have a mass of 15 g plus mounting. Momentum wheel 3 momentum wheels must be mounted in the satellite, perpendicular to each other. Each momentum wheel can have a mass of 25 g, including flywheel, motor and sensor.

166 154 Chapter C Interface Control Document Attitude Control System for AAUSAT-II The total mass budget of the ACS is shown in table C.2. Component Weight Component Weight Motor 1 7 g Mountings 10 g Motor 2 7 g Magnetorquer 1 15 g Motor 3 7 g Magnetorquer 2 15 g Flywheel 1 6 g Magnetorquer 3 15 g Flywheel 2 6 g Electronics 40 g Flywheel 3 6 g Tacho 1 2 g Tacho 2 2 g Tacho 3 2 g Total 140 g +10% Margin 160 g Table C.2: Mass budget for the ACS system. House keeping interface House keeping data is necessary to evaluate the performance of the ACS. There is a list of different informations, which is to be broadcasted on the INSANE (see [19]), together with the frequency of the broadcasts. Message ID Description Value frequency ACS.TM.1 Angular velocity of momentum wheels ω mw 20 Hz ACS.TM.2 Angular momentum h mw 10 Hz ACS.TM.3 Control torque N mw 1 Hz ACS.TM.4 Control dipole moment m mt 1 Hz ACS.TM.5 Fault message f i Table C.3: Telemetry package list including ID, description, value name and frequency of the telemetry message. C.2.2 ADS external IF ADS needs 2 power channels from EPS: 3.3 V channel for the MCU 5 V channel to supply the necessary voltage for the magnetometer, gyros and the amplifiers. The expected power usage is shown in table C.4.

167 Group 830a C.3 Internal Interfaces 155 Operation mode Power usage mw Component supply voltage V average minimum peak Detumbling Micro controller Magnetometer 5.0 Amplifiers 5.0 Pointing Micro controller Gyros 5.0 Magnetometer 5.0 Amplifiers 5.0 Off Table C.4: Power budget ADS, approximated number with a small margin of 5 10% C.3 Internal Interfaces The definitions of all internal interfaces between ACS and ADS, together with a description of interfaces inside each subsystem, is explained in this section. C.3.1 ADS to ACS interfaces ACS requires different types of data from ADS to be able to perform the control task. All the necessary internal and external interfaces to provide these data are stated in this section. The interfaces are divided in two groups, due to different operating modes of ADS and ACS, see section C.1.1. Detumbling To use the Ḃ-algorithm, it is necessary for ACS to be provided with the strength of the Earths magnetic field, which must be measured by the magnetometer in ADS. The interface from ADS to ACS in this mode is shown on table C.5. Pointing The ACS needs the attitude of the satellite and the angular velocity of the spacecraft in each axis to control the satellite in this mode. A full description is shown in table C.6.

168 156 Chapter C Interface Control Document Attitude Control System for AAUSAT-II Interface name Description Implementation ACS.Detumble.1 The Earths magnetic field strength S B, measured in the spacecraft frame. The interface is implemented by a message on the INSANE, consisting of S B. The measurement of S B must be done only with magnetometers, and with a sampling frequency of 0.1 Hz. If the magnetometer fails, then the S B must be estimated by all means possible. This is the last resort as it is a critical issue in the mission to get the satellite detumbled. Table C.5: IF specification for the detumble operating state of the ACS. Interface name Description Implementation Interface name Description Implementation Interface name Description Implementation ACS.Pointing.1 To point at a given object in the space, the subsystem is given a quaternion T q. Then it is necessary to know the quaternion S Iq, which de- I scribes the rotation of the spacecraft in relation to the inertial reference frame. The ADS system estimates the position and rotation of the spacecraft and transfers the quaternion S q to the ACS through the INSANE. The I sample rate is 1 Hz. ACS.Pointing.2 The spacecrafts angular rate S ω, is needed to stabilize the satellite around 3 axes. The angular rate measurement is transfered through the INSANE with an accuracy of rad and a sample rate of 1 Hz. s ACS.Pointing.3 The vector S R target to the target is needed to make autonomous pointing control of the satellite. The vector is continuously calculated by ADS and transfered to the supervisor controller, which uses the reference to generate the right control quaternion for the attitude controller. Table C.6: IF specification for the pointing operating state of the ACS.

169 Group 830a C.3 Internal Interfaces 157 C.3.2 ACS to ADS interfaces All necessary internal interfaces from ACS to ADS, are stated in this section. The IFs are divided into groups of operating modes of ADS. Detumbling It is needed for this mode to schedule the usage of the magnetometer and the magnetorquers. This is done by the IF defined in table C.7. Interface name Description Implementation ADS.Detumbling.1 To make sure the magnetometer measurements are not corrupted, due to detumbling by the magnetorquers, a scheduling message is sent from ACS on the INSANE. An INSANE message sent from the ACS saying when ACS is not using the magnetorquers. ADS will respond to ACS when it has measured S B. Table C.7: The Interface description for detumbling mode. Pointing Some measurements from the ACS are needed for the Kalman filter to work properly. In table C.8, there is a description of the different requirements the ADS has towards the ACS.

170 158 Chapter C Interface Control Document Attitude Control System for AAUSAT-II Interface name Description Implementation Interface name Description Implementation Interface name Description Implementation ADS.Pointing.1 To be able to estimate the attitude,the EKF needs to know h mw and ḣmw for the kinematic and dynamic linearization. The ACS observes h mw and ḣmw generated with the momentum wheels and transfers it to ADS through the INSANE. ADS.Pointing.2 To predict, the Extended Kalman Filter (EKF) needs to know h mw for the Kinematic/Dynamic linearization (see linearization chapter). The ACS obeserves the angular momentum h mw in the momentum wheels and transfers it to ADS through the INSANE. ADS.Pointing.3 To predict, the Extended Kalman Filter (EKF) needs to know N mw the motor torque. The torque is determined and ACS transfers it to ADS through the INSANE. Table C.8: IF specification between ACS and ADS for the Pointing mode.