Optimal Thruster Actuation in High Precision Attitude and Orbit Control Systems

2005:310 CIV EXAMENSARBETE

Optimal Thruster Actuation in High Precision Attitude and Orbit Control Systems

HENRIK JOHANSSON

MASTER OF SCIENCE PROGRAMME in Space Engineering

Luleå University of Technology Department of Space Science, Kiruna

2005:310 CIV • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 05/310 - - SE I e c h n i k

Optimal Thruster Actuation in High Precision Attitude and Orbit Control Systems

Master’s thesis by Henrik Johansson Master of Science in Space Engineering Department of Space Science, Kiruna Lule˚aUniversity of Technology, Sweden

September 2005

Examiner: Priya Fernando Department of Space Science, Kiruna Lule˚aUniversity of Technology, Sweden

Supervisors: Dipl.-Phys. Silvia Scheithauer ZARM University of Bremen, Germany

Dipl.-Ing. Alexander Schleicher ZARM University of Bremen, Germany

ABSTRACT

The increasing demand for high precision spacecraft attitude and orbit maneuvers puts very strict requirements on thrusters and their control. Best suited for this task are proportion- al thrusters able to produce precise micronewton thrust levels with very little noise. Such thrusters under current development are cold gas microthrusters, field emission electric propul- sion (FEEP), colloid thrusters and micro-resistojets, with ion engines and Hall thrusters having attractive properties should a miniaturization be possible. Optimal utilization of proportional thrusters can be achieved by minimizing the 1-norm, 2-norm or ∞-norm of the thrust command vector, resulting in a minimum flow rate controller, a minimum power controller or a minimum force controller respectively. The first and last are found by solving linear programs, the middle by using the pseudoinverse of the thruster configuration matrix along with a bias. The control authority, which is the maximum performance of a thruster system, can be found by maximizing the force and torque output. A single number, referred to as the minimum control authority, measures the weakest output of the thruster system. All of these concepts are given a thorough review, and the thesis rounds off by implementing them on the LISA Pathfinder mission. Cal- culations show the algorithms to work well, but a more efficient way of finding the minimum control authority is desirable.

Page I

SAMMANFATTNING

(Abstract in Swedish)

Det ¨okande kravet p˚ah¨ogprecisions attityd- och banreglering av rymdfarkoster s¨atter str¨anga krav p˚astyrraketer och deras styrning. B¨ast l¨ampade f¨or denna uppgift ¨ar proportionella styr- raketer som noggrant kan producera krafter p˚amikronewtonniv˚amed mycket lite brus. S˚adana styrraketer under nuvarande utveckling ¨ar kallgas-mikrostyrraketer, elektrisk f¨altemissionsfram- drivning (FEEP), kolloid-styrraketer och mikroelektrotermiska styrraketer, med jonmotorer och Hall-styrrakter som har attraktiva egenskaper om en f¨orminskning ¨ar m¨ojlig. Optimalt utnytt- jande av proportionella styrraketer kan uppn˚asgenom att minimera 1-, 2-, eller ∞-normen av styrvektorn, vilket respektive resulterar i en minsta fl¨odeskontroller, en minsta effektkontroller och en minsta kraftkontroller. Den f¨orsta och sista erh˚allsgenom att l¨osa linj¨ara program, den mittersta genom att anv¨anda pseudoinversen av styrraketernas konfigurationsmatris tillsammans med en bias. Kontrollauktoriteten, vilket ¨ar den maximala prestandan fr˚anett styrraketssystem, kan hittas genom att maximera kraft- och momentprestationen. Ett enda nummer, kallat den minsta kontrollauktoriteten, m¨ater den svagaste prestationen fr˚anstyrraketssystemet. Alla dessa koncept ges en noggrann genomg˚ang,och avhandlingen avrundar genom att implementera dem p˚aLISA Pathfinder-missionen. Ber¨akningar visar att algoritmerna fungerar bra, men en mer effektiv metod f¨or att hitta den minsta kontrollauktoriteten ¨ar ¨onskv¨ard.

Page III

Contents

1 INTRODUCTION 1

2 PROPORTIONAL THRUSTERS 3 2.1 Cold Gas Thrusters ...... 3 2.1.1 Current Status ...... 4 2.1.2 Evaluation ...... 4 2.2 Field Emission Electric Propulsion ...... 5 2.2.1 Current Status ...... 6 2.2.2 Evaluation ...... 6 2.3 Colloid Thrusters ...... 6 2.3.1 Current Status ...... 7 2.3.2 Evaluation ...... 8 2.4 Ion Engines ...... 8 2.4.1 Current Status ...... 8 2.4.2 Evaluation ...... 9 2.5 Hall Thrusters ...... 10 2.5.1 Current Status ...... 11 2.5.2 Evaluation ...... 11 2.6 Resistojets ...... 11 2.6.1 Current Status ...... 12 2.6.2 Evaluation ...... 12

3 COMMONLY USED CONTROL METHODS 15 3.1 Minimum Power Controller ...... 17 3.1.1 Finding a Null Space Vector ...... 17 3.1.2 Fixed Bias ...... 20 3.1.3 Dynamic Bias ...... 20 3.1.4 Iterative Approach ...... 21 3.2 Minimum Flow Rate Controller ...... 23 3.3 Minimum Force Controller ...... 23 3.4 Numerical Examples in 2-D ...... 24 3.4.1 Test Case 1 ...... 24 3.4.2 Test Case 2 ...... 28 3.4.3 Test Case 3 ...... 31

4 THE CONTROL AUTHORITY 35 4.1 Definition ...... 35 4.1.1 Control Authority Surface ...... 36 4.1.2 Finding the Control Authority ...... 37

Page V Contents

4.2 Control Authority Plot ...... 38 4.2.1 Examples of the Control Authority Plot ...... 40 4.3 Minimum Control Authority ...... 44

5 IMPLEMENTATION - LISA PATHFINDER 47 5.1 About LISA Pathfinder ...... 47 5.2 Actuator System ...... 47 5.3 Implementation and Results ...... 48 5.3.1 Profile 1 ...... 49 5.3.2 Profile 2 ...... 50 5.4 Evaluation ...... 50

6 CONCLUSIONS 55 6.1 Limitations ...... 55 6.2 Future Work ...... 55

Page VI List of Figures

2.1 A very simple cold gas thruster, consisting of a propellant tank, valve and nozzle. 4 2.2 The two types of emitter designs for FEEP thrusters...... 5 2.3 A simple schematic of the principle behind the single capillary colloid thruster. . 7 2.4 A schematic showing the principle behind the operation of an electron bombard- ment ion thruster. Magnetic rings are incorporated into the design in order to reduce primary electron mobility...... 9 2.5 Cross section of an axis-symmetric Hall thruster. B denotes the magnetic field. . 10 2.6 A simple schematic of a conventional resistojet...... 12

3.1 The iterative control scheme. The letter z denotes the shift operator...... 21 3.2 Gain simulation for the test case 1 discussed in section 3.4.1. The plot was made by recording the number of iterations required for each gain to reach convergence for 2000 randomized force and torque vectors, whereafter the average number of iterations for each gain was plotted against the gain...... 22 3.3 Test case 1. The directions of the resulting forces from the three thrusters are ◦ denoted by T1, T2 and T3. The angle between each thruster is 120 ...... 25 3.4 Test case 2. Four thrusters capable of producing both forces and torques. The arrows point in the direction of the resulting forces from thrusters T1 through T4, and the length of the lever arms is 1 length units...... 28 3.5 Test case 3. Five thrusters in a non-symmetric configuration, capable of producing both forces and torques. The arrows point in the direction of the resulting forces from thrusters T1 through T5, and the length of the lever arms is 1 length units. 31

4.1 The triangle represents the control authority of the thruster configuration in test case 1. It was made assuming that the thrusters could be turned completely off, Tlow = 0, and that the flow limit was equal to 1...... 36 4.2 The control authority for test case 1 using a minimum force controller with a force limit. The three shapes represent three different algorithms for finding the thrust command vector. Calculations were made assuming that the thrusters could be turned off completely, Tlow = 0, and that the force limit was equal to 1...... 38 4.3 The control authority for test case 1 using a minimum flow rate controller with a flow rate limit, T1,1, a minimum power controller with a power limit, T2,2, and a minimum force controller with a force limit, T∞,∞. The figure was made assuming that the thrusters could be turned completely off, Tlow = 0, and that the limit for each controller was equal to 1...... 39 4.4 The control authority surface of test case 2 in section 3.4.2 using a minimum flow rate controller with a flow rate limit of 1...... 41

Page VII List of Figures

4.5 The control authority plot made from the control authority vectors used to make the control authority surface in figure 4.4. Each cross represents a control author- ity vector. A line marking the minimum control authority plot has been manually added to the plot...... 41 4.6 The control authority surface of test case 2 in section 3.4.2 using a minimum power controller with a power limit of 1 and the fixed bias algorithm...... 42 4.7 The control authority plot made from the control authority vectors used to make the control authority surface in figure 4.6. Each cross represents a control au- thority vector, and a line marking the minimum control authority plot has been manually added to the plot...... 42 4.8 The control authority surface of test case 2 in section 3.4.2 using a minimum force controller with a force limit of 1...... 43 4.9 The control authority plot made from the control authority vectors spanning the control authority surface in figure 4.8. Each cross represents a control authority vector. The minimum control authority plot is marked by straight lines, and has been manually added to the plot...... 44 4.10 Simple overview of the principle behind finding the minimum control authority. The inner loop finds the control authority with a maximizing linear program, while the outer loop finds the minimum of the control authority...... 45

5.1 Artist conception of the LISA mission, which will consist of three spacecraft, fly- ing a formation in the shape of a triangle having a side length of five million kilometers. LISA Pathfinder, on the other hand, will be just two spacecraft, and the distance between them a lot shorter. (Courtesy of ESA.) ...... 48 5.2 The position and orientation of the FEEP thrusters on LISA Pathfinder...... 49 5.3 The distribution and relative position of the FEEP and cold-gas thrusters. . . . . 49 5.4 The control authority plot for LISA Pathfinder, assuming each thruster opera- tional and having a thrust generating capability between 0.1 µN and 150 µN. The minimum control authority is 1.5 · 10−4, and in the plot it is marked by an arc. . 50 5.5 The length of the force and torque command vector in the first disturbance pro- file compared with the control authority of the thruster system in that direction. Vertical bars mark regions where the control authority is exceeded...... 51 5.6 The demanded forces and torques in the first disturbance profile plotted with the achieved forces and torques. The vertical bars cover the same regions as the ones in figure 5.5. Clearly the thrusters are unable to supply the thrust required when- ever the control authority is exceeded...... 52 5.7 The length of the force and torque command vector in the second disturbance profile compared with the control authority of the thruster system in that direction. Vertical bars mark regions where the control authority is exceeded...... 53 5.8 The demanded forces and torques in the second disturbance profile plotted with the achieved forces and torques. The vertical bars cover the same regions as the ones in figure 5.7. Note that the thrusters are unable to supply the thrust required whenever the control authority is exceeded...... 54

Page VIII List of Tables

2.1 A summary of the approximate span of performance for the various thrusters. Sources: a: [12] [20]; b: [12] [16]; c: [12] [16] [20]; d: [12] [20]; e: [12] [20]; f : [12] [20] [22] 3

3.1 Results for test case 1 when using the minimum power controller and a fixed bias. Note that some thrust vector elements are below the lower boundary, Tlow. . . . . 26 3.2 Results for test case 1 when using the minimum power controller and a dynamic bias...... 26 3.3 Results for test case 1 when using the minimum power controller and the iterative approach. The tolerance used in the control loop is 1E-6...... 27 3.4 Results for test case 1 when using the minimum flow rate controller...... 27 3.5 Results for test case 1 when using the minimum force controller...... 28 3.6 Results for test case 2 when using the minimum power controller and a fixed bias. 29 3.7 Results for test case 2 when using the minimum power controller and a dynamic bias...... 30 3.8 Results for test case 2 when using the minimum power controller and the iterative approach. The tolerance used in the control loop is 1E-6...... 30 3.9 Results for test case 2 when using the minimum flow rate controller ...... 30 3.10 Results for test case 2 when using the minimum force controller ...... 31 3.11 Results for test case 3 when using the minimum power controller and a fixed bias. 32 3.12 Results for test case 3 when using the minimum power controller and a dynamic bias...... 33 3.13 Results for test case 3 when using the minimum power controller and the iterative approach. The tolerance used in the control loop is 1E-6...... 33 3.14 Results for test case 3 when using the minimum flow rate controller...... 33 3.15 Results for test case 3 when using the minimum force controller...... 34

4.1 The nine possible combinations of controllers and limits...... 37

Page IX

List of Acronyms

AOCS Attitude and Orbit Control Systems ARCS Austrian Research Center in Seibersdorf Cs Caesium EADS European Aeronautic Defence and Space Company ESA European Space Agency FEEP Field Emission Electric Propulsion FMMR Free Molecule Micro-Resistojet Hz Hertz In Indium kV Kilovolts kW Kilowatts LISA Laser Interferometer Space Antenna MEMS microelectromechanical systems MIT Massachusetts Institute of Technology µN Micronewton µNm Micronewtonmeter mN Millinewton N Newton N/A Not Available NASA National Aeronautics and Space Administration Rb Rubidium RF Radio Frequency s Seconds SMART Small Missions for Advanced Research in Technology TPF Terrestrial Planet Finder W Watts VLM Vaporizing Liquid Microthruster Xe Xenon

Page XI

Nomenclature

0 Vector of zeros α Angle β Angle A Thruster configuration matrix B Magnetic flux F Force and torque vector

Fc Commanded force and torque vector

Fd Desired force and torque vector

Fr Resulting force and torque vector f Vector of ones

Fx Force along x-axis

Fy Force along y-axis G Gain I Vector of ones I Identity matrix

Isp Specific impulse k Iteration step

Mz Moment around z-axis m Number of degrees of freedom

m1 Flow rate limit

m2 Power limit

m∞ Force limit n Number of thrusters R Rescaling factor T Thrust command vector

T1 Thrust command vector with a minimized 1-norm

T2 Thrust command vector with a minimized 2-norm

T∞ Thrust command vector with a minimized ∞-norm

Tc Thrust command vector with all vector elements ≥ 0

Tp Non-biased thrust command vector

Tpc Pseudoinverse biased thrust command vector

Page XIII List of Symbols

Tlow Lower thruster force boundary

Tup Upper thruster force boundary

Tn Null space vector V A vector z A scalar

(−)i Index

(−)F Force part of vector

(−)M Torque part of vector

(−)max Maximum vector element

(−)min Minimum vector element

(−)p Type of thruster limit

(−)s Type of controller

Page XIV 1 INTRODUCTION

Attitude and orbit control is a vital part of spacecraft operation. Without it, the spacecraft would be unable to face the direction it is supposed to face, and it would not be able to maintain its intended orbit or trajectory.

There are different means of controlling the attitude of a spacecraft. Assuming a three-axis- stabilized spacecraft, there are several ways of achieving a complete rotational control around all axes. These are thrusters, which act externally, and reaction wheels, momentum wheels or control moment gyroscopes, which act internally. Orbit adjustments can be done by using thrusters, as well as by utilization of aerodynamic and solar pressure forces. Precise orbit con- trol, however, can only be done by thrusters, and they are thus the only option for performing both attitude and orbit control in a system that requires a very high precision.

In the past, thrusters have often been of a pulse type. By exerting a short burst of thrust, the momentum of the spacecraft was altered. Indeed, this method of operation did not make for instance a geosynchronous communication satellite truly stationed in space relative to the Earth, but rather it moved around in a pattern within a confined volume of space.

Lately, there have been a growing need for higher pointing accuracy and drag-free control within the space community. This necessity has mainly arisen from several upcoming high precision requiring missions, such as LISA (detection of gravitational waves), Darwin (looking for signs of life on Earth-like planets) and the Terrestrial Planet Finder (TPF) (studying planets outside our solar system). All of these use interferometry payloads with attitude and position maintenance requirements that far exceed conventional limits. The demand lies not only in fine pointing maneuvers, but also in very low noise levels, and the requirements are strict enough to rule out the use of internal wheels and gyros.

When applying thrusters to high precision spacecraft control, a pulse mode of operation is no longer adequate to fulfill the task. Even if a very short pulse could be achieved, generating a very small impulse bit, the pulses themselves may induce enough vibration to make the payload unable to perform its task. And to perform drag-free control, the thrusters need to counteract various disturbances, such as the solar radiation pressure and solar wind pressure, which act upon the spacecraft for a longer period of time. Therefore, proportional thrusters, able to pro- duce a continuous thrust within a given thrust interval, are required. And the force exerted by the thrusters must be extremely small, on the scale of micronewtons. Since currently available hardware generally supplies a thrust much larger than that, this creates a demand for the de- velopment of smaller and more precise thrusters.

Not only the thrusters themselves are important for precise maneuvers, but how they are con- trolled is too. Deciding what combination of thrusters to fire, and how much force they should apply, is vital for achieving the intended purpose. It is highly desirable to optimize this pro- cedure so that energy is not needlessly wasted. Recognizing that there are physical limitations

Page 1 to how much force and/or torque a thruster system can exert, it is necessary to analyze the boundary conditions of that system, so that it can be ascertained whether or not it is adequate for performing the envelope of expected maneuvers. Investigating the boundaries also makes it possible to compare different controllers, and view their respective advantages and disadvantages.

There has been previous work done in the area of thruster control, and this thesis is mainly based on the PhD theses by Jeng-Heng Chen [1], Peter Wiktor [29] and HaiPing Jin [10].

This thesis will give an overview of available proportional thrusters under development for high precision attitude and orbit control, as well as a couple that would be most attractive for such a use, provided that the design can be modified to accommodate the required thrust levels. After that, in chapter 3, there will be complete review of commonly used control methods that optimize the thruster utilization with respect to either propellant use, power or the maximum thrust. Chapter 4 follows by discussing the control authority - a valuable tool for examining the boundaries of a thruster system - as well as the minimum control authority, which constitutes the most important figure of merit when determining if a thruster system can accomplish all of its intended maneuvers. The last chapter, chapter 5, will implement knowledge from the previous chapters on a reference mission - the LISA Pathfinder.

Page 2 2 PROPORTIONAL THRUSTERS

There are many different kinds of thruster concepts out there. Some are well tested and have flown on several missions, while others are in various test phases or perhaps still undergoing feasibility studies. In order to limit the scope of thrusters covered in this thesis, only propor- tional thrusters will be discussed, and specifically those that have the potential to be used in high precision attitude and orbit control systems (AOCS) in the not too distant future. The proportionality constraint immediately rules out the various concepts working in a pulse mode. Pulsed thrusters may also give rise to induced vibrations that are intolerable on high precision missions. Proportional thrusters, on the other hand, are here defined as being able to supply a continuous thrust within a given interval, and in most cases there is also the option of turning the thruster off completely. The demand for high precision also calls for small thrust levels, millinewtons and below [20], as well as small impulse bits, tens of micronewtonseconds and be- low [20].

A table of the thrusters discussed in this chapter, as well as their approximate span of perfor- mance, can be seen in table 2.1.

Table 2.1: A summary of the approximate span of performance for the various thrusters. Sources: a: [12] [20]; b: [12] [16]; c: [12] [16] [20]; d: [12] [20]; e: [12] [20]; f : [12] [20] [22]

2.1 Cold Gas Thrusters

The basic notion behind the design and function of cold gas thrusters is very simple: A gas is stored under pressure and opening a valve expels it through a nozzle, producing thrust, see figure 2.1. No chemical reaction takes place and the only electrical power required is basically for thermal control and for control of the valve mechanism. The propellant used can be virtually any gas, but the by far most frequently used one is nitrogen, which represents a reasonable trade-off between propellant storage density, performance and lack of contamination concerns [20]. Using nitrogen, the specific impulse (Isp) lies roughly in the area between 60 and 70 seconds [20]. Higher Isp is possible to achieve using lighter gases, such as helium (∼ 150 s) [20] [24], but the low molecular weight of such gases puts higher constraints on storage, resulting in a heavier

Page 3 2.1. COLD GAS THRUSTERS tank. Indeed, leakage concerns are one of the main drawbacks of cold gas systems, and leakage may even increase with time as various contaminations accumulate in sensitive areas, particulary in the valve mechanism.

P r o p e l l a n t T a n k

V a l v e N o z z l e

Figure 2.1: A very simple cold gas thruster, consisting of a propellant tank, valve and nozzle.

2.1.1 Current Status Several cold gas thrusters are available and they have been flown on several missions. These, however, generally supply too much thrust and low Isp for high precision missions. A develop- ment of interest here are cold gas microthrusters. In this area there are a couple of promising developments currently underway. One of those is currently formed by Marotta U.K. It is a piezo-actuated nitrogen propelled proportional thruster with a thrust range from 0 to 1000 µN and a specific impulse of about 70 s [15]. Another interesting example is a microelectromechan- ical systems (MEMS)-based cold gas thruster under development by ACR in Sweden. MEMS technology offers new and exciting possibilities, as well as many challenges, for miniaturizing thrusters.

Both the ACR and the Marotta thruster heat the propellant before expelling it. This way it is possible to increase the efficiency [11] of the thruster, as well as its Isp [12], which can be almost doubled. A summary of cold gas microthruster performance specifications can be seen in table 2.1

2.1.2 Evaluation Cold gas thrusters are considered to be a reliable technology. Some of their main advantages are their low power consumption, cleanness (if a benign propellant is used) and relative ease of miniaturization. Though it should be noted that in the case of MEMS-based cold gas mi- crothrusters, the development is uncomplete, and many challenges are still left to overcome.

When considering the disadvantages of a cold gas thruster system, two issues immediately stand out. These are propellant leakage and low specific impulse performance, both resulting in a large and heavy tank. The tank itself is also a source for spacecraft contamination, as microscopic metal flakes left over from fabrication are released from the tank surface and settle on sensitive spacecraft surfaces. For high precision applications another big disadvantage is varying thrust levels caused by pressure and temperature changes in the gas container during operation. These

Page 4 CHAPTER 2. PROPORTIONAL THRUSTERS unwanted characteristics can be counteracted by using pumps or a pressurant, with the cost of an increased system complexity.

Despite the concerns involved in cold gas technology it presents an attractive option for high precision AOCS. Especially the MEMS-based technology shows promise, provided it can over- come the design and scale challenges involved. As of yet, the data available on their performance and gas leakage is scarce.

2.2 Field Emission Electric Propulsion

Field emission of liquid metal ion sources have been studied since the late 1960s and have found a variety of applications. In a Field Emission Electric Propulsion (FEEP) system for space ap- plication thrust is generated by accelerating ions through an electrostatic field. The propellant used is a liquid metal, which is fed by capillary forces to an emitter tip, where the potential dif- ference causes a cone to form. When the electric field becomes high enough, electrons are ripped from the propellant atoms and the resulting ions are released from the emitter tip. Because the voltage required for this field emission is high, above 10 kV, the ions are expelled at a very high velocity, leading to specific impulses up to 10000 s [20], as well as an electric efficiency that is into the 90 % range [5]. Because positive ions are emitted from a FEEP thruster, a neutralizer needs to be added in order to avoid spacecraft contamination issues as well as spacecraft charging.

Currently there are two main approaches to FEEP design: A pin or needle type thruster, and a slit emitter. See figure 2.2 for a schematic overview of the different FEEP concepts. Slit emitters caries the advantage of an increased emitter area, which in turn yields higher thrust levels, while pin or needle type thrusters have to be clustered in order to obtain a comparable thrust level [19]. The pin or needle type of emitter can either use external wetting, as shown in figure 2.2a, or it can use internal wetting similar to the colloid thruster in figure 2.3.

Accelerator

P r o p e l l a n t Accelerator

P r o p e l l a n t E m i t t e r E m i t t e r

+ - - + + - - + (a) Pin or needle type emitter (b) Slit emitter

Figure 2.2: The two types of emitter designs for FEEP thrusters.

Page 5 2.3. COLLOID THRUSTERS

2.2.1 Current Status There are two main institutions working on FEEP thruster development: Austrian Research Center in Seibersdorf (ARCS) and ALTA in Italy. The two differ in design approach as well as the propellant used. ARCS favors the needle design with indium (In) propellant, while ALTA pursues the slit emitter using caesium (Cs) or rubidium (Rb) propellant. An In propellant carries an advantage due to its benign properties. In contrast, caesium is extremely reactive, making it difficult to handle.

A prototype testing of a multi-emitter In needle assembly showed a thrust of 0.1 - 100 µN per emitter with a minimum impulse bit of 5 nNs [5], which is a very small number. The specific impulse lied between 8000 and 12000 s, and the electrical efficiency was 95 % [5].

Cs slit emitters operate between similar specific impulses, 6000 to 12000 s [16]. A simple rule of thumb for Cs slit emitter thrust is 15 to 20 µN per millimeter slit [5]. Cs also has a slightly higher thrust-to-power ratio, 17 µN/W, compared to In thrusters with 15 µN/W [16].

Table 2.1 summaries the performance of FEEP thrusters.

2.2.2 Evaluation FEEPs offer several attractive features for high precision spacecraft control. They offer a high degree of throttling over the entire thrust range by adjusting the electrostatic field, as well as a precise thrust level control along with the ability to generate very small impulse bits. The performance is also excellent, with a very high specific impulse and electrical efficiency (about 90 %). The overall system efficiency is, however, lower (around 50 %).

The ability to store propellant in a solid state allows for not only compact storage, it also elim- inate sloshing disturbances and leakage concerns.

Something to be counted as a disadvantage for FEEP thrusters are their high voltage levels, which should preferably use dedicated power supplies. If a Cs or Rb propellant is used there is also the issue of spacecraft contamination to be considered. In test facilities, there has been reported an excessive Cs back-scattering [5].

2.3 Colloid Thrusters

The history of colloid thrusters dates back to the 1960s and 1970s, when they were first studied for spacecraft attitude control and drag makeup. They were, however, found to require too much power in order to produce a satisfactory thrust, which caused them to fall out of favor with the space community [20]. But with the recently developed need for micropropulsion for small spacecrafts, as well as precise attitude control with drag-free motion, colloid thrusters have sparked a new interest for space applications. See figure 2.3 for an illustration of a single capillary colloid thruster.

Colloid thrusters operate in a similar way to FEEP thrusters. But instead of accelerating in- dividual ions, fine charged liquid droplets are accelerated electrostatically and expelled from a

Page 6 CHAPTER 2. PROPORTIONAL THRUSTERS capillary. The liquid can be either positively or negatively charged. In fact, it is possible to design a cluster of colloid thrusters featuring both charges, causing them to cancel each other out, and thus eliminate the need for a neutralizer. This concept is called a bipolar thruster.

The performance of colloid thrusters is determined by the charge state of the droplets, the ac- celerating voltage, the propellant flow rate and the ion beam divergence. A commonly used propellent is glycerol (C3H5(OH)3) solvent doped with a solute, such as sodium iodine (NaI), which yields positive droplets. Negative droplets can for instance be produced using a sulfuric acid (H2SO4) dopant. The beam expelled from a colloid thruster has been found to consist of not only charged droplets, but also glycerine molecules, molecular ions and electrons [20]. A new and particularly attractive propellent option lies in ionic liquids, which are composed completely of ions and thus highly conductive.

Another similarity with FEEPs is that one single colloid thruster emitter usually is insufficient to produce an adequate thrust, and clustering in arrays is therefore necessary.

Positive Pressure Feed Extractor Plate

e - e -

+ - C a t h o d e

Figure 2.3: A simple schematic of the principle behind the single capillary colloid thruster.

2.3.1 Current Status In the early colloid thruster development, when high thrust was the main criteria, voltage differ- ences up to 100 kV were required. When the intended application is changed to fine maneuvers that require a small thrust, the acceleration voltage is lowered to orders of 4-20 kV [20]. There are, however, developments underway that aim for an even lower voltage. With the use of MEMS technology it has for instance been demonstrated to be possible to get a single emitter thrust from 1.36 µN to 4.85 µN out of a voltage between 1400 V and 2800 V [30]. Another project using microfabrication is underway at Massachusetts Institute of Technology (MIT). Their goal is to design a compact unit able to deliver thrust ranging from less than a µN up to one mN with the use of a bipolar ionic liquid ion source [18]. Tests have been carried out on externally wetted emitters (like the FEEP emitter shown in figure 2.2a) showing promise in the concept [14].

Work on micropropulsion colloid thrusters is also carried out in Europe, for example there is a collaboration between the Queen Mary University of London and the Rutherford Appleton Laboratory working on colloid thruster arrays [27].

Page 7 2.4. ION ENGINES

See table 2.1 for a summary of colloid thruster performance.

2.3.2 Evaluation

Colloid thrusters offer a high efficiency (70 to 75 %) and good performance in terms of Isp, even though it is not as high as for FEEPs. On the other hand, colloids generally have a higher thrust-to-power ratio. The thrust values supplied are well suited for precise AOCS.

Since the electrostatic fields in colloid thrusters carry a high voltage, it will typically require a dedicated power supply, which in many cases is a heavy affair. Colloids are also quite com- plicated to operate and build. In most designs, for example, a micro-pump is needed to put a positive pressure feed on the propellant.

But the possibility to eliminate a separate neutralizer in the bipolar concept offers several advan- tages compared to other electronic propulsions. And the use of ionic liquids open up possibilities of an even better performance in the future.

2.4 Ion Engines

An ion engine, see figure 2.4, is composed of two stages: An ionizing stage and an accelerating stage. In the ionizing stage the propellant atoms are typically ionized in one of three ways. The first is an electron bombardment process where the electrons are emitted from a hollow cathode. The second is a radio frequency (RF) type in which the ionizing electrons are accelerated in a RF field. And the third way to ionize the propellant is by a microwave process.

Hollow cathode electron sources have been shown to be one of the major life-limiting compo- nents of an electron bombardment system, and thus RF and microwave techniques carry a big advantage in not requiring an electron source. [23].

The accelerating stage of an ion engine is composed of grids to accelerate the ionized propellant. Since positive ions are emitted, a neutralizer is required to prevent spacecraft charging.

Xenon (Xe) is a typical propellant used in ion engines. It bares several advantages, such as being virtually noncontaminating and allowing for compact propellant storage. In the early developments in the 1960s it was common to use more reactive and/or toxic materials, such as caesium or chlorine fluorides [20], but their negative properties have caused them to loose favor in space propulsion applications.

2.4.1 Current Status Ion engines have been flown successfully on several missions, for instance as primary propulsion on smaller scientific spacecraft and for station keeping purposes onboard communication satel- lites. A big advantage of ion engines is their high specific impulse - values of 3000 s are not uncommon. The thrust range on current available hardware is, however, too high for precise pointing maneuvers. Values typically range from 5 mN up to several hundred mN [12] [20].

Page 8 CHAPTER 2. PROPORTIONAL THRUSTERS

M a g n e t i c R i n g s

P r o p e l l a n t - I o n I n j e c t i o n B e a m - + 0

- Electrons Em itted by Hollow Cathode - Traverse Discharge Acceleration F i e l d N eutralizer Figure 2.4: A schematic showing the principle behind the operation of an electron bombardment ion thruster. Magnetic rings are incorporated into the design in order to reduce primary electron mobility.

When considering thrust in the micronewton range, there was some early work conducted in the 1960s and 1970s, yielding thrusts from about 12 µN up to a couple of hundred micronewton. But they carried disadvantages in terms of poor efficiency and/or difficult to manage propellants. Modern micronewton ion engines are in very early stages of development, and considerable work is still left to be done. It is, for instance, highly desirable to reduce the thruster size, which presents a great challenge in terms of maintaining a plasma in a small space, as well as minia- turizing the thruster components [20]. In this area there is still a substantial amount of work to be done.

Table 2.1 shows a summary of the performance of ion engines.

2.4.2 Evaluation

The technology behind ion engines is well tested. They carry a high Isp and efficiency as well as a high thrust compared to other electrical propulsion systems. The high thrust, usually in the millinewton range, makes ion engines unsuited for high precision AOCS. It is possible that ion engines will never reach low enough thrusts for such an application, as micronewton ion engines are still at very early stages of development. If, however, such a feat can be achieved, a micro- ion engine will offer an attractive alternative to FEEP and colloid thrusters. A potential higher thrust-to-power ratio is particularly appealing, and the benign properties of the Xe propellant carry a lot of advantage.

An issue that ion engines typically suffer from is sputtering, caused by slowed down ions impact- ing on solid surfaces. The negative accelerator grid and neutralizer emitter tip is particulary prone to damage due to sputtering, which can severely alter their characteristics. Indeed, sput- tering is one of the major life limiting factors of ion engines.

Page 9 2.5. HALL THRUSTERS

2.5 Hall Thrusters

The Hall thruster is an electrostatic type of thruster and, like most of its kind, its origin can be dated back to the 1960s. Most of the research thereafter was conducted in Russia, and the Russians have flown hall thrusters on several spacecrafts since, their function mainly being tra- jectory insertions and east-west station-keeping of satellites [13].

Thrust from a Hall thruster is generated by accelerating an ionized propellant, typically Xenon. The ionization is done in two ways [17]. One way is by electrons emitted from an external hollow cathode. Some of these electrons enter the thruster and collide with neutral propellant atoms injected at the back of the thruster, which also acts as the anode. The collision gives rise to more electrons, and positive propellant ions that are accelerated axially out of the thruster by the potential difference between the internal anode and the external cathode, producing thrust. The second way ionization takes place involves electron trapping. Electrons entering the thruster from the external cathode, as well as those generated inside the thruster due to the previously mentioned ionization process, are on their way to the anode deflected and accelerated azimuthally by a radial magnetic field. This effect, often called the Hall effect, is due to the Lorentz force acting on the electrons. By trapping the electrons in a gyrating motion, even more collisions with propellant gas occur, and thus more ions are created and can be used to produce thrust. Furthermore, this also gives rise to thrust through a magnetic pressure force exerted by the electrons on the magnets [17].

The high electron density in the magnetic field region allows for a denser ion beam to be formed compared to ion engines, enabling a more compact design for the same amount of thrust [20]. But it should be noted that Hall thrusters typically deliver smaller specific impulses than ion engines do. See figure 2.5 for a simple schematic of a Hall thruster.

N eutralizer, C athode

A n o d e - -

0 I o n + B e a m P r o p e l l a n t I n j e c t i o n

M a g n e t i c C o i l s B

Figure 2.5: Cross section of an axis-symmetric Hall thruster. B denotes the magnetic field.

Page 10 CHAPTER 2. PROPORTIONAL THRUSTERS

2.5.1 Current Status Having flown on over a hundred missions the Hall thruster has proven itself in space over and over again. It has not only been used successfully on telecommunication satellites for station keep- ing purposes, but also as primary propulsion on the European SMART-1 spacecraft to the moon.

Typical performance specifications on commercial available Hall thrusters are specific impulses between 1500 and 2500 s with 50 to 60 % total efficiency [9]. Studies in the USA have shown no fundamental restraints to achieving specific impulses above 3000 s with the same efficiency as previously mentioned [8], something which could make Hall thrusters an attractive alternative to ion engines.

Thrust values range from a few mN up to a couple of hundred mN [12] [20]. Even larger thrusters are under development in the USA, where a request for offer was released in 2001 for the design and development of a Hall thruster with a minimum thrust of 2.5 N at 50 kW and at least 60 % efficiency [3]. Micronewton Hall thruster development is, however, scarce at best. There are several scaling issues inherent to miniaturization of Hall effect thrusters. These issues lie mainly in smaller thruster channel dimensions requiring larger magnetic field strengths in order to reduce the electron gyration radius [20]. The physics of a Hall thruster also hinders it from achieving a high accuracy between desired and achieved thrust. Current precision is around 0.3 mN [12].

Table 2.1 lists different specifications for Hall thrusters.

2.5.2 Evaluation

Hall thrusters offer a high performance in terms of a high Isp and have a relatively high efficiency in the area of 50 %. The use of the benign Xe propellant is also very attractive.

When comparing to the much similar ion engine, Hall thrusters offer an advantage due to its more compact design for the same delivered thrust. Hall thrusters usually also have a lower power consumption than ion engines, but on the other hand they suffer from a lower efficiency and shorter lifetime. Another concern is a quite large plume divergence.

As previously mentioned, there are currently poor alternatives available for micronewton thrust levels. Many scaling issues will need to be solved before a Hall thruster with low enough thrust and a high enough precision for very fine manoeuvrers can be achieved.

2.6 Resistojets

Work on resistojets began back in the 1960s, and the concept behind how they function is quite simple. A propellant, which is stored in either a gaseous or a liquid phase, is heated to va- porization through means of conduction and convection from a heater element. By thermally expanding the vaporized propellant through a nozzle, thrust is produced. Virtually any pro- pellant can be used, even a urine concept has been investigated as a potential propellant for the Space Station [25], but more common ones are, for example, water (H2O), nitrogen (N2), ammonia (NH3) and nitrous oxide (N2O) [20] [28].

Page 11 2.6. RESISTOJETS

When choosing a propellant for a resistojet, it is important to consider some of the propellant properties. A low molecular weight yields a higher specific impulse, but a high heat of vapor- ization results in a higher heating requirement.

Some resistojets operate in pulses, but the ones of interest here use a continuous mode.

T h e r m a l R adiation Shielding R e s i s t i v e Heater Assem bly

P r o p e l l a t E x h a u s t

Heat Exchanger P o w e r S u p p l y

Figure 2.6: A simple schematic of a conventional resistojet.

2.6.1 Current Status A conventional resistojet is characterized by a high efficiency, around 80 % [12], an attractive thrust-to-power ratio up to 1 mN/W [12], but a generally low specific impulse in the area of 127 to 1000 s [12] [20]. Thrust values typically exist in the millinewton range, and as such they have been flown on several missions [28].

There are projects underway to scale down the resistojet concept to micronewton thrust levels. Two of them are the Free Molecule Micro-Resistojet (FMMR) and the Vaporizing Liquid Mi- crothruster (VLM). The difference between these is that the FMMR relies on molecular flow in the heat exchanger region, while the VLM is a laminar flow concept. The development of both are mainly being carried out in the USA [12], and both are based on MEMS technology. The principle behind their operation is the same as for the conventional resistojet, except it is done on a much smaller scale. Their intended use is for upcoming microspacecraft missions and for fine pointing maneuvers on spacecrafts requiring a very high accuracy [21]. Presently, the thrust for these thrusters range down to about 25 µN [12], but the aim is to reach even lower values.

Performance data on micro-resistojets can be seen in table 2.1.

2.6.2 Evaluation A great advantage of micronewton resistojets concepts is their potential to provide very small impulse bits. Target values of 0.1 to 1 µNs have been reported. And provided that their devel- opment can be successfully completed, they will also feature attractive properties such as a lower

Page 12 CHAPTER 2. PROPORTIONAL THRUSTERS power consumption and a low mass. However, initial tests show relative low specific impulse values, typically between 50 and 100 s.

There is still a long way to go in the development of micronewton resistojets, but when/if fin- ished, they should make up an attractive alternative for very precise spacecraft control.

Conventional resistojets are judged unsuitable for high precision AOCS.

Page 13

3 COMMONLY USED CONTROL METHODS

There are various ways of determining the configuration of thrusters to fire in order to produce a desired net force and/or torque. It is important that the control concept chosen for a specific task or mission is such that it maximizes the performance of the system of thrusters, while optimizing the use of the resources at hand. The factors which must be taken into consideration when making this choice are mainly the computational resources available, the thrusters limita- tions and the design constraints.

• The computational efforts should naturally always be strived to be kept at a minimum. How- ever, different controllers require different amounts of calculations, and sometimes it may be necessary to choose a ”slower” controller in order to obtain the desired overall performance. But it is also important to consider how often it will be necessary to issue thrust commands in order to maintain the required control over the spacecraft. If this frequency is higher than the time taken to calculate the commands, then the resulting forces and torques will become inadequate by the time they are executed, resulting in an unstable system.

• Within the thruster limitations lies for instance the minimum and maximum thruster force obtainable. Also included here are the minimum and maximum amount of time that the thruster can fire. All these limitations depend mainly on the type of thruster used.

• Design constraints may limit the maximum accessible torque due to the length of the lever arms, which are not always possible or practical to be as long as wished for. Solar panels, instruments and other protruding or free line of sight requiring components might also restrict the position of the thrusters, because the exhaust could easily contaminate spacecraft surfaces or instrument readings. Constraints such as limited power available or thermal control re- strictions could also limit the number of simultaneously fireable thrusters, just as propellant limits the accumulated time they are used.

In the end, the method of control chosen will be a trade-off between all the properties mentioned above, as well as others, along with their benefits and detriments.

How to control thrusters also depends on the method of its operation. Many are only able to supply a constant thrust, and in such a case controlling them involves varying the length of time (pulse width) or how often (frequency) they are fired [26]. When using proportional thrusters it is also possible to alter the level of thrust used. For this case (the only one which will be dealt with in this thesis) the general approach in determining what thrusters to fire and what level of force each should produce in order to achieve the desired net force and torque is

F = AT (3.1)

Page 15 where F = force and torque vector, size (m × 1) A = thruster configuration matrix, size (m × n) T = thrust command vector, size (n × 1).

This equation is based on a body fixed coordinate system, and the thruster configuration matrix specifies the position and direction of each thruster. The number of degrees of freedom is denoted by m, and in the general three dimensional case there will be 3 forces and 3 torques, in or around each x-, y and z-direction or axis. That is, m will be equal to 6. The number of thrusters used is n. In order to achieve a six degree of freedom control Chen [1] showed that at least 6 two-sided or 7 one-sided thrusters are needed. A two-sided thruster has exhausts in two opposite directions, and if one direction is denoted positive the thruster will also be able to supply a negative thrust using the opposite exhaust. The one-sided thruster, on the other hand, is able to give thrust in just one direction. This thesis will focus on one-sided thrusters only, which means that each element in T must be larger or equal to some lower thruster force boundary, denoted Tlow. That is

T ≥ Tlow where the inequality holds for each element of T. This lower boundary is 0 if the thrusters can be turned off completely. Some thrusters are not able to supply a continuous thrust all the way down to zero, but have a step from zero to some minimum force value. This must also be taken into consideration when designing the controller algorithms.

Unfortunately, equation (3.1) is not as simple as it might appear. Due to for instance redundancy issues, the number of thrusters, n, will generally be greater than the degrees of freedom, m. This makes the equation underdetermined, and additional constraints are required in order to solve it. A common way to do this is by optimizing the thrust command vector with respect to one of three vector norms; the 1-norm, the 2-norm or the ∞-norm (pronounced infinity-norm).

1. The 1-norm of a vector is the sum of the absolute values of each element of a vector. Min- imizing this norm results in an optimum thrust command vector from a fuel consumption point of view, and it is called a minimum flow rate controller. For electrical propulsion this is analogous to optimizing the current.

2. The 2-norm is defined as the square root of the sum of the squared absolute values of each element of a vector. If this norm is minimized the result will be an optimum thrust command vector from a power consumption point of view. This is called a minimum power controller.

3. The ∞-norm is the maximum value of the absolute values of each element of a vector. By minimizing this norm the thrust command vector will be optimized with respect to the maximum force that is utilized by an individual thruster, making it a minimum force controller. In the case of electrical propulsion this is the same as minimizing the maximum voltage applied.

The following sections will go through each of the three different types of controllers, and finish off with numerical examples to show some of the differences between the three.

Page 16 CHAPTER 3. COMMONLY USED CONTROL METHODS

3.1 Minimum Power Controller

The 2-norm of a vector is the square root of the sum of the squared absolute values of each element in a vector, and it is sometimes also called its length. Minimizing it is denoted by

T2 = minkTk2 (3.2) T which is to say that T2 is that specific thrust command vector that minimizes the 2-norm of T. The subscript 2 signifies that the 2-norm has been minimized. Fortunately, the above equation has a very simple solution (see work by Chen [1] for a derivation) since the thrust command vector, T2, in this case is a linear function of the desired force and torque vector, F. The solution is

† T2 = A F (3.3)

where A† = the pseudoinverse of A = AT (AAT )−1 AT = transpose of A.

This type of controller is called a minimum power controller and it is one of the most widely used controllers, mainly due to its simplicity. It is also very fast since the pseudoinverse, A†, needs only be computed once for a given thruster configuration, and then remains constant for each thrust command. However, equation (3.3) contains no guarantee against negative elements in T2, and this makes it unsuited for direct application to one-sided thrusters. One way to get around this issue is to bias the thrusters about a middle point. This bias, which is called Tn, must not add any forces or torques to the equation. That is to say it must satisfy the condition

F = ATn = 0 (3.4) where 0 is a vector of zeros. This condition is known as the null space criteria, and a vector that satisfies it is called a null space vector. By adding this new vector, equation (3.3) becomes

† T2 = A F + Tn. (3.5) The key of applying the simple pseudoinverse solution that is the minimum power controller to one-sided thrusters is found in the null space vector. Finding the correct one is crucial, and there are numerous ways to do so. The following four subsections will first discuss how to find a null space vector, and then discuss three different approaches for finding T2.

3.1.1 Finding a Null Space Vector When looking for a null space vector the goal is to find the one that maximizes the minimum control authority (see chapter 4) of the thruster system. Such a vector would constitute the optimum solution, and finding it depends on two things: The thruster configuration and the limits of the system.

If the thruster configuration is symmetric the so called null space, the space within which all thrust combinations satisfy the null space criteria of equation (3.4), will consist of a simple straight line. For this case firing all thrusters with the same amount of thrust will result in no net force or torque being applied on the spacecraft. This simple linearity makes it easier, often

Page 17 3.1. MINIMUM POWER CONTROLLER even intuitive, to find the optimum null space vector.

If the thruster configuration on the other hand is non-symmetric it becomes more complicated. In this case the null space is defined by more than one base vector, making the number of possi- ble thrust combinations virtually infinite. Therefore, the best way to find a suitable null space vector may be through ways of simulations and/or trail and error.

The limits of the system are the lower and upper thrust boundaries but also the overall total limit, which can be either a flow rate limit, m1, power limit, m2, or thruster force limit, m∞. Note that these three limits have the same names as the three different types of controllers. It is very important to keep apart the controller and the limit! If a controller for instance is flow rate limited, the total flow of the thrust command vector must not exceed a given limit, which is the physical limitation of the system. The analogy is the same for the power and thruster force limit. All these limits are addressed further in chapter 4 ”The Control Authority”.

Next, ways to find a null space vector for the three different limits are shown. It should be noted that the algorithms discussed here find a null space vector that will work, but they do not guarantee it being an optimum null space vector.

Flow Rate Limit For the flow rate limit the optimum null space vector is one where the 1-norm equals the limit. That is

kTnk1 = m1 (3.6)

where m1 = the flow limit. The reason for this is that such a null space vector added to the non-biased thrust command vector gives a new command vector where the 1-norm is always at the limit. And that is the highest possible performance the system can give.

Applying the relation of equation (3.6) for a symmetric thruster configuration is straight forward. The force each thruster should exert is simply the limit divided by the number of thrusters. But it is not as easy for the non-symmetric case.

Finding the null space vector for non-symmetric configurations could be expressed as a min-max problem according to

min max Vi(Tn) (3.7) Tn Vi

such that ATn = 0 (3.8)

Tn ≥ Tlow (3.9)

kTnk1 = m1 (3.10) · ¸ Tlow − (Tn)min where Vi = . (3.11) kTnk1 − m1

Page 18 CHAPTER 3. COMMONLY USED CONTROL METHODS

In words, the above equation says to find the null space vector, Tn, that minimizes the maximum element of the vector Vi, under the constraints that the null space vector satisfies the null space criteria, has all vector elements above the lower thruster boundary and has a 1-norm at the flow rate limit. The vector Vi is a two element vector that contains functions of the null space vector. These functions are the difference between the lower thruster boundary and the minimum element of the null space vector, and the difference between the 1-norm of the null space vector and the flow rate limit.

Power Limit If a power limit is used it is very important that the 2-norm of the null space vector does not exceed the power limit of the system, nor should it be exactly at the limit. Finding an optimum null space vector for a power limited system is the most difficult case, and it may sometimes best be done by trial and error. However, it could also be looked at as a min-max problem and could therefor be stated as

min max Vi(Tn). (3.12) Tn Vi

such that ATn = 0 (3.13)

Tn ≥ Tlow (3.14)

kTnk2 < m2 (3.15) · ¸ Tlow − (Tn)min where Vi = . (3.16) kTnk2 − m2 The above equation can be used for both a symmetric and a non-symmetric thruster configura- tion.

Force Limit An appropriate null space vector for the force limit with a symmetric thruster configuration is found by simply biasing the thrusters about a middle point between the lower and upper thruster force boundary.

If it is non-symmetric, though, finding a suitable middle point is not as intuitive as in the symmetric case. But it can be found relatively easy in three steps. The first step is to find the minimum and maximum null space vector that satisfies the null space criteria. This can be done with the aid of linear programming.

T Tnmin = min f Tn (3.17) Tn T Tnmax = max f Tn (3.18) Tn

such that ATn = 0 (3.19)

Tn ≥ Tlow (3.20)

Tlow ≤ Tn ≤ Tup (3.21)

Page 19 3.1. MINIMUM POWER CONTROLLER

where f = vector of ones, the same size as Tn

Tup = the upper thruster force boundary.

The notations min and max in equations (3.17) and (3.18) mean that Tnmin and Tnmax are those Tn Tn T specific null space vectors which minimize and maximize f Tn. This results in the minimum and maximum null space vector possible. The second step is then to take the average of these

T + T T = nmin nmax (3.22) navg 2 followed by the third step which is to rescale Tnavg into a null space vector so that the difference between the minimum null space vector element and the lower thruster force boundary, and the difference between the upper thruster force boundary and the maximum value of the null space vector, are the same. So the rescaled null space vector must satisfy

(Tn)min − Tlow = Tup − (Tn)max. (3.23) This way of finding a null space vector for a force limited system may be used for a symmetric thruster configuration as well.

3.1.2 Fixed Bias

A fixed bias is the simplest way to utilize a bias. In the fixed bias approach Tn is computed only once, and it remains constant for each thrust command vector calculation. This makes it very quick and easy to use equation (3.5).

3.1.3 Dynamic Bias Instead of having a constant bias that is computed only once, as in the case of the fixed bias, the dynamic bias rescales the null space vector for each thrust command calculation. The rescaling is done based on the relations between the vector elements of the non-biased thrust command vector and the null space vector that is to be rescaled. That is

Tlow − (Tp)i Ri = (3.24) (Tnold )i

th where Ri = relation between the i element of Tp and Tnold † Tp = non-biased thrust command vector = A F

Tnold = the old null space vector to be rescaled. The index i indicates that the ith element of the vector is used in the calculation. The old null space vector can be either the Tn from equation (3.23) in the case of a force limit, or the Tn from the optimization in equation (3.12) in the case of a power limit. It is not possible to use a dynamic bias for a flow rate limit, since it there needs to be fixed with a 1-norm equal to the limit in order to function properly.

After the relation in equation (3.24) has been calculated for each vector element, the highest relation is chosen for rescaling of the old null space vector

Tndyn = Rhighest · Tnold . (3.25)

Page 20 CHAPTER 3. COMMONLY USED CONTROL METHODS

This way, when having added the dynamic null space vector, Tndyn , to the non-biased thrust command vector, it will place the minimum element of the biased thrust command vector at the lower thruster force boundary. In other words

T = Tp + Tndyn (3.26) makes (T)min = Tlow. A dynamic bias must be calculated for each thrust command vector.

In overall performance the dynamic bias is more flexible and fuel efficient than the fixed bias, yet it is only marginally slower.

3.1.4 Iterative Approach An iterative approach of finding the thrust command vector was devised by Jin [10], and it can be viewed in figure 3.1. The iterative approach is based on the fixed bias, with the control loop being initiated only if the fixed biased thrust command vector still contains vector elements below the lower thruster force boundary. In such a case it attempts to find a new command vector which satisfies the lower boundary.

T n 1 / 2 + + F d + F c + T p c + T c F r - 1 + G z A + A - + I · I / 2

Figure 3.1: The iterative control scheme. The letter z denotes the shift operator.

The calculations made in the control loop are

† Tpc = A Fc(k) + Tn (3.27) T + |T | T = pc pc (3.28) c 2 Fr = ATc (3.29)

Fc(k + 1) = Fc(k) + (Fd − Fr) · G (3.30)

where Tpc = pseudoinverse biased thrust command vector

Tn = null space vector

Tc = thrust command vector, all elements ≥ 0

Fr = resulting force and torque vector

Fc = commanded force and torque vector

Fd = desired force and torque vector G = gain k = iteration step.

Page 21 3.1. MINIMUM POWER CONTROLLER

The loop is initiated with Fc = Fd, and it stops when the error between the resulting and the desired force and torque vector is below a specified value. A gain can also be used to speed up the convergence, and an appropriate value for the gain is best found by running a large number of simulations with various gains. Figure 3.2 shows an example of such a simulation. The gain which on average results in the least amount of iterations required to reach a convergence is the one best suited for use. In the figure it is clearly seen that for this case the most appropriate gain is 3.

The Tc last calculated in the control loop when meeting the error requirement is the thrust command vector that is outputted from the loop.

It should be noted that the algorithm above does not guarantee a convergence. To avoid infinity loops a maximum number of iterations needs to be set. In a case where this number is exceeded, an alternative method of calculating Tc needs to be used.

It is also possible to use the iterative control loop without the bias in equation (3.27). But this makes the process slower, since it will require more iterations to find a solution.

14

12

10

8

6 Number of Iterations 4

2

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Feedback Gain

Figure 3.2: Gain simulation for the test case 1 discussed in section 3.4.1. The plot was made by recording the number of iterations required for each gain to reach convergence for 2000 ran- domized force and torque vectors, whereafter the average number of iterations for each gain was plotted against the gain.

Page 22 CHAPTER 3. COMMONLY USED CONTROL METHODS

3.2 Minimum Flow Rate Controller

Finding the thrust command vector using the minimum flow rate controller is done by minimizing its 1-norm.

T1 = minkTk1 (3.31) T The 1-norm is the sum of the absolute values of each element of the thrust command vector. Solving it can be done with the use of linear programming, expressed as

0 T1 = min I T (3.32) T

bound by AT = F (3.33)

T ≥ Tlow (3.34)

where I = vector of ones, same size as T.

Equation (3.32) says that the thrust command vector, T1, which has a minimized 1-norm is that specific T which has the minimum sum of all the elements in T, bound by (3.33) and (3.34). Solving this type of optimizations can be done using the simplex method. Or if the mathemat- ical software Matlab is available, it incorporates a function called linprog in the optimization toolbox that makes it very easy to solve equations of this sort.

The minimum flow rate controller is an optimum controller from a propellant consumption point of view. In case of an electrical propulsion system, the flow of propellant can be interpreted as the current.

It should be kept in mind that the main disadvantage of solving linear programs, like the one in equation (3.32), is that they are relatively slow, seeing as they require a lot of computations. Depending on the input the time taken to perform an optimization can also vary, and it is not even guaranteed that the algorithm will find a solution. In such a case an alternative controller needs to be used, but that is done with the cost of increased usage of propellant.

3.3 Minimum Force Controller

The minimum force controller finds the optimum thrust command vector from a thruster force point of view. That is to say, it finds the thrust command vector that has the lowest possible value on the maximum thrust vector element. Finding this vector is done by minimizing the ∞-norm of the thrust command vector

T∞ = minkTk∞. (3.35) T As for the minimum flow rate controller, the solution for the minimum force controller can be found with the aid of linear programming · ¸ £ ¤ 0 T T∞ = min 01 1 . (3.36) T z

Page 23 3.4. NUMERICAL EXAMPLES IN 2-D

· ¸ £ ¤ T bound by −I I ≥ 0 (3.37) z 1 · ¸ £ ¤ T A 0 = F (3.38) 2 z

T, z ≥ Tlow (3.39)

where I = identity matrix, size (n × n) I = vector of ones, size (n × 1)

01 = vector of zeros, size (n × 1)

02 = vector of zeros, size (m × 1) z = a scalar m = number of degrees of freedom n = number of thrusters.

A way of putting this into words is to say that z should be as small as possible and no element of T may be larger than z (3.37) while T still fulfils the basic requirement of providing the desired net forces and torques (3.38), with only one-sided thrusters being used (3.39). This linear program results in the lowest possible ∞-norm on T. It is a slightly modified version of the one devised by Wiktor [29].

Being a linear program, the above equation has the disadvantages of requiring a lot of compu- tations and of there being no guarantee that a solution will be found.

Minimizing the maximum force of the thrust command vector can in the case of electrical propulsion be interpreted as minimizing the maximum voltage supplied to the thrusters.

3.4 Numerical Examples in 2-D

Three test cases were devised in order to test and verify the algorithms from the previous sec- tions. All test cases were in two dimensions which made them easier to visualize. Each algorithm was implemented and tested in Matlab.

Two different types of thrusters were used in the examples. One had a lower thruster force boundary of 0 force units, and the other had the same boundary at 0.3 force units. No consid- eration was here taken to an upper thruster force boundary, unless otherwise stated.

3.4.1 Test Case 1

The first test case was the simplest, featuring three thrusters that were able to supply forces only and no torques. Test case 1 can be viewed in figure 3.3.

Page 24 CHAPTER 3. COMMONLY USED CONTROL METHODS

y a = 1 2 0 °

T 2

a T 1 x

T 3

Figure 3.3: Test case 1. The directions of the resulting forces from the three thrusters are denoted ◦ by T1, T2 and T3. The angle between each thruster is 120 .

In this case equation (3.1) became   · ¸ T1 Fx = A T2 (3.40) Fy T3 where the desired force vector is expressed in terms of forces along the x- and y-axis.

The thruster configuration matrix was · ¸ 1 cos(α) cos(2α) A = (3.41) 0 sin(α) sin(2α) with α = 120◦. Four different desired force vectors were used as inputs to test each algorithm. These were · ¸ · ¸ · ¸ · ¸ 1 0 −0.2 0.3 F = , F = , F = , F = . a 0 b 1 c 0.8 d −0.4

The results of the Matlab test runs are shown in tables 3.1 through 3.5. In order to compare the computational effort required by each algorithm, the CPU time taken to perform the calculations was measured. Naturally, these values depend greatly on the hardware available, and the CPU time should therefore only be used for interrelation comparison between the different algorithms. The max|Fc − Fd| column shows the maximum error when comparing the desired force vector and the calculated force vector. The latter vector was found by taking the thruster configuration matrix times the calculated thrust command vector. In the case of the iterative solution there are two additional columns for the number of iterations required to reach convergence and the gain used in the control loop.

Page 25 3.4. NUMERICAL EXAMPLES IN 2-D

Table 3.1: Results for test case 1 when using the minimum power controller and a fixed bias. Note that some thrust vector elements are below the lower boundary, Tlow.

It is worth noticing that in the case of a fixed bias, the algorithm fired more thrusters than necessary. From figure 3.3 it can for instance easily be seen that if a force is desired in the x-direction only, the most efficient solution would be to just fire thruster 1. Still, for force Fa, the fixed bias fired thrusters 2 and 3 as well, resulting in wasted energy. This is an inherent disadvantage of using a fixed bias. Also notice that the lower thruster force boundaries were broken for both boundaries in test case 1.b, and the boundary of 0.3 in test case 1.c. This means that the desired forces lied outside of the control authority for this algorithm. The control authority is further discussed in chapter 4. A larger null space vector could have solved the issue of a broken lower boundary, but might also have given rise to new ones such as a lower overall control authority. The null space vectors used in all minimum power controllers in test case 1 were

  0.5 Tn = 0.5 when Tlow = 0.0 0.5   0.65 Tn = 0.65 when Tlow = 0.3. 0.65

Also note when comparing the tables that the fixed bias was the only algorithm that yielded vector elements below the lower boundary. All other algorithms contain conditions that prevent this from occurring.

Table 3.2: Results for test case 1 when using the minimum power controller and a dynamic bias.

Page 26 CHAPTER 3. COMMONLY USED CONTROL METHODS

The dynamic bias was in this test case more intelligent than the fixed bias, as it rescaled the null space vector so that the lowest element of the thrust command vector was at the lower boundary. This is the fastest and most efficient solution in the case of a symmetric thruster configuration.

Table 3.3: Results for test case 1 when using the minimum power controller and the iterative approach. The tolerance used in the control loop is 1E-6.

Since the control loop of the iterative approach was only initiated if applying the fixed bias yielded vector elements below the lower boundary, the two algorithms gave similar results. Therefore, even the iterative approach may waste energy by firing more thrusters than necessary. However, where the fixed bias failed by breaking the lower boundary, the iterative still found a solution. This can been seen when comparing table 3.3 with table 3.1.

Table 3.4: Results for test case 1 when using the minimum flow rate controller.

The results for the minimum flow rate controller in table 3.4 and those of the minimum force controller in table 3.5, are very similar. This is not a general result, but was due to the simple and symmetric nature of test case 1. But it should be noticed that the algorithm of the minimum force controller was slightly slower than the minimum flow rate controller.

Page 27 3.4. NUMERICAL EXAMPLES IN 2-D

Table 3.5: Results for test case 1 when using the minimum force controller.

3.4.2 Test Case 2

In test case 2 a fourth thruster was added and the configuration changed so it also produces a torque. Figure 3.4 shows the thruster configuration.

y a = 4 5 ° M Z L = 1 l . u .

T T 2 L 1 a x

T 4 T 3

Figure 3.4: Test case 2. Four thrusters capable of producing both forces and torques. The arrows point in the direction of the resulting forces from thrusters T1 through T4, and the length of the lever arms is 1 length units.

For this configuration equation (3.1) became

    T1 Fx T2  Fy  = A   (3.42) T3 Mz T4 with a thruster configuration matrix of

Page 28 CHAPTER 3. COMMONLY USED CONTROL METHODS

  cos(α) − cos(α) − cos(α) cos(α) A = − sin(α) − sin(α) sin(α) sin(α) (3.43) −1 1 −1 1

where α = 45◦. The five different desired force and torque vectors used as inputs to the algorithms were

          1 0 0 −0.2 −0.1 Fa = 0 , Fb = 1 , Fc = 0 , Fd =  0  , Fe =  0.4  . 0 0 1 0.5 −0.2

The null space vectors used were

  0.5 0.5 T =   when T = 0.0 n 0.5 low 0.5   0.65 0.65 T =   when T = 0.3. n 0.65 low 0.65

Table 3.6: Results for test case 2 when using the minimum power controller and a fixed bias.

Page 29 3.4. NUMERICAL EXAMPLES IN 2-D

Table 3.7: Results for test case 2 when using the minimum power controller and a dynamic bias.

The results for test case 2 were pretty much the same as for those in test case 1. That is to say that the fixed bias and the iterative approach yielded the same results, except for when the lower boundary was broken for the fixed bias, in which case the control loop was initiated for the iterative approach. The dynamic bias was still the smarter one with the highest performance.

Table 3.8: Results for test case 2 when using the minimum power controller and the iterative approach. The tolerance used in the control loop is 1E-6.

Table 3.9: Results for test case 2 when using the minimum flow rate controller

Page 30 CHAPTER 3. COMMONLY USED CONTROL METHODS

Table 3.10: Results for test case 2 when using the minimum force controller

Since test case 2, just like test case 1, was symmetric, the results from the minimum flow rate controller and the minimum force controller were the same. The difference between these two controllers will become more apparent in test case 3, and in chapter 4 where the control authority is discussed.

3.4.3 Test Case 3

Yet another thruster was added in test case 3, amounting to a total of five thrusters. Their configuration was also changed to become non-symmetric in order to better investigate the different properties between the algorithms. Test case 3 can be seen in figure 3.5.

y M a = 3 0 ° Z T 2 b = 4 5 ° L = 1 l . u .

T T 3 L 1 b a x

T 4

T 5

Figure 3.5: Test case 3. Five thrusters in a non-symmetric configuration, capable of producing both forces and torques. The arrows point in the direction of the resulting forces from thrusters T1 through T5, and the length of the lever arms is 1 length units.

This thruster configuration made equation (3.1)

Page 31 3.4. NUMERICAL EXAMPLES IN 2-D

    T1   Fx T2     Fy = A T3 (3.44) Mz T4 T5 where the thruster configuration matrix is   0 1 −1 0 − cos(β) A =  1 0 0 −1 − sin(β) (3.45) cos(α) −1 sin(β) cos(α) −1 with α = 30◦ and β = 45◦. The five different desired force vectors used as inputs to the algorithms were the same as for test case 2. That is           1 0 0 −0.2 −0.1 Fa = 0 , Fb = 1 , Fc = 0 , Fd =  0  , Fe =  0.4  . 0 0 1 0.5 −0.2 The null space vectors used were

  0.6587   0.6587   Tn = 0.3413 when Tlow = 0.0 0.3413 0.4487   0.7894   0.7894   Tn = 0.4091 when Tlow = 0.3. 0.4091 0.5378

Both null space vectors were based on a force limited system, and they were found with the use of equations (3.17) through (3.23), where an upper thruster force limit of 1 was assumed.

Table 3.11: Results for test case 3 when using the minimum power controller and a fixed bias.

Page 32 CHAPTER 3. COMMONLY USED CONTROL METHODS

Worth noticing in all the tables for test case 3 is that many thruster force elements were above 1. Since an upper thruster force boundary of 1 was assumed when finding the null space vector, the elements above this value would have been impossible for the thrusters to achieve. Therefore, these thrust vectors lied outside of the control authority.

Table 3.12: Results for test case 3 when using the minimum power controller and a dynamic bias.

Table 3.13: Results for test case 3 when using the minimum power controller and the iterative approach. The tolerance used in the control loop is 1E-6.

Table 3.14: Results for test case 3 when using the minimum flow rate controller.

Page 33 3.4. NUMERICAL EXAMPLES IN 2-D

When comparing table 3.14 and table 3.15 it can be seen that for this non-symmetric test case, the minimum flow rate controller and the minimum force controller may yield different results.

Table 3.15: Results for test case 3 when using the minimum force controller.

Page 34 4 THE CONTROL AUTHORITY

When implementing the different types of controllers in chapter 3 it was seen that they may result in different thrust command vectors. Since there are various physical properties that limit the performance of thrusters, it is easily recognized that some controllers will work bet- ter than others under given constraints. Therefore, it is highly desirable to somehow calculate a performance envelope for the controllers. A tool that enables this is called the control authority.

This chapter will give a complete definition of the control authority, as well as provide methods on how to find it. The chapter will also introduce tools to help visualize the control authority, something that is very useful to be able to do. A new concept - the minimum control authority - will also be defined as a single scalar value that illustrates the performance of a controller under the most conservative conditions.

4.1 Definition

The control authority is defined as the surface made up by the maximum possible output forces in all directions. This surface exists in an m-dimensional space, where m is the number of degrees of freedom. The size and shape of the control authority surface depends on the thruster configuration, the controller used and the limits it is bound by. In mathematical terms, the control authority is defined as the set of forces

Fs,p = ATs (4.1)

such that kTskp = mp (4.2)

where Ts = minkTks (4.3) T

and s, p = 1, 2, ∞ s = type of controller p = type of thruster limit.

In words, these equations say that the thrust vector, Ts, is the thrust vector that minimizes the s-norm, equation (4.3). That is, it is either a minimum flow rate, a minimum power or the minimum force controller. And the p-norm of that thrust vector must be at the p-limit, mp, equation (4.2). This basically means that the system supplies the highest possible output. By then multiplying the configuration matrix with this thrust vector, equation (4.1), the result will be a force and torque vector which is at the so called control authority surface.

Page 35 4.1. DEFINITION

4.1.1 Control Authority Surface

To help understand the concept of control authority, consider again the example of figure 3.3, called test case 1. It had three thrusters arranged in the xy-plane, where each thruster was separated by 120 degrees. What would be the maximum force that the configuration can supply in an arbitrary direction? To answer this question it must first be known which type of controller is used to calculate the thrust command vector, and what the limit is when using that controller.

The perhaps easiest controller to visualize intuitively for test case 1 is the minimum flow rate controller. Since this controller deals with the sum of the absolute values of each element in the thrust command vector, the maximum force it can apply is in the direction of one of the thrusters, if all the thrust is directed through that thruster only. Changing from full thrust on one thruster to full thrust on its neighboring thruster would be done in one smooth transition if maximum performance is desired of the whole system. The maximum performance would come from constantly keeping the 1-norm of the thrust command vector at its limit. If this way of reasoning is applied to vectors in a plane, and if a line was drawn at the head of the force vector found when adding the forces from all the thrusters, it would draw straight and linear lines between the three points that constituted the force when firing just one of the three thrusters at its full thrust. In other words, it would assume the shape of a triangle according to figure 4.1. Any point on the boundary of this triangle represents the control authority in that direction. The whole of the triangle makes up the control authority in all directions, that is the control authority surface, and the thruster configuration can supply all forces at and beneath the boundary of this triangle, but none above.

é 0 ù ê 1 ú é T ù ê ú 1 0 T ê T ú ëê ûú = ê 2 ú T ëê 3 ûú

é 1 ù é 5.0 ù ê 0 ú é 0 ù ê ú ê 0 ú ê 0 ú ê ú ëê 0 ûú ê ú ëê 5.0 ûú ëê 1 ûú

Figure 4.1: The triangle represents the control authority of the thruster configuration in test case 1. It was made assuming that the thrusters could be turned completely off, Tlow = 0, and that the flow limit was equal to 1.

If another controller is used, the control authority will have a different shape. The shape is also decided by the type of limit used when calculating the control authority. In total there are nine ways to combine controllers and limits, as shown in table 4.1. Not all nine are reasonable to use though. To use a different limit than the type of controller, for instance a flow limit for a minimum force controller, makes little sense as it does not carry much support from reality. However, for the minimum power controller there is some prudence in applying one of the other limits. This is due to the simplicity of its pseudoinverse nature. For some special thruster

Page 36 CHAPTER 4. THE CONTROL AUTHORITY configurations, mainly symmetric ones with a small number of thrusters, the minimum power controller yields the same result when using a flow rate or force limit as the minimum flow rate and minimum force controller does, only it is done much faster. Furthermore, the three different algorithms that were shown for the minimum power controller in the previous chapter, may each give a different control authority. This fact is demonstrated in figure 4.2, where the three different algorithms are applied using a force limit. Figure 4.3 shows the three different controllers using their respective limits.

Table 4.1: The nine possible combinations of controllers and limits.

4.1.2 Finding the Control Authority Finding the control authority surface is best done by looking in one direction at a time. The length of the vector in that direction is then maximized so that it is as long as possible without breaking the limit.

x = max x (4.4) x

such that F = x · eF (4.5)

where F = ATs (4.6)

and Ts = minkTks (4.7) T

bound by x ≥ 0 (4.8)

Ts ≥ Tlow (4.9)

kTskp ≤ mp (4.10)

where x = length of the vector

eF = unit vector.

When using a minimum power controller with a flow rate limit there is also the additional constraint that the minimum element of the thrust vector must be at the lower thruster force boundary.

Page 37 4.2. CONTROL AUTHORITY PLOT

Iterative Approach F i x e d B i a s Dynam ic Bias

M i n i m u m C o n t r o l A u t h o r i t y

Figure 4.2: The control authority for test case 1 using a minimum force controller with a force limit. The three shapes represent three different algorithms for finding the thrust command vector. Calculations were made assuming that the thrusters could be turned off completely, Tlow = 0, and that the force limit was equal to 1.

(T)min = Tlow (4.11) This procedure for finding the control authority is repeated for all directions. The directions in which to look for the control authority are vectors to points on the surface of a unit sphere, or a hypersphere if the surface exists in a higher dimension than three.

4.2 Control Authority Plot

In order to understand and analyze the control authority it is desirable to have some way of visualizing it. The best and most direct way to do this is simply to plot it, which works well when the control authority exists in a two or three dimensional space. In the previous section simple examples of two dimensional surfaces were given. Three dimensional control authorities, on the other hand, arise for instance when translation and rotation control are decoupled on a spacecraft. In such a case it is possible to plot two control authority surfaces: one for the force space and one for the torque space. But in the general case, where firing one thruster affects both the translation and rotation of the spacecraft, the control authority will exist in a six dimensional space. Of course, such large dimensions can not be plotted and are impossible to visualize intuitively. It is therefore necessary to use another tool, and this tool is called the

Page 38 CHAPTER 4. THE CONTROL AUTHORITY

T T 1 , 1

T 2

T 1

T 3

T 2 , 2

Figure 4.3: The control authority for test case 1 using a minimum flow rate controller with a flow rate limit, T1,1, a minimum power controller with a power limit, T2,2, and a minimum force controller with a force limit, T∞,∞. The figure was made assuming that the thrusters could be turned completely off, Tlow = 0, and that the limit for each controller was equal to 1.

control authority plot.

The control authority plot separates the control authority vector into two parts, where one is a vector containing all the force elements and the other is a vector containing all the torque elements. That is

· ¸ (Fs,p)F Fs,p = (4.12) (Fs,p)M

where (Fs,p)F = force part of the control authority, Fs,p

(Fs,p)M = torque part of the control authority, Fs,p.

The magnitudes of these two new vectors are then plotted against each other,

k(Fs,p)F k2 versus k(Fs,p)M k2, with the force magnitude on the x-axis and the torque magnitude on the y-axis. This way it becomes possible to see the extreme boundaries of the control authority surface in a simple two dimensional plot. From the plot it can for instance be seen how much torque the system can

Page 39 4.2. CONTROL AUTHORITY PLOT supply if no forces are required, or vice versa, as well as the maximum force and torque the system can supply simultaneously. It should be noted that the plot does not tell the direction in space to any of its points. But the controller can handle any and all disturbances, in any direction, that are below the lower left boundary of the control authority plot. All points there are completely enclosed by the control authority surface, and this boundary is called the min- imum control authority plot. The control authority plot described in this section is based on the minimum control authority plot that was originally developed by Wiktor [29]. Finding the minimum control authority plot is really desirable, and it is what the designer of a thruster control system will have the most use of. Unfortunately this requires algorithms that are able to find the global minimum of the control authority, which the functions in the Matlab opti- mization toolbox used in making this thesis did not guarantee. Therefore a control authority plot was made instead, with the main disadvantage being that it takes a lot more time to produce.

To better help understand the control authority plot and the minimum control authority plot there will now follow a few examples based on test case 2 in section 3.4.2.

4.2.1 Examples of the Control Authority Plot

Test case 2, seen in figure 3.4, features a thruster configuration with three degrees of freedom: Two forces in a plane and one torque. This makes it possible to plot the surface of the control authority because it exists in three dimensions. It also makes it possible to produce a control authority plot by splitting up the force and torque part of the control authority vector. This procedure will be illustrated for test case 2 using three different controllers.

First, consider a minimum flow rate controller with a flow rate limit. Both the control authority surface and the control authority plot will be plotted in order to better understand the relation- ship between them.

Starting off by plotting the control authority surface, it is constructed using the method de- scribed in section 4.1.2. If a flow rate limit of 1 is assumed for simplicity, and if the control authority is calculated in fifteen degree intervals of the azimuthal and polar angle in a spherical coordinate system, plotting the three dimensional surface made up by the control authority vectors will end up looking like figure 4.4. It is clearly in the shape of a tetrahedron with in- equilateral faces, one side√ of its constituent triangles having the length of 2 and the other two sides having the length of 6.

Making a control authority plot of the vectors that make up figure 4.4 results in a plot such as the one in figure 4.5, in which each cross represents a point on the control authority surface, in other words a control authority vector. Studying the figure it is possible to draw some conclusion that can be confirmed when studying the thruster configuration of test case 2. One such conclusion is that the maximum possible torque achievable with this configuration and controller is 1, and with that torque applied it is possible to also get a force component absolute value ranging from 0 to 1. In other words by firing two opposite thrusters with equal amounts of thrust yields no net force as the two thruster will counteract each other, but firing only one thruster at its maximum also gives a net force component of 1. The line marked ”Minimum Control Authority Plot” has been added manually. All disturbances below this line can be handled by the controller.

Page 40 CHAPTER 4. THE CONTROL AUTHORITY

Figure 4.4: The control authority surface of test case 2 in section 3.4.2 using a minimum flow rate controller with a flow rate limit of 1.

M inim um Control Authority Plot

Figure 4.5: The control authority plot made from the control authority vectors used to make the control authority surface in figure 4.4. Each cross represents a control authority vector. A line marking the minimum control authority plot has been manually added to the plot.

Page 41 4.2. CONTROL AUTHORITY PLOT

Figure 4.6: The control authority surface of test case 2 in section 3.4.2 using a minimum power controller with a power limit of 1 and the fixed bias algorithm.

M inim um Control Authority Plot

Figure 4.7: The control authority plot made from the control authority vectors used to make the control authority surface in figure 4.6. Each cross represents a control authority vector, and a line marking the minimum control authority plot has been manually added to the plot.

Page 42 CHAPTER 4. THE CONTROL AUTHORITY

Secondly, consider instead a minimum power controller with a power limit. The control authority surface in this case has the shape of an ellipsoid, as can be seen figure 4.6. It is not a pure ellipsoid, however, which can be seen when looking at the control authority plot in figure 4.7. This imperfection arises from the null space vector used, which had all elements equal to 0.43, equal because of the symmetric thruster configuration. In some directions this null space vector is inadequate for maintaining the ellipsoid form, and that results in the imperfection. One way to overcome this is to choose a null space vector with a 2-norm closer to the power limit, which in this case was equal to 1. However, for this example, such an approach was found to make the control authority surface smaller.

And as a third example, consider a minimum force controller with a force limit. This results in a slightly more complex shape as seen in figure 4.8. Note that the minimum control authority plot in figure 4.9 is higher than for the previous two controllers. The reason for this is that the thrust level from one thruster is not linked to the thrust level on any of the other thrusters, except that they are all bound to the constraint that none of them may exceed the maximum thrust level. This property works in the favor of the minimum force controller in this specific example.

Figure 4.8: The control authority surface of test case 2 in section 3.4.2 using a minimum force controller with a force limit of 1.

Page 43 4.3. MINIMUM CONTROL AUTHORITY

M inim um Control Authority Plot

Figure 4.9: The control authority plot made from the control authority vectors spanning the control authority surface in figure 4.8. Each cross represents a control authority vector. The minimum control authority plot is marked by straight lines, and has been manually added to the plot.

4.3 Minimum Control Authority

The minimum control authority is a scalar value defined as the shortest distance to the control authority surface [29]. In other words it is the weakest output direction of the thruster system. Finding the minimum control authority can thus be expressed as

minkFs,pk2, s, p = 1, 2, ∞ (4.13)

where s = type of controller p = type of limit.

That is, it is the minimum force and torque vector of the force and torque vectors that constitute the control authority surface.

A straight forward approach of finding the minimum control authority involves two linear pro- grams organized as an inner and an outer loop, see figure 4.10. The inner loop is nothing more than the maximization stated in equations (4.4) through (4.10), and the outer loop simply looks for the minimum value that can be obtained using that maximization.

Page 44 CHAPTER 4. THE CONTROL AUTHORITY

C onstraints

Linear Program for finding the M inim um C ontrol A uthority

Linear Program for finding the C ontrol A uthority

C onstraints m a x i m i z e

m i n i m i z e

Figure 4.10: Simple overview of the principle behind finding the minimum control authority. The inner loop finds the control authority with a maximizing linear program, while the outer loop finds the minimum of the control authority.

Figure 4.2 illustrates what the minimum control authority is for different algorithms using the minimum force controller with a force limit in test case 1.

Because this method involves two linear programs it is relatively slow in finding a solution. Another concern discovered when implementing the algorithm in Matlab was the difficulty of finding a global minimum, which is, after all, the desired result. The Matlab functions executed only found a minimum, but bared no implications on the result being local or global. Further- more, certain cases were found to return a false minimum, arisen from round-off errors and other computer imperfections. These issues need to be studied in further detail.

Page 45

5 IMPLEMENTATION - LISA PATHFINDER

The LISA Pathfinder mission was chosen as a reference mission in order to test and verify the algorithms developed in the previous chapters. This chapter will begin with an overview of the background and purpose of LISA Pathfinder, and then proceed with suggesting a suitable controller and analyzing its control authority under given assumptions. The controller will also be tested against different disturbance profiles and, to round things off, there will be an evaluation of the results.

5.1 About LISA Pathfinder

LISA Pathfinder (old name SMART-2) is the precursor of the LISA mission (figure 5.1). LISA is a joint ESA and NASA mission with the goal of detecting and observing gravitational waves as predicted by Albert Einstein’s general theory of relativity [6]. As opposed to Sir Isaac Newton’s law of gravity, which states that gravity forces act instantaneously, Einstein’s theory of relativity declares that nothing can move faster than light, not even gravity [2]. Gravitational waves arise when, for instance, massive black holes interact with each other, causing alternately stretching and squeezing in the space-time continuum. Detecting these ”ripples” requires extremely precise measurements in the order of picometers (10−12 m) in the 10−3 to 10−1 frequency band [7]. This not only requires an advanced measurement system, but also an exceptionally accurate attitude and orbit control system. It is the task of the LISA Pathfinder to test, demonstrate and evaluate the advanced technology under development for the LISA mission.

5.2 Actuator System

At the time this thesis was written there were two actuator systems proposed to be placed on LISA Pathfinder [4]. One was a FEEP micropropulsion system, and the other a cold-gas mi- cropropulsion system. Both were designed to operate independently of each other and provide a full six degree of freedom control even with one thruster failed. The configuration of the two systems was also the same: Twelve thrusters placed in three identical modules of four thrusters each, see figures 5.2 and 5.3. Looking at the figures it is clear that the thruster configuration is symmetric. Only a phase difference exists between the FEEP system and the cold-gas system. Having two propulsion systems on a spacecraft with limiting space available and with mass constraints enforced requires an efficient utilization of the propellant available. Therefore, a minimum flow rate controller was chosen as the baseline for this implementation.

Exact specifications on the performance of the thrusters were not available when this thesis was written, but they were assumed to be able to provide a continuous thrust from 0.1 µN to 150 µN. This limited maximum force of the thrusters puts a stringer constraint on the thruster system than the overall consumption of propellant. The force is in other words the limit of the thruster

Page 47 5.3. IMPLEMENTATION AND RESULTS

Figure 5.1: Artist conception of the LISA mission, which will consist of three spacecraft, flying a formation in the shape of a triangle having a side length of five million kilometers. LISA Pathfinder, on the other hand, will be just two spacecraft, and the distance between them a lot shorter. (Courtesy of ESA.)

system.

Knowing the thruster configuration, the thruster limits, and that a minimum flow rate controller with a force limit is to be used, it is possible to calculate a control authority plot. The result of this can be seen in figure 5.4. A 90◦ arc has been added to the plot, marking the lower boundary of the control authority plot. The radius of the arc is 1.5 · 10−4, which is consequently also the minimum control authority.

5.3 Implementation and Results

In order to validate the functionality of the controller, it was tested against two disturbance profiles as might be expected during the first 40 seconds after deployment. The results from these calculations can be seen in two different plots. One plot shows the length of the force and torque command vector along with the control authority of the thruster system in that direction. The other plot shows the desired forces and torques projected onto each of the six degrees of freedom, that is force in x-, y- and z-direction and torque around x-, y- and z-axis, along with the achieved forces and torques in those directions.

For simplicity, only the case where all 12 thrusters are functioning has been considered.

Page 48 CHAPTER 5. IMPLEMENTATION - LISA PATHFINDER

Figure 5.2: The position and orientation of the FEEP thrusters on LISA Pathfinder.

Figure 5.3: The distribution and relative position of the FEEP and cold-gas thrusters.

5.3.1 Profile 1

Immediately after being deployed the attitude and orbit control systems (AOCS) called for a lot of authority from the thruster system in order to adjust the spacecraft into its correct posi- tion. As seen in figure 5.5 this authority initially exceeded the control authority of the system, a situation that may result in instability if unchecked. After about 16 seconds, however, the control authority became higher than the command vector, and the system was able to correct the disturbance. It should be noted that the length of the command vector for the most part surpassed the minimum control authority of 1.5 · 10−4 for another 14 seconds. But because the control authority in those directions happened to be larger, the system was able to compensate anyway.

Figure 5.6 shows the difference between the desired and the achieved forces and torques. When the AOCS demanded more thrust than the thrusters were able to give, they saturated, resulting in a different end result than the one called for. This is a cause for concern as it may produce an instability. A way to manage this situation is to scale down the thrust command vector whenever a thruster is in danger of being saturated. The scaling would then be done so as to

Page 49 5.4. EVALUATION

Figure 5.4: The control authority plot for LISA Pathfinder, assuming each thruster operational and having a thrust generating capability between 0.1 µN and 150 µN. The minimum control authority is 1.5 · 10−4, and in the plot it is marked by an arc.

result in the force and torque vector having the same direction as the one desired, but being of a smaller magnitude.

5.3.2 Profile 2

The results from the second profile can be seen in figures 5.7 and 5.8. These are in general the same as for the first profile. Initially, the thruster system was unable to achieve the high authority demanded by the AOCS, but after about 11 seconds it fell below the control authority, and after 15 seconds it was below the minimum control authority.

5.4 Evaluation

The implementations made in this chapter underline the importance of tools like the control au- thority when evaluating the sufficiency of a thruster system. In short, all forces and torques that are expected to be compensated by the thruster system should be below the control authority, and preferably below the minimum control authority to be on the safe side. If the system is required to function fully with one or more thruster failures it is also important to consider the control authority and minimum control authority under those conditions.

Page 50 CHAPTER 5. IMPLEMENTATION - LISA PATHFINDER

−4 x 10 7 Control Authority Length of Command Vector 6

5

4

3

2 Length of Force and Torque Vector 1

0 0 5 10 15 20 25 30 35 40 Time [s]

Figure 5.5: The length of the force and torque command vector in the first disturbance profile compared with the control authority of the thruster system in that direction. Vertical bars mark regions where the control authority is exceeded.

No consideration was in this implementation taken to how long it took to solve the linear pro- gram of the minimum flow rate controller. The reason for this is that the time depends greatly on the hardware and software available, and thus bare no general quantity. But on a (year 2004) modern laptop computer, with a force and torque command vector frequency of 10 Hz, the controller was able to find a solution with time to spare. However, in the event that the controller does not find a solution within a given time frame, it is prudent to have a back-up controller based on the pseudoinverse controller.

A back-up solution is also a good idea to implement when the controller requests a force and torque command vector that exceeds the control authority. Such a back-up should scale down the thrust command vector so as to produce a force and torque vector in the same direction as the one called for, but of a magnitude at or below the control authority.

Page 51 5.4. EVALUATION

Achieved a b c Demanded 200 60 150 400 40 100 N] 200 µ 50 20 0 0

Force [ 0 −50 −200 −20

−100 −40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40

d e f 200 60 200 150 40 150 100 20

Nm] 100 µ 50 0 50 0 −20 0 Torque [ −50 −40 −50 −100 −60

0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Time [s] Time [s] Time [s]

Figure 5.6: The demanded forces and torques in the first disturbance profile plotted with the achieved forces and torques. The vertical bars cover the same regions as the ones in figure 5.5. Clearly the thrusters are unable to supply the thrust required whenever the control authority is exceeded.

Page 52 CHAPTER 5. IMPLEMENTATION - LISA PATHFINDER

−4 x 10 7 Control Authority Length of Command Vector 6

5

4

3

2 Length of Force and Torque Vector 1

0 0 5 10 15 20 25 30 35 40 Time [s]

Figure 5.7: The length of the force and torque command vector in the second disturbance profile compared with the control authority of the thruster system in that direction. Vertical bars mark regions where the control authority is exceeded.

Page 53 5.4. EVALUATION

Achieved a b c Demanded 150 1000 100 50

N] 50 500 µ 0 0

Force [ −50 0 −100 −50 −150 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40

d e f

400 40 200 300 20

Nm] 200 0 µ 100 100 −20 −40 0

Torque [ 0 −60 −100 −80 −200 −100 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Time [s] Time [s] Time [s]

Figure 5.8: The demanded forces and torques in the second disturbance profile plotted with the achieved forces and torques. The vertical bars cover the same regions as the ones in figure 5.7. Note that the thrusters are unable to supply the thrust required whenever the control authority is exceeded.

Page 54 6 CONCLUSIONS

The current development in thruster design is very exciting and display a great potential for future application on scientific missions in particular. Keywords are high specific impulses along with low and precise thrust levels. Several concepts are under way to meet these requirements, though many have a substantial amount of obstacles to overcome before an actual implementa- tion in an AOCS can be achieved. Foreseeing which concept to be the most successful one is of course impossible, but two worth keeping an eye on are in the author’s opinion FEEP thrusters and cold gas microthrusters. The first because of its highly attractive specific impulse, thrust level and efficiency. The second for having a very low power consumption, a proven reliability and a relatively low cost.

This thesis has also shown three ways to optimize the thruster actuation − the minimum flow rate controller, the minimum power controller and the minimum force controller. The controller that perhaps carries the most practical value is the minimum flow rate controller, as it in space always is of paramount importance to optimize the utilization of the limited resources at hand. The minimum power controller, on the other hand, has a big advantage in being very fast and simple. Its reliable characteristics also makes it well suited as a back-up controller for the other two. But it is important to keep in mind the the actual controller chosen should always be the one best suited under the actual conditions. The goal when designing a controller should always be to maximize the minimum control authority.

6.1 Limitations

The work done in this thesis is theoretical, and several practical issues have intentionally been left out of the calculations for the three types of controllers. For instance, thrusters generally suffer from imperfections such as noise, plume divergence and an adjustment time taken to reach the desired thrust level. All these give rise to an error between the desired thrust and the thrust that is actually achieved. They are therefore important to consider when designing the whole AOCS, of which the controllers described in this thesis are but one part.

Another important practical concern not discussed in this thesis is the conversion from physical thruster parameters to thrust values. This is judged to be best known by the manufacturer of the thruster.

6.2 Future Work

There are several areas in which to proceed the work begun in this thesis. Finding the minimum control authority, for instance, is a vital tool in evaluating a thruster system. It would be highly desirable to develop a more time efficient and reliable method for finding the minimum control authority than the one described in chapter 4. And if such a thing can be achieved it could also be used to construct the much informative minimum control authority plot without first having

Page 55 6.2. FUTURE WORK to calculate the larger control authority plot, and thus save even more time for the designer.

It would also be of interest to explore alternative approaches to the linear programs that in this thesis were used to solve the minimum flow rate controller and the minimum force controller. A simpler method would be most advantageous and would lessen the lead in computational effort currently held by the minimum power controller.

Finally, it would be of great value to investigate general relations between the physical thruster parameters and the actual thrust values they give rise to.

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