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Optimal Thruster Actuation in High Precision Attitude and Orbit Control Systems

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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 LuleºaUniversity of Technology, Sweden September 2005 Examiner: Priya Fernando Department of Space Science, Kiruna LuleºaUniversity 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, ¯eld 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 1-norm of the thrust command vector, resulting in a minimum flow rate controller, a minimum power controller or a minimum force controller respectively. The ¯rst and last are found by solving linear programs, the middle by using the pseudoinverse of the thruster con¯guration 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 o® by implementing them on the LISA Path¯nder mission. Cal- culations show the algorithms to work well, but a more e±cient way of ¯nding the minimum control authority is desirable. Page I SAMMANFATTNING (Abstract in Swedish) Det Äokande kravet pºahÄogprecisions attityd- och banreglering av rymdfarkoster sÄatter strÄanga krav pºastyrraketer och deras styrning. BÄast lÄampade fÄor denna uppgift Äar proportionella styr- raketer som noggrant kan producera krafter pºamikronewtonnivºamed mycket lite brus. Sºadana styrraketer under nuvarande utveckling Äar kallgas-mikrostyrraketer, elektrisk fÄaltemissionsfram- drivning (FEEP), kolloid-styrraketer och mikroelektrotermiska styrraketer, med jonmotorer och Hall-styrrakter som har attraktiva egenskaper om en fÄorminskning Äar mÄojlig. Optimalt utnytt- jande av proportionella styrraketer kan uppnºasgenom att minimera 1-, 2-, eller 1-normen av styrvektorn, vilket respektive resulterar i en minsta flÄodeskontroller, en minsta e®ektkontroller och en minsta kraftkontroller. Den fÄorsta och sista erhºallsgenom att lÄosa linjÄara program, den mittersta genom att anvÄanda pseudoinversen av styrraketernas kon¯gurationsmatris tillsammans med en bias. Kontrollauktoriteten, vilket Äar den maximala prestandan frºanett styrraketssystem, kan hittas genom att maximera kraft- och momentprestationen. Ett enda nummer, kallat den minsta kontrollauktoriteten, mÄater den svagaste prestationen frºanstyrraketssystemet. Alla dessa koncept ges en noggrann genomgºang,och avhandlingen avrundar genom att implementera dem pºaLISA Path¯nder-missionen. BerÄakningar visar att algoritmerna fungerar bra, men en mer e®ektiv metod fÄor att hitta den minsta kontrollauktoriteten Äar ÄonskvÄard. 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 De¯nition . 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 Path¯nder . 47 5.2 Actuator System . 47 5.3 Implementation and Results . 48 5.3.1 Pro¯le 1 . 49 5.3.2 Pro¯le 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 ¯eld. 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 con¯guration, 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 con¯guration in test case 1. It was made assuming that the thrusters could be turned completely o®, 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 di®erent algorithms for ¯nding the thrust command vector. Calculations were made assuming that the thrusters could be turned o® 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, T1;1. The ¯gure was made assuming that the thrusters could be turned completely o®, 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 ¯gure 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 ¯xed bias algorithm. 42 4.7 The control authority plot made from the control authority vectors used to make the control authority surface in ¯gure 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 ¯gure 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 ¯nding the minimum control authority. The inner loop ¯nds the control authority with a maximizing linear program, while the outer loop ¯nds 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 ¯ve million kilometers. LISA Path¯nder, on the other hand, will be just two spacecraft, and the distance between them a lot shorter.
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