Technical Note. Secondary AOCS Design. Laszlo Szerdahelyi, Astrium GmbH. Walter Fichter, Astrium GmbH. Alexander Schleicher, ZARM Date:

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1 Technical Note HYPER Title: Secondary AOCS Design Laszlo Szerdahelyi, Astrium GmbH Walter Fichter, Astrium GmbH Prepared by: Alexander Schleicher, ZARM Date: Project Management: Ulrich Johann Distribution: See Distribution List Doc. No: HYP-2-01 Page 1

2 astrium GmbH Technical Note HYPER Copying of this document, and giving it to others and the use or communication of the contents thereof, are forbidden without express authority. Offenders are liable to the payment of damages. All rights are reserved in the event of the grant of a patent or the registration of a utility model or design. Doc. No: HYP-2-01 Page A-2

3 Change Record Issue Date Sheet Description of Change Release xx all all all all all all all all some editorial changes Correction of the thruster force command table in ch on p. 14, X and Y axis only Complete revision, including change requests from PM4 Complete revision, new chapters and sections added Modifications according to meeting in FN Complete and final revision, including the chapter controller algorithms design Some modifications according to final review by Phil Airey Doc. No: HYP-2-01 Page A-I

4 Table of Contents 1 INTRODUCTION SCOPE ABBREVIATIONS AND ACRONYMS 2 2 DOCUMENTS APPLICABLE DOCUMENTS REFERENCE DOCUMENTS 4 3 SYSTEM REQUIREMENTS AND CONSTRAINTS RELATED SYSTEM REQUIREMENTS SPACECRAFT PARAMETERS SPACECRAFT GEOMETRY MASS PROPERTIES SURFACE PROPERTIES SOLAR ARRAY PROPERTIES FORCES AND TORQUE FORCES AND TORQUE WORST CASES DISTURBANCE FORCES SUMMARY DISTURBANCE TORQUE SUMMARY 21 4 ARCHITECTURE OPERATIONAL MODES SECONDARY AOCS MODES SET-POINTS CONTROL LOOPS TIMING AND SAMPLING FREQUENCIES SENSOR AND ACTUATOR CHARACTERISTICS STAR TRACKER DRAG-FREE SENSORS PRECISION STAR TRACKER MICRO-PROPULSION SENSOR ACQUISITION DRAG-FREE SENSOR ACQUISITION PRECISION STAR TRACKER ACQUISITION REORIENTATION MANOEUVRES SLEW DURATION SENSOR ACQUISITION TOTAL REORIENTATION TIME BETWEEN GUIDE STAR POINTING 32 Doc. No: HYP-2-01 Page B-I

5 5 MEASUREMENT CONCEPT LINEAR ACCELERATION MEASUREMENT OF ONE DRAG-FREE SENSOR CONFIGURATION WITH ONE DRAG-FREE SENSOR CONFIGURATION WITH TWO DRAG-FREE SENSORS LINEAR ACCELERATION MEASUREMENT ANGULAR ACCELERATION MEASUREMENT CONFIGURATION WITH THREE DRAG-FREE SENSORS LINEAR ACCELERATION MEASUREMENT ANGULAR ACCELERATION MEASUREMENT CONFIGURATION WITH FOUR DRAG-FREE SENSORS DRAG-FREE SENSOR SUMMARY AND BASELINE CONFIGURATION ATTITUDE MEASUREMENT ATTITUDE MEASUREMENT TRANSVERSE TO POINTING DIRECTION ATTITUDE MEASUREMENT AROUND POINTING DIRECTION 38 6 CONTROL LOOP AND DISTURBANCE REJECTION CONTROLLING THE ASU PHASE OVERVIEW DRAG-FREE CONTROL TRANSFER FUNCTIONS DRAG-FREE DISTURBANCE REJECTION REJECTION OF LINEAR ACCELERATION DISTURBANCE REJECTION OF ANGULAR ACCELERATION DISTURBANCE DRAG-FREE SENSOR DISTURBANCES DRAG-FREE SENSOR BIAS DRAG-FREE SENSOR NOISE 45 7 CONTROLLER, SENSOR, AND ACTUATOR REQUIREMENTS SENSORS AND ACTUATORS RESIDUAL ACCELERATION REQUIREMENTS DISTURBANCE REJECTION OF DETERMINISTIC ACCELERATIONS NOISE REJECTION ANGULAR RATE REQUIREMENTS POINTING REQUIREMENTS AXES TRANSVERSE TO PST BORESIGHT PST BORESIGHT AXIS PST ACQUISITION REQUIREMENT 49 8 CONTROLLER DESIGN SUMMARY OF IMPORTANT REQUIREMENTS AND ASSUMPTIONS ALGORITHM DEVELOPMENT ATTITUDE CONTROLLER FOR THE X-AXIS ATTITUDE CONTROLLER FOR THE Y/Z-AXES OPTION: Y/Z AXIS ATTITUDE CONTROLLER WITH ACCELERATION MEASUREMENTS 53 Doc. No: HYP-2-01 Page B-II

6 8.2.4 DRAG-FREE CONTROL FOR THE X/Y/Z-AXES PRELIMINARY PERFORMANCE ANALYSIS NOMINAL PERFORMANCE ROBUSTNESS WITH RESPECT TO MASS PROPERTIES DRAG-FREE SENSOR BIAS FILTERING MICRO-PROPULSION ACTUATION 61 9 SIMULATION PLAN OBJECTIVES AND SCENARIOS CONTROLLER CONFIGURATION PARAMETERS, INITIAL CONDITIONS, AND NOMINAL DATA SET PARAMETERS AND INITIAL CONDITIONS NOMINAL DATA SET TEST SCENARIO ATTITUDE CONTROL WITHOUT DFS TEST SCENARIO FINDING INDIVIDUAL EFFECTS REALISTIC AND WORST CASES ALTERNATIVE CONTROLLER DESIGNS 70 Doc. No: HYP-2-01 Page B-III

7 1 INTRODUCTION 1.1 Scope This document describes the Secondary AOCS design of HYPER. The Secondary AOCS is the control system that is used during the science mode of HYPER, i.e. when scientific measurements are taken. This technical note does not cover any aspects of the classical AOCS functions, such as acquisition modes, orbit raise manoeuvres, etc. Basically, this technical note can be divided into the following parts: 1. System architecture and control system design trades 2. Summary of Secondary AOCS requirements, in particular requirements for the estimator and controller design 3. Design of the actual controller algorithms 4. Test plan and test cases The results of the simulation campaign are documented in the Secondary AOCS Performance Analysis, HYP Abbreviations and Acronyms AA ASU BOL CoM DFACS DFP DFS DoF FEEP FI FOV HYPER LT MLI Atomic Assembly Atomic Sagnac Unit Begin-Of-Life Centre of Mass Drag-Free and Attitude Control System Drag-Free Point Drag-Free Sensor Degree of Freedom Field Emission Electric Propulsion Fiber Injector Field of View Hyper-Precision Cold Atom Interferometry in Space Lense Thirring Multi Layer Insulation Doc. No: HYP-2-01 Page 2

8 OB PM PST RRM SA SC SD SSM Optical Bench Proof Mass Precision Star Tracker Retro-Reflecting Mirror Solar Array Spacecraft Spectral Density Secondary Surface Mirror Doc. No: HYP-2-01 Page 3

9 2 DOCUMENTS 2.1 Applicable Documents The following documents are applicable according to the extent as specified within chapter 3 ff. AD Doc. No. Title 01 HYP-ESA-RS-SPA-3 Hyper Payload Requirement Specification, Issue 2, 7 June HYP-2-05 Performance Requirements Breakdown, version HYP-4-01 HYPER Drag-Free Sensor Requirements Specification, version HYP-5-05 HYPER FEEP Requirements Specification 2.2 Reference Documents RD Doc. No. Title 01 HYP-ASU-DD-CST-1 Description of the Atomic Sagnac Unit, Issue 1, 31 January CDF-09 Hyper CDF Study Report (as ammended by Errata Corrigum ref HYP-CDF-E/C-1) 03 HYP-LTE-DD-CST-1 Explanation of the Lense Thirring Effect 04 HYP-1-01 Orbit Selection Report 05 HYP-3-01 PST and OB Design Report 06 HYP-2-04 HYPER Simulator User and Maintenance Manual Doc. No: HYP-2-01 Page 4

10 3 System Requirements and Constraints 3.1 Related System Requirements The system requirements for the HYPER satellite are given in AD02. In the following table, the requirements relevant to the design of the Secondary AOCS (drag-free and attitude control system) are listed. Derived requirements, as they result from (control) system design described in this document, are given in the chapter Controller, Sensor, and Actuator Requirements in this document. Requirement # Requirement 3 Value 1 Value R2-ENL R2-ENL R2-ENL R2-ENL dragfree point (each axis, G 1 X, G 2 X, GY) angular acceleration around transverse axes (axes G 1 X and G 2 X) rate around PST boresight rigid body rate around transverse axes G 1 Z and G 2 Z) < /2 < m/sec 2 m/sec 2 < /2/0.5 < rad/sec 2 rad/sec 2 < 10-6 rad/sec < rad/sec < rad/sec < rad/sec Table 3-1: Summary of system requirements applicable to 2 nd AOCS design. For the computation of the 3 values the following frequency characteristics shall be taken into account: Relative gain for the rotation (R1-ENL-01) and linear acceleration (R1-ENL-02) as it is specified in RD01. These relative gains can be considered as low pass filters. The above requirements are applicable after passing the signals through the corresponding filter function (relative gains). Low frequency ASU phase control loop as it is sketched in the figure below. If the signal/value under consideration increases with less than second order below the corner frequency of the ASU phase control loop, the variance contribution from zero frequency to the ASU phase correction corner frequency can be neglected. Doc. No: HYP-2-01 Page 5

11 0 Bode Diagram -10 Phase (deg) Magnitude (db) Frequency (rad/sec) Figure 3-1: Transfer function of the ASU phase correction loop. 3.2 Spacecraft Parameters Doc. No: HYP-2-01 Page 6

12 3.2.1 Spacecraft Geometry Figure 3-2 shows the SC geometry model that is used for design and analysis purposes of the Secondary AOCS.. Ø Ø y z Ø 860 Ø x z 100 Ø 860 Ø Figure 3-2: SC geometry It is noted that the SC geometry model is currently taken from the HYPER CDF Report Mass Properties The following SC mass properties are applicable for AOCS analysis and design purposes, where the moments and cross products of inertia are referred to the spacecraft reference frame S (AD02) but shifted to the CoM. The moments of inertia in Table 3-2 are preliminary estimates and include margin. Note that the mass properties remain constant during the period in which the Secondary AOCS is in operation. Doc. No: HYP-2-01 Page 7

13 Mission Phase Initialisation of 2 nd AOCS Nominal Mission Mass [Kg] with 20% margin CoM (x,y,z) [m] Ix, Iy, Iz [kgm²] Ixy, Ixz, Iyz [kgm²] / 0 / / 0 / / 270 / / 270 / / 0.0 / / 0.0 / 0.0 Table 3-2: Mass properties Surface Properties Figure 3-3 shows the SC surface model. Note that the interface ring to the launcher adapter is neglected in Figure 3-3. GaAs z MLI MLI x y SSM MLI SSM Figure 3-3: SC surface materials Doc. No: HYP-2-01 Page 8

14 Table 3-3 lists up typical BOL reflection coefficients of materials for infrared and visible light that are used for analysis purposes. Infrared Visible (albedo) Material C a absorption C s specular C d diffuse C a absorption C s specular C d diffuse MLI Solar Array (Si) Solar Array (GaAs) SSM Table 3-3: Spacecraft surface properties Solar Array Properties The solar array (SA) has a circular shape of a diameter of 2.2m. The SA surface area of 3.8m² is used for radiation pressure force and torque analysis purposes. The SA cell material is Gallium Arsenide (GaAs). 3.3 Forces and Torque Forces and Torque Worst Cases Worst Case Profiles Worst case disturbance profile were generated by simulation. In these simulations the seasonal variations of the environment over the year as well as the expected operational attitudes of the spacecraft were considered. Daily variations of the environment disturbances were not considered. For each spacecraft axis and for each disturbance type (non-gravitational, solar radiation, etc., see below) those time history is plotted that showed the maximum peak value (over one orbit) from the complete set of simulation runs. This means that the X,Y and Z force and torque profile for a particular disturbance type does not correspond necessarily to the same simulation run. The following disturbance types are shown: total environmental force and torque solar radiation pressure force and torque air drag force and torque gravity-gradient and magnetic disturbance torques albedo and infrared radiation disturbance force and torque Doc. No: HYP-2-01 Page 9

15 The superimposed saw-tooth profile on the albedo force and torque profiles is due to the angular discretisation of the earth radiation map. Parameters Orbit parameters: inclination deg argument of perigee 90.0 deg semimajor axis m eccentricity e-004 right ascension of ascending node depends on date (sun-synchronous orbit) SC parameters: SC attitude: +-30 deg variation about Y,Z at 10 deg steps. Nominal SC attitude: SC X axis pointing off the Sun in opposite direction and it lies in the equatorial plane. SC Z axis is parallel to Z J2000. The SC Y axis completes the right-handed ref. frame. The attitude variations have to be considered from this nominal attitude. SC moment of inertia: Ixx=300, Iyy=250, Izz=250, Ixy=Ixz=Iyz=0 kg m² SC mass: 770 kg MSIS86 density model parameters: pessimistic values F (mean) F10.7_Prev 380 (previous day) AP 300 (magnetic activity index) Doc. No: HYP-2-01 Page 10

16 Figure 3-4: Total environmental torque. Doc. No: HYP-2-01 Page 11

17 Figure 3-5: Total environmental force. Doc. No: HYP-2-01 Page 12

18 Figure 3-6: Solar radiation torque. Doc. No: HYP-2-01 Page 13

19 Figure 3-7: Solar radiation force. Doc. No: HYP-2-01 Page 14

20 Figure 3-8: Aerodynamic torque. Doc. No: HYP-2-01 Page 15

21 Figure 3-9: Aerodynamic force. Doc. No: HYP-2-01 Page 16

22 Figure 3-10: Albedo and IR torque. Doc. No: HYP-2-01 Page 17

23 Figure 3-11: Albedo and IR force. Doc. No: HYP-2-01 Page 18

24 Figure 3-12: Gravity gradient torque. Doc. No: HYP-2-01 Page 19

25 Figure 3-13: Gravity gradient torque. Doc. No: HYP-2-01 Page 20

26 3.3.2 Disturbance Forces Summary The deterministic disturbance forces are summarised in Table 3-4. Disturbance Force Level Frequency Remarks Gravity-Gradient radial (DFP offset from COM 10 cm in the YZ plane) 150 N double orbit frequency in +/- Y and +/-Z Air drag 34 N (average at MSIS max) 120 N (daily peak at MSIS max) double orbit frequency and higher average value from HYP-1-01 Solar Pressure Force N ~ constant force Earth IR and Albedo 4 N constant and higher frequency parts, negligible High altitude winds (modulation of air drag) +/- 3.4 N negligible Thermal emission < 2 N approximately constant, assuming radiation power of 500 W, negligible RF emission negligible assuming a radiation power of several W Table 3-4: Disturbance forces Disturbance Torque Summary The deterministic disturbance torque is summarised in Table 3-5. Disturbance Torque Level Frequency Remarks Magnetic moment (= 1 A m 2 ) up to 50 N m dependent on the orientation of the magnetic moment vector double orbit frequency in all Doc. No: HYP-2-01 Page 21

27 axes Gravity gradient torque 80 Nm double orbit frequency drives thrust requirement Air drag induced torque 6.5 N m (average) up to 100 N m (max) double orbit frequency and higher, depends on CoM-CoP offset Solar pressure torque up to 10 N m constant, depends on CoM- CoP offset Thrust vector misalignment and angular stability about 2.5 % of the torque for X-axis control assumption: 1.5 deg thruster misalignment Table 3-5: Disturbance torque. Doc. No: HYP-2-01 Page 22

28 4 Architecture 4.1 Operational Modes Secondary AOCS Modes According to the ESA CDF report the Secondary AOCS has basically two operational modes: 1. Inertial Pointing Mode. This mode includes also inertial re-orientation manoeuvres, i.e. a slew from one guide star to another 2. Science Mode In this mode the science measurements are performed. All performance requirements must be fulfilled here. In Science Mode 6 DoF are to be controlled, i.e. 3 rotational DoF (attitude, rate, and angular acceleration) and 3 translational DoF (linear accelerations). In the Inertial Pointing Mode only the rotational DoF are controlled. In the following, 6 DoF control (Science Mode) is considered unless mentioned otherwise. The Science Mode consists of two sub-modes: Hold Mode. In this sub-mode the spacecraft is attitude controlled with respect to the attitude that is measured at sub-mode entry. Control Mode. In this sub-mode the science measurements are taken during steady state conditions. The spacecraft is controlled with respect to the set point given below Set-Points During the Science Mode (Control Sub-Mode) the following set-points (reference value for drag-free and attitude control) apply: a. ref,dfp Non-gravitational linear acceleration at drag-free point: T ref,t Inertial attitude: T, where ref, T is the reference attitude with respect to the Observation Target Frame (T frame) given in AD02. With other words the attitude deviation with respect to the T frame shall vanish. 4.2 Control Loops The Secondary AOCS provides basically a 6 DoF control therefore, loosely speaking 6 control loop are required. Doc. No: HYP-2-01 Page 23

29 In Table 4-1 the sensors used for the Secondary AOCS and the associated measurement information is shown. For Table 4-1 a baseline with two drag-free sensors is assumed. Position Attitude Rate Angular Acceleration Linear Accel. X/Y/Z X Y/Z X/Y/Z X Y/Z X/Y/Z DFSs estimated option (not used) x PST x estimated STR x estimated GPS Rx x Table 4-1: Sensors and associated measurement information. The detailed configuration and performance parameters are given in the paragraphs below. In Figure 4-1 a schematic view of the controllers and the use of the sensor information is shown. In Science Mode the precision star tracker (PST) provides 2-axis inertial attitude information. The third axis attitude (around the PST boresight) is extracted from the 3-axis off-the-shelf star tracker (STR). Furthermore, full 3- axis attitude information is the STR is used during re-orientation manoeuvres and PST measurement acquisition. Drag-free sensors (DFS) are mainly used to provide 3-axis linear acceleration measurement. The number and configuration of DFSs have an impact on the optical bench configuration and the amount of information (placement of drag-free point, angular acceleration, gravity gradient) that can be obtained. For the current mission baseline it turns out that the use of the DFSs for angular control purposes and for gravity gradient computation can be avoided. This is the preferable solution because it simplifies the control system. Depending on the required bandwidth of the angular control loops, the DFSs may be used to compute angular acceleration information. A GPS receiver is required since precise position information is the basis for Earth gravity gradient estimates. The latter is used for phase compensation purposes of the ASU, and for angular acceleration estimation in case of 2 or 3 DFSs. From Figure 4-1 it can be seen that the angular and linear control loops are decoupled and that nominally the control system does not rely on gravity gradient model information. Doc. No: HYP-2-01 Page 24

30 GPS Receiver Gravity Gradient Estimate Estimated gravity gradient To ASU for compensation purposes 3-axis attitude Star Tracker x-axis attitude X-Axis Attitude Controller PST DFS 1 DFS 2 y/z-attitude 3-axis linear accelerations Y,Z-Axis Attitude Controller X,Y,Z-Axis Linear Acceleration Controller Thruster Selection FEEPs 4 sets of 4 thrusters each Actuation signal for each FEEP thruster Figure 4-1: Controllers for 6 degrees of freedom control. 4.3 Timing and Sampling Frequencies The required sampling frequency of the Secondary AOCS is primarily dependent on the closed loop control bandwidth. For the study a basic sampling frequency of 10 Hz is assumed for the control system. This affects the sensors (DFS, PST, STR), actuators (FEEPs) and the computer. The operation of these components with 10 Hz is definitely feasible. Probably the typical closed loop bandwidth is in the range of rad/sec. Thus, the basic sampling frequency could be reduced at a later stage of the development, if this turns out to impose unnecessary limitations on component selection and cost reduction. The sensor data age (delay between data acquisition) and beginning of the computational interval of the computer is specified below, for each sensor. The delay between thruster command and actuation shall be assumed to be 100 msec (one computation interval). 4.4 Sensor and Actuator Characteristics Star Tracker Output 3-axis inertial attitude. Specification Sampling frequency: 10 Hz (could be operated at 2 Hz, this is not a feasibility issue) Delay time: 0.1 sec (could be larger, this is not a real feasibility issue) Bias: approximately orbit frequency Doc. No: HYP-2-01 Page 25

31 10 arcsec (3) transverse to boresight 30 arcsec (3) around boresight Noise: at sampling frequency 6 arcsec (3) transverse to boresight 60 arcsec (3) around boresight Configuration An off-the-shelf star tracker has to be used for the attitude control of the angular motion around the PST boresight. No particular configuration is defined within this study, since it does not have an impact on feasibility. Instead, worst case noise and bias characteristics around boresight must are taken into account properly. The alignment knowledge with respect to PST boresight is assumed to be better than 1 arcmin Drag-Free Sensors Output 1. Relative 3-axis linear acceleration of each DFS, expressed in the DFS frame 2. Relative 3-axis angular acceleration of each DFS, expressed in the DFS frame Nominally, only linear acceleration measurements are processed. Specification Sampling frequency: 10 Hz Further specification see AD03 and explanations/derivations below in this document Configuration The connection line between the two DFS coincides with the intersection of the two ASU planes. The nominal drag-free point is in the middle of the intersection line. Expressed in the optical bench system, the drag-free proof masses have the following coordinates: DFS1: r mm DFS1 DFS1: r ( ) 0 0 mm DFS2 T T The 350 mm is the half length of the optical bench, see Figure 4-2, the 100 mm is the half length of the edge of a DFS. Doc. No: HYP-2-01 Page 26

32 Figure 4-2: Optical bench dimensions Precision Star Tracker Output Inertial 2-axis attitude, expressed in the P frame Specification A sampling frequency: 10 Hz Field of view: 25 arcsec Further specifications see RD05 Configuration See RD05 and AD Micro-Propulsion Output Force and torque along all 3 spacecraft axes Specification Doc. No: HYP-2-01 Page 27

33 Command frequency: 10 Hz Further performance specification of one FEEP thruster see AD04 and explanations/derivations below in this document. Configuration Y Z X 24 Solar Array Figure 4-3: HYPER FEEP Thruster Arrangement (option 1 of Ref. HYP-5-02). The arrows show the ion thrust direction. Figure 4-3 shows the thruster arrangement. There are 4 thruster branches. In the following table the branches and the normalised thrust directions of the FEEP thrusters are listed. Thruster Branch Normalised force direction cosine vector Thruster position wrt SC reference frame S [m] Normalised torque vector [m] * Branch IV Branch III Branch II Branch I FEEP 1 -X [-1, 0, 0] 0.4 [ ] FEEP 11 +Y [0, 1, 0] 0.75 [ ] FEEP 21 -Z [0, 0, -1] 0.75 [ ] FEEP 2 -X 0.4 [ ] FEEP 12 -Y 0.75 [ ] FEEP 22 +Z 0.75 [ ] FEEP 3 +X 1.45 [ ] FEEP 13 -Y 0.75 [ ] FEEP 23 -Z 0.75 [ ] FEEP 4 +X 1.45 [ ] FEEP 14 +Y 0.75 [ ] FEEP 24 +Z 0.75 [ ] Doc. No: HYP-2-01 Page 28

34 * the SC COM position is assumed at [-0.925; 0; 0] m in the S reference frame. Direction Torque Force Thrusters Amplitude [µnm] * Thrusters Amplitude [N] * +X F21+F F3+F X F23+F F1+F Y F2+F3 150 F11+F Y F1+F4-150 F12+F Z F2+F4 150 F22+F Z F1+F3-150 F21+F * 100 µn FEEP thruster is assumed. 4.5 Sensor Acquisition Drag-Free Sensor Acquisition The major contributions to an acceleration measurement are drag gravity gradient linear acceleration caused by angular acceleration (times level arm) All other contributions such as centrifugal accelerations and self gravity are comparably small, i.e. negligible. The numerical values based on rough estimations are listed in Table 4-2. The sum of these is in the order of 10-6 m/sec 2. Taking into account a margin factor of 5 the measurement range of a single accelerometer should be larger than m/sec 2. Effect Value Derivation Remarks drag m/sec 2 given from RD=4, HYP-1-01, para for solar max, altitude 1000 km gravity gradient 10-6 m/sec 2 gravity gradient = /sec 2 level arm 0.5 m angular acceleration 10-7 m/sec 2 ang. accel rad/sec 2 level arm 0.5 m Sum 10-6 m/sec 2 Doc. No: HYP-2-01 Page 29

35 Table 4-2: Maximum acceleration measured by a single accelerometer Precision Star Tracker Acquisition The acquisition of the precision star tracker is determined by the relative accuracy between PST and STR, which includes all measurement errors and the mechanical misalignment. For successful PST acquisition, an angular area must be scanned that is larger than the total relative accuracy between PST and STR. The following contributors are considered for the relative accuracy of the STR with respect to the PST (half cone angles, 3): STR measurement accuracy noise : typically 6 arcsec (3). Actually, this is a measurement noise, the resulting pointing noise due to the measurement noise will be smaller. Therefore, the assumption is conservative. Misalignment of the CCD with respect to the STR baseplate: 10 arcsec (3) This value includes also thermal-mechanical deformation of the STR, etc. Alignment of the STR baseplate with respect to the P system (PST co-ordinate system) 30 arcsec (3) Margin of 4 arcsec (3) Any contributions from the PST are considered to be negligible The (linear) sum of these contributions is 50 arcsec. Thus, a potential guide star can be within an angular area of 50 arcsec centred around the nominal PST pointing direction. This is denoted as search area. The field of view of the PST is 25 arcsec. Scanning the search area 9 scans allows an overlap of 12.5 arcsec (half PST FOV) between scans. A guide star (calibration star) acquisition comprises then two steps: 1. Scan of the search area of 50 arcsec, with a FOV of 25 arcsec. This results in pointing in 9 directions. 2. When the guide star appears in the FOV, a re-orientation drives the guide star within a reduced PST FOV (15 arcsec). The latter is required in order to avoid that the guide star leaves the real FOV (25 arcsec) due to transients after controller witch. 4.6 Reorientation Manoeuvres Slew Duration Assuming a bang-bang profile of a reorientation from one guide star to another guide star, the slew time is approximately Doc. No: HYP-2-01 Page 30

36 t / I max where I is the moment of inertia, is the reorientation angle, and max is the maximum and constant torque applied for acceleration and deceleration. With a maximum torque of Nm and a moment of inertia of 350 kgm 2, the reorientation time as a function of the slew angle is plotted in Figure 4-4. This is just a rough estimation and gives only an indication of the slew time interval Time [h] Slew Angle [deg] Figure 4-4: Duration of a bang-bang reorientation maneuver Sensor Acquisition Three-Axis Star Sensor The reorientation manoeuvre can be implemented in a closed loop manner, since the 3-axis star sensor is switched on and in use continuously during Secondary AOCS operation. The maximum rate in case of a bang-bang slew of 40 deg (without rate limitation) is about deg/sec rad/sec. This is compatible with the operational rate range of an off-the-shelf star tracker. In any case a rate limitation can be implemented in the feedback control if necessary. Drag-Free Sensor In case of a 40 deg bang-bang reorientation manoeuvre the angular acceleration is about rad/sec 2 and the maximum rate is about rad/sec. Both, centrifugal terms and angular acceleration terms contribute therefore in the order of 10-7 m/sec 2. Thus, linear acceleration measurements are even possible during reorientation manoeuvres. Doc. No: HYP-2-01 Page 31

37 4.6.3 Total Reorientation Time between Guide Star Pointing The total time required for reorientation between steady state pointing to two different guide stars si composed of: Slew time according to Figure 4-4 (Inertial Mode) PST sensor acquisition (Inertail Mode), 10 minutes Settling time in PST FOV (Science Mode, Hold Submode), 10 minutes Transition from hold to fine pointing with PST around y and z (Science Mode, Control Submode), 10 minutes The total time that is needed for a guide star change is then in the order of 1 hour. Doc. No: HYP-2-01 Page 32

38 5 Measurement Concept In this chapter the different drag-free sensor configurations and the related information content is described. The baseline configuration is defined and justified. At the end of the chapter the (relatively simple) attitude measurement concept is described. 5.1 Linear Acceleration Measurement of One Drag-Free Sensor One drag-free proof mass say the first DFS, index 1- provides the measurement a ( ~ ~ Ur d s m 2 1 ω ω ) r1 under the assumption that thermo-elastic motion (Coriolis and relative acceleration) can be neglected. This assumption is usually fulfilled, since the DFS is servo-controlled (Coriolis acceleration) and the relative acceleration (cage motion) is made negligible by spacecraft design. Here, is the inertial rate of the frame where the drag-free sensor is mounted (G-system), U is the gravity gradient tensor, r 1 is the vector from the spacecraft CoM to the proof mass CoM, and d denotes the nongravitational forces (drag). s and m are self gravity acceleration and magnetic coupling, respectively. The impact of s and m must also be negligible (by spacecraft design), therefore the measurement can be modelled as a 2 ( ω ~ ω ~ r1 Ur1 d 1 ) Configuration with One Drag-Free Sensor With the realistic assumption that the centrifugal (quadratic rate) term is negligible, the acceleration at the PM is The acceleration at any drag-free point is given by a DFP a ωr ~ Ur d ω ~ ( r l / 2) U( r l / 2) d 1 12 where l 12 is the vector from the DFS to the drag-free point. This can be obtained by adding additional terms to the acceleration a 1 provided by the DFS, i.e. y1 a ~ ωl12 / 2 Ul12 / 2 d : The gravity gradient term can be modelled, l 12 is precisely known. However, since the angular acceleration cannot be measured, the related term (second right hand side term) of the above equation cannot be added and thus, results directly in an acceleration determination error. This error must be kept as small as possible by the DFS configuration. a DFP Doc. No: HYP-2-01 Page 33

39 Considering the angular motion around x is controlled much worse than around y/z (3-axis star tracker versus precision star tracker), it can be assumed that it is. This means the harmful angular acceleration component is around the x-axis. This implies that the connecting line from the DFS to the DFP shall be parallel to the x-axis. Since the DFP has to be on the ASU plane s intersection line (see AD02), the same applies for the DFS. x y, z 5.3 Configuration with Two Drag-Free Sensors Linear Acceleration Measurement A configuration with two DFSs is shown in Figure 5-1. In order to compute the linear acceleration y 1 of the drag-free point, the common-mode signal of the two drag-free sensors has to be computed a11 y1 1 ~ a 2 2 ~ 2 2 ( ω ω ) rdfp d UrDFP : a DFP where r DFP is the vector from the spacecraft CoM to the drag-free point. The drag-free point is defined by r DFP r r i.e. it is on the line between the two drag-free proof masses. The coefficients 1, 2 can be used to place the DFP on this connecting line. Usually the DFP is exactly between the first and second DFS, then and the linear acceleration is simply the sum of both DFS measurements: y 1 a 1 a DFS2 Drag-Free Point l 12 DFS1 r DFP r 2 r 1 S/C Centre of Mass Figure 5-1: Drag-free sensor configuration Angular Acceleration Measurement Doc. No: HYP-2-01 Page 34

40 The angular acceleration around y and z is of primary interest. It is recovered using differential measurements. In general a differential measurement of the two drag-free sensors is given by y 2 1a1 2a 2 ( ω ~ ω ~ 2 U)( 1r1 2r2 ) ( 1 2 ) d the last term must be eliminated, therefore it must be 0 12, e.g. 1, y 2 ( ω ~ ω ~ )( r2 r1 ) U( r2 ~ ~ 2 ( ω ω ) l Ul 12 r ) Then it is This information contains several terms. Now the following assumptions are used, based on mission requirements that have to be fulfilled for other reasons: 1. The last term can be compensated due to accurate modelling of the gravity gradient. The vector l 12 is known accurately since it is defined over the optical bench, i.e. a very stable structure. 2. The rate around the axes transverse to the PST boresight is controlled accurately, i.e.,. x y z Then, the differential measurement can be re-written as y 2 0 z y 0 x z y 0 x 0 x x y z x 0 y 2 x xz 0 l 2 x 12 The second and third elements of y 2 contain basically the angular acceleration information around y and z. In order to zero out angular accelerations terms around the x-axis, the y-, and z-components of l 12 must be zero (or very small). Thus, the vector l 12 has to be parallel to the x-axis. Since the drag-free point is on the ASU plane s intersection line (AD02), the drag-free sensors also have to be accommodated on this intersection line. 5.4 Configuration with Three Drag-Free Sensors Linear Acceleration Measurement In case of three DFSs the linear acceleration of the drag-free point is computed by y 1 1a1 2a 2 1 ( ω ~ ω~ ( ω ~ ω ~ U) r 3 3 DFP a 3 1r1 2r2 3r U) 1 2 d : a The drag-free point lies in a plane that is spanned by the drag-free sensor locations. Two of the three parameters i can be used to place the DFP within that plane. DFP 3 3 d Doc. No: HYP-2-01 Page 35

41 5.4.2 Angular Acceleration Measurement Two differential measurements can be formed, each of them as it is done in case of two DFSs. Assuming that the gravity gradient terms are compensated by means of a model, the following equations can be used to obtain angular acceleration information: y y ~ ~ 2 2 ( ω ω ) l12 ~ ~ 2 3 ( ω ω ) l13 Assuming the quadratic rate term to be negligible, this yields y y 2 3 ~ l ~ l This equation is over-determined and can be solved e.g. using least squares or weighted least squares. Note: The assumption of negligible rate is based on the fact that the rate is controlled using angular acceleration measurements. Thus, the rate is a feedback disturbance in the measurement process ω 5.5 Configuration with Four Drag-Free Sensors In case of four DFSs the angular acceleration as well as the gravity gradient can be measured. The processing scheme is outlined in the following: Step 1: Form the full differential acceleration matrix, similar to a gravity gradient matrix. The coordinate system for the differential measurements is defined by the connection line between the fourth DFS and the DFS 1 to DFS 3, i.e. in general it is not orthogonal. The result of the differential measurement matrix is denoted D s. Step 2: Transform D s to a Cartesian co-ordinate system. 1 Dc TcsDsTcs ~ ( ω ω ~ 2 ) U D s contains a symmetric (centrifugal term and gravity gradient) and skew-symmetric part (angular acceleration) Step 3: Extract symmetric and skew-symmetric part A ( D 2 D c A S 1 T ), T c Dc c Dc 1 S ( D 2 Step 4: Recover the angular acceleration from the skew-symmetric part A ~ ω A ) Doc. No: HYP-2-01 Page 36

42 Step 5: Estimate the rate based on the angular acceleration (from step 4) and attitude information (from star sensor). This is a dynamic estimation process. Step 6: Recover the gravity gradient information from the symmetric part ~ 2 U S ω 5.6 Drag-Free Sensor Summary and Baseline Configuration One Drag-Free Sensor: The drag-free point can be placed outside of the DFS only by using a gravity gradient model. The DFS shall be on the ASU plane s intersection line. Two Drag-Free Sensors:. The drag-free point, i.e. the point of nominally zero linear acceleration can be placed anywhere on the connection line between the two DFSs, without relying on a gravity gradient model. The angular acceleration can be obtained only about 2 axes transverse to the connecting line of the two DFSs. Since the angular motion around the Y/Z axes shall be controlled precisely, the DFS connecting line shall be parallel to (coincident with) the ASU planes intersection line. Three Drag-Free Sensors: The DFP can be placed within two dimensions, i.e. a plane spanned by the three DFS locations, without relying on a gravity gradient model. In this case full 3-axis angular acceleration can be obtained and thus, there is no special configuration requirement resulting from angular motion. Angular acceleration computation requires a gravity gradient model. Four Drag-Free Sensors: In addition to the previous case of 3 DFSs, the gravity gradient can be measured and the DFP can be placed anywhere. Also, the angular acceleration is computed without the assumption of small angular rates. Baseline Configuration: Table 5-1 summarises the properties of the different configurations. From a mass, power, and cost point of view the one DFS solution is preferable. In order to have some margin with respect to the angular control authority (bandwidth) around the y and z axes and with respect to redundancy (DFS failure), the configuration with 2 DFS is selected as baseline. 1 DFS 2 DFS 3 DFS 4 DFS Linear acceleration measurement Drag-free point placement w/o GG Not possible On line defined by 2 DFS On plane defined by 3 DFS Angular acceleration measurement Anywhere in the ASU volume Information n/a 2-axis transverse to 3-axis 3-axis content connection line (Y/Z) Measurement disturbance n/a Quadratic rate term Quadratic rate term None, independent of rates Doc. No: HYP-2-01 Page 37

43 Configuration requirements Requirement DFS on ASU plane s intersection line DFSs connection line coincident with ASU plane s intersection line Plane defined by 3 DFSs shall contain ASU plane s intersection line none Table 5-1: Summary of different DFS configurations. 5.7 Attitude Measurement Attitude Measurement Transverse to Pointing Direction The precision star tracker directly provides inertial 2-axis attitude information transverse to the PST boresight. Thus, no data processing except scaling, etc. is required. In order to exploit the full PST capability (with respect to accuracy) it must be operated with the star spot centred in the FOV. Thus, any aberration effects or any other data processing must not be considered, except simple scaling. This procedure will lead to a pointing error that is periodical, since the aberration effect will be periodical. The corresponding rate is estimated in the following: The magnitude of the aberration effect is approximately v ˆ ab rad / sec c where v is the orbital velocity, about m/sec and c is the speed of light, about m/sec. Then the maximum attitude error is about rad. Since this attitude error occurs at orbit frequency the corresponding rate error is ˆ ab ˆ For the orbit frequency 0 = rad/sec the rate error becomes rad/sec. This is already about 60% of the 3 rate error requirement of rad/sec. ab Attitude Measurement Around Pointing Direction The attitude component around the PST boresight is derived from the 3-axis attitude information provided by the off-the-shelf star tracker. The algorithm for this computation is outlined in the following: (1) Attitude provided by the star tracker is denoted by TPI, i.e. the attitude of the P system with respect to the inertial system. The inertial system is e.g. J2000. (2) The T-system (target system) can be computed on board. It is the reference system for attitude control. The attitude of the T system with respect to the inertial system is denoted by T TI. (3) The actual attitude of the of the P system with respect to the T system is simply obtained by Doc. No: HYP-2-01 Page 38

44 T PT T PI T T TI It can also be represented by a quaternion q PT. (4) Since the transverse attitude components of q PT are controlled precisely with a PST and thus, are very small by definition, the attitude can be approximated as q T PT sin( / 2) 0 0 cos( / 2) where f is the roll attitude. From the first and fourth component of the quaternion of q PT, the roll angle can be extracted. Note that this can be done even for large roll angles (again, under the assumption that the transverse axes are precisely controlled). Doc. No: HYP-2-01 Page 39

45 6 Control Loop and Disturbance Rejection 6.1 Controlling the ASU Phase Overview In Figure 6-1 the three effects that attenuate the phase that is seen by the ASU or each interferometer, respectively: 1. Drag-free control, which is the major topic of this report. 2. The low pass behaviour of the ASU discrete time operation. The low-pass becomes effective at frequencies higher that 0.3 Hz (sampling frequency of the ASU): For acceleration, it has a slope of 40dB/decade, corresponding to a second order system. For rate, it has a slope of 20 db/decade, corresponding to a first order system. See also RD The phase correction loop as it is part of the ASU. This control loop becomes effective at frequencies typically lower than rad/sec, corresponding to about 8 hours. A second order rejection characteristics is assumed, as shown in Figure 3-1. DFS Noise Drag-Free Control Acceleration ASU Sampling Acc@ASU ASU Phase Control Equivalent ASU Figure 6-1: Attenuation effects of input disturbances to the ASU phase. The two filter functions other than drag-free control have to be taken into account in the design and specification of the drag-free and attitude control system Drag-Free Control Transfer Functions In Figure 6-2 a generic drag-free control closed loop is shown. Under the assumption that the accelerometer dynamics is very fast compared to the drag-free control closed loop dynamics, the transfer function G can be considered as unity. The closed loop transfer functions are: a (1 K) Sa Ta n d a (1 K) 1 1 a d Ka n Doc. No: HYP-2-01 Page 40

46 S and T are usually denoted as Sensitivity Function and Complementary Sensitivity Function. For low frequencies (lower than closed loop bandwidth), T has a gain of 0 db. For high frequencies (higher than closed loop bandwidth), S has a gain of 0 db. In the low (high) frequency range it is possible to attenuate input disturbances (sensor noise). a d a a n _ K G Figure 6-2: Generic drag-free control loop. 6.2 Drag-Free Disturbance Rejection Rejection of Linear Acceleration Disturbance Rejection of Deterministic Disturbances The (relative large) deterministic disturbance accelerations occur typically with double orbit frequency. At this frequency it is simple to achieve a rejection ratio such that the residual acceleration is negligible with respect to the overall 3 requirement. The maximum disturbance force is less than 250 N, corresponding to m/sec 2. A disturbance attenuation to a level where it is negligible, i.e m/sec 2, corresponds to 50 db. Including a margin of -10 db leads to a disturbance rejection requirement of -60 db. Note that a large rejection in low frequency ranges is not critical. Thus, this requirement may be changed without much consequence. Rejection of Noise Disturbances For the definition of the residual acceleration requirement the following approach is taken: 1. Define a qualitative spectral density requirement for the residual acceleration. The assumption is a constant value from zero frequency to ASU sampling frequency, and a drop with second order above ASU sampling frequency. 2. Assess if this requirement can be met realistically, given the FEEP and drag acceleration noise. The 1 requirement for the linear acceleration is m/sec 2. The constant spectral density requirement can then be computed by Doc. No: HYP-2-01 Page 41

47 SD f ASU 4 / m / sec 2 / Hz where SD is the spectral density value between zero frequency and ASU sampling frequency. In the following it is verified that this specification is justified given the noise disturbances of FEEPs and drag. In Figure 6-3 the noise spectra of drag and micro-propulsion are shown. Moreover, a closed loop sensitivity function with a bandwidth of 0.01 Hz as shown in Figure 6-4 is assumed. From Figure 6-3 the following can be concluded: 1. Drag disturbances do not even need to be actively controlled, since their spectral density is below the requirement. 2. FEEP noise needs to be controlled only at relatively low frequencies, typically below 0.01 Hz, as selected here (closed loop bandwidth). Spectral Densities m/sec 2 /sqrt(hz) Linear Acceleration: Requirement and Disturbance Rejection Requirement FEEPs AeroDistMax Accel@ASU Frequency [Hz] Figure 6-3: Residual acceleration: specification, disturbances, and expected performance. Doc. No: HYP-2-01 Page 42

48 10 0 Sensitivity Functions of the Closed Loop (Translation) Sensitvity Function Magnitude [db] Frequency [Hz] Figure 6-4: Specification (typical) of closed loop sensitivity function Rejection of Angular Acceleration Disturbance Rejection of Deterministic Disturbances Here, basically the same is applicable what was stated for the linear acceleration case. The maximum disturbance torque is less than 160 Nm, corresponding to m/sec 2 (a moment of inertia of 250 kgm 2 is used for this conversion). In order to attenuate his disturbance to a level where it is negligible, i.e m/sec 2, a rejection ratio of about 50 db is required. An additional margin of -10 db results in a disturbance rejection requirement of -60 db. Note that a large rejection in low frequency ranges is not critical. Thus, this requirement may be changed without much consequence. Rejection of Noise Disturbance In this paragraph the effect of the two torque noise contributions originated by the FEEPs and the drag are evaluated. This is done by expressing torque noise requirements in terms of force noise requirements and comparing it to the requirements derived for the translation. The angular acceleration requirement can be expressed as req a r r ASU is the characteristic length of the ASU, say 0.5 m. The torque requirement can be expressed as a force requirement as follows req ASU Doc. No: HYP-2-01 Page 43

49 f req r I req a I r req ASU f req I r r ASU a req f req is the thrust noise requirement and r is the lever arm. In case of drag it is the distance between the CoM and CoP. In case of FEEPs it is the distance between the CoM and the location of the FEEP. On the other side the noise consideration for the translation resulted in f m req a req Drag: In the case of drag the coefficient I/(r CoM-CoP r ASU ) is larger than the coefficient m, since r CoM-CoP is small and r ASU is relatively small. This means that the translation is the more driving requirement. However, since drag force noise is uncritical (see above), drag torque noise will be even more uncritical. FEEPs: In the case of FEEPs the coefficient I/(r r ASU ) is about the same order of magnitude than the coefficient m. This means that the rotation requires about the same bandwidth that translation. However, this bandwidth is small since the FEEP noise has to be rejected only at very low frequencies (see above). Conclusion: As in the case of translation a bandwidth of typically 0.01 Hz will be sufficient to reject noise contributions. 6.3 Drag-Free Sensor Disturbances Drag-Free Sensor Bias The bias error of the drag-free sensor(s) and any very low frequency error is passed through the closed loop with gain one. This has two effects: 1. There will be a control error in acceleration at zero and low frequencies, i.e. at the frequency of the DFS bias. This is not relevant as long as it is in the frequency range that is controlled by the ASU phase control loop. 2. There will be a thruster actuation with the magnitude corresponding to the DFS bias. For instance assuming a bias of 10-7 m/sec 2 and a spacecraft mass of 850 kg, the additional force authority required will be 85 N. This means, the propulsion system has to provide additional thrust in the axes with large drag-free sensor biases. The second effect has a system design impact, since the drag-free bias drives the sizing micro-propulsion system. There are two options to cope with the situation: Option 1: Reduce the DFS bias as far as possible and design the propulsion system accordingly. The adaptation of existing (GOCE) accelerometers can result in biases in the range of 10-7 m/sec 2. This value is usually given in one axis only, the two other axes have smaller biases. This limitation is caused by the goldwire for discharging and thus, cannot be reduced any further. Doc. No: HYP-2-01 Page 44

50 Option 2: Introduction a filter in the controller algorithm that suppresses any control action at low frequencies. In that case all natural disturbance will be seen and rejected by the ASU phase control loop. This possibility is addressed in the chapter Controller Design further below Drag-Free Sensor Noise The drag free sensor noise has an impact on the residual acceleration. In this section the noise performance requirement for the drag-free sensor is derived. The basic approach is to require negligible impact of the drag-free sensor noise with respect to the residual acceleration requirements. The derivation is based on the following worst case assumptions: (1) The drag-free control closed loop bandwidth is > ASU sampling frequency. This means that the drag-free control does not average (attenuate) the DFS noise. Attenuation is only accomplished by the low pass behaviour of the ASU. (2) At low frequencies, the ASU internal phase correction loop just compensates the increasing (with lower frequency) acceleration noise density, resulting at a constant ( horizontal ) spectral density even at low frequencies. The first assumption is justified since the closed loop bandwidth will be lower than ASU sampling frequency and thus, there will be an additional attenuation effect of the DFS noise. The second assumption is justified since a second order disturbance characteristics of the ASU internal phase correction is assumed. The slope of the DFS noise spectral density at lower frequencies is usually lower (first order). Under the above assumptions the spectral density (square root of the power spectral density) of the DFS can be expressed as SD f 4 3 ASU where f ASU = 0.3 Hz is the ASU sampling frequency. In order to have a negligible DFS noise, the one sigma value is required to be m/sec 2. Since two sensors are used for acceleration computation, a factor of 1/2 has to be introduced. This results in a spectral density value of m/sec 2 /Hz. In Figure 6-5 the noise requirement for one drag-free sensor is shown, taking into account the ASU phase correction loop (relaxation at low frequency with first order slope) and ASU low pass behaviour (relaxation above ASU sampling frequency with second order slope). For comparison, a typical example of a drag-free sensor is also shown. Doc. No: HYP-2-01 Page 45

51 10-6 Spectral Density m/s 2 /sqrt(hz) Requirement Typical DFS Example Frequency [Hz] Figure 6-5: DFS requirement and typical example. Doc. No: HYP-2-01 Page 46

52 7 Controller, Sensor, and Actuator Requirements 7.1 Sensors and Actuators The subsequent set of requirements is by no means complete. Only those requirements are listed that are considered critical with respect to feasibility. Precision Star Tracker The precision star tracker shall fulfil performance requirements as specified in the Performance Requirements Breakdown document, AD02. Star Sensor The star sensor shall fulfil the performance requirements as listed in chapter Accelerometer The accelerometer shall fulfil the measurement noise requirement as specified in Figure 6-5. FEEP A FEEP thruster (i.e. one of four emitter sets of an assembly) shall fulfil the thrust noise requirement as specified in Figure Residual Acceleration Requirements Acceleration requirements are basically disturbance rejection requirements of linear and angular disturbance accelerations Disturbance Rejection of Deterministic Accelerations Deterministic disturbance accelerations occur typically around orbit frequency. Since the ASU phase correction control loop operates at lower frequency and the low-pass filtering of the ASU is effective at higher frequency, the complete disturbance must be attenuated by the drag-free control. Linear Acceleration The closed control loop shall attenuate the linear disturbance acceleration by 60 db in the frequency range from 0 to rad/sec (twice orbit rate) Justification: See explanation above. Angular Acceleration Doc. No: HYP-2-01 Page 47

53 The closed control loop shall attenuate the angular disturbance acceleration by 60 db in the frequency range from 0 to rad/sec (twice orbit rate) Justification: See explanation above Noise Rejection Linear Acceleration The linear acceleration shall be as specified in R2-ENL in AD02. Angular Acceleration The angular acceleration shall be as specified in R2-ENL in AD02. Justification: Since the rejection of deterministic acceleration disturbances is specified such that they become negligible, and the sensor noise impact is specified to be negligible, the complete portion of the Level-2 requirements summarised in Table 3-1 can be allocated to the noise disturbance contribution. 7.3 Angular Rate Requirements Rate Around PST Boresight The rate around the PST boresight shall be as specified in R2-ENL in AD02. Rate Around PST Transverse Axes The rate around the transverse PST axes shall be as specified in R in AD02. Justification: These are basically Level-2 requirements that are justified in AD Pointing Requirements Axes Transverse to PST Boresight The 3 pointing accuracy around the PY and PZ axis shall be 1/2 pixel of the PST, i.e. < arcsec. Justification: The criterion for pointing accuracy is to guarantee sufficient accuracy of the PST measurements, which includes the requirement of normal distribution, since this assumption was used to derive system level requirements. When the accuracy is within of 1/2 pixel, the centroiding error is virtually linear. This means the qualitative error distribution will not be modified. Doc. No: HYP-2-01 Page 48

54 7.4.2 PST Boresight Axis The 3 pointing accuracy around the PX axis shall be less than 120 arcsec. Justification: This value is chosen about the same as the required inertial attitude knowledge. It has only an impact on projection errors of the Lense-Thirring effect. These projection errors are much smaller than those caused by the guide star direction. Therefore, this requirement could be easily relaxed. 7.5 PST Acquisition Requirement It shall be possible to acquire pointing with the PST, with a field of view of the PST of 50 arcsec (half cone). For the initial conditions of the acquisition it shall be assumed that the spacecraft is attitude controlled only with the 3-axis star tracker. For the 3-axis star tracker accuracy the specification as given in this document shall be taken into account. Justification: No drag-free sensor information shall be taken into account for the PST acquisition, since this eliminates any potential saturation problems. Doc. No: HYP-2-01 Page 49

55 8 Controller Design In this chapter the controller design and preliminary analysis of the closed loops performance is described. A full test and verification of the control loops is performed within a dedicated simulation campaign, based on sophisticated environment, sensor, and actuator models. The latter is not part of this document. 8.1 Summary of Important Requirements and Assumptions The requirements for the Secondary AOCS controllers are summarized intable 8-1. The performance of the control algorithms will be tested with respect to these requirements. Remember that the acceleration and angular rate requirements are valid for the filtered closed-loop system response, i.e. the output signal of the system has to be filtered with an appropriate filter function in order to take into account the effects of the internal ASU servo-loop and the ASU's sampling frequency. The frequency response of the filter function is shown in Figure 8-1. Requirement 3-Value 1-Value Acceleration at DFP a x, a y, a z < m/s 2 < m/s 2 Attitude around PST boresight Attitude around transverse axes Attitude < 120 arcsec < 40 arcsec Angular velocity x < 10-6 rad/s < rad/s Attitude, < arcsec < arcsec Angular velocity y, z < rad/s < rad/s Angular acceleration y, z < rad/s 2 < rad/s 2 Table 8-1: Requirements for the Secondary AOCS control loops. Doc. No: HYP-2-01 Page 50

56 Figure 8-1: Frequency response of filter function for control lops requirements. The assumptions made for controller design are in some cases slightly different from what is the baseline. However, these differences do not have a significant impact. For completeness, all assumptions for controller design are listed in following: 1. A linear approach is chosen because the displacements from the working point are expected to be small and the nonlinearities in the system are expected to be negligible. 2. The cross-coupling between different axes is expected to be negligible, so the algorithms are derived separately for each axis using simple one-dimensional systems. 3. The system parameters defined in Table 8-2 will be assumed for the design of the algorithms and the performance analysis. Parameter Value Description M 770 kg Mass of the satellite I x 300 kg m 2 Mass moment of inertia around x I y 270 kg m 2 Mass moment of inertia around y I z 270 kg m 2 Mass moment of inertia around z o rad/s Orbit rate Table 8-2: System parameters. Doc. No: HYP-2-01 Page 51

57 4. The FEEP noise is assumed to be that of the GOCE FEEPs. The noise profile is taken from RD The delay between controller command and thruster action is assumed to be 200 ms. This is a margin factor 2 with respect to the specification above. 6. The maximum thrust that can be produced by a single FEEP thruster is assumed to be 200 N. 7. The readout noise of the off-the-shelf Star Tracker (STR) is assumed to be white with a 3 value of 10 arcsec. This value is chosen higher than the STR accuracy transverse to the boresight. 8. The model for the Drag-Free Sensor (DFS) is equivalent to the model described in RD03. The constant part of the bias is assumed to be zero. 9. A simplified model of the Precision Star Tracker (PST) described in RD03 is used here. The model from RD03 is simplified in the way that only the different noise and error sources from the original model are extracted and added to the real attitude angle. The effect of the aberration is not considered. 8.2 Algorithm Development Attitude Controller for the X-Axis The attitude controller around the PST boresight relies on the measurement that is extracted from the 3-axis "off-the-shelf" star tracker. The system under consideration here is a very simple second order system that can be modeled by a double integrator that is driven by the external torques consisting of disturbance torques and the control torques. It is shown in Figure 8-2. T c +T d d/dt 1 1 I x s s Figure 8-2: One-dimensional system for attitude control synthesis. This corresponds to the following state space description: where T 0 x 0 y 1 x, u Tc / I x and d Td / I x. 1 0 x x u d The controller for the attitude around x is derived using a discrete augmented Linear Quadratic Gaussian (LQG) regulator design. This type of controller has been applied to similar problems and has shown good results. The advantage of this method is that system and sensor noise can be considered as white noise during the design process, making it well suited for systems with many different random error sources. Doc. No: HYP-2-01 Page 52

58 The standard LQG controller consists of an optimal Linear Quadratic Regulator (LQR) that uses full state feedback in combination with a Kalman filter used for state estimation. The standard form of the LQG is augmented in the way that a model for the dominant disturbance is included in the Kalman filter in order to estimate this disturbance. This estimated disturbance is then used in a feedforward control to cancel the real disturbance, i.e. the augmented control variable is then defined as u K x e d e where K is the LQR gain, x e is the estimated state and d e is the estimated disturbance. The estimated disturbance for the x-axis is assumed to be sinusoidal with a frequency of roughly twice the orbit rate. The augmented system used for the Kalman filter then has the following state space description x~ d d where T 0 0 x~ 0 0 y u T / I. and c x o ~ x x~ u Both the LQR and the Kalman filter are designed using discrete design methods such that the resulting LQG controller is sampled with 10 Hz Attitude Controller for the Y/Z-Axes Since this case is very similar to the attitude control around the x-axis, the discrete augmented LQG design is chosen again. The derivation follows the derivation for the x-axis, i.e. the standard LQG design is augmented such that a model of the dominant disturbance is included in the Kalman filter design and the estimated disturbance is used in a feed-forward control. Like for the x-axis it is assumed that the dominant disturbance is sinusoidal with a frequency of roughly twice the orbit rate. The final LQG controller is again discrete with a sampling frequency of 10 Hz Option: Y/Z Axis Attitude Controller with Acceleration Measurements This is an option and not the baseline. It is described here for completeness only and it is not tested in the simulation campaign. From a design point of view this case is more complex and the standard design techniques have to be modified in order to implement the acceleration information into the controller design. The benefit of the additional information is that the state estimation should be improved. Two different methods of implementing the acceleration information into the controller design have been investigated. The first method applies state augmentation, namely the state of the system used for the design of the LQR and the Kalman filter is augmented to include the angular acceleration. This means basically to add an additional integrator at the scaled plant input (scaled in the sense that accelerations are fed into the plant Doc. No: HYP-2-01 Page 53

59 and not torques). In addition to the inclusion of the angular acceleration in the system state, the Kalman filter is augmented further to include a model of the dominant disturbance. This yields the following state-space description for the Kalman filter system x x 1 u ωo y x where T x φ ω ω d d and u u d/dt(t c /I x). Using this method the derived controller will not give a torque command, but a torque derivative with respect to time. In order to implement the controller, the output of the controller has to be filtered through an additional integrator. The second method makes full use of the fact that the angular acceleration can be accessed directly. The angular acceleration is used as the input for the Kalman filter system thus making an estimation of the dominant disturbance obsolete since the disturbances are present in the measurement of the angular acceleration. This leads to the following state-space description for the Kalman filter system where x ω T and u ω x x u y 1 0 x The resulting controller is not a "true" LQG controller since the output of the LQR gain is not fed back into the Kalman filter because the true angular acceleration is available. This second method is preferable since the resulting controller is less complex than the controller derived with the first method so the first method will not be considered further at this point. The structures of the standard LQG controller and the modified LQG controller are shown in Figure 8-3. As before the controller is discrete with a sampling frequency of 10 Hz. Doc. No: HYP-2-01 Page 54

60 u LQG LQR x e Kalman Filter Filter u y u LQR Modified LQG x e Kalman Filter Filter d/dt y Figure 8-3: Standard LQG structure and modified LQG structure Drag-Free Control for the X/Y/Z-Axes Since the accelerometer dynamics is very fast compared to the drag-free control closed loop dynamics, the plant can be considered to be unity for design purposes. A drag-free controller is derived for each individual axis that has to meet the linear acceleration requirements. The drag-free requirements are virtually requirements on the disturbance rejection of the closed loop. Therefore the drag-free controllers are derived using H -optimization algorithms where frequency-dependent weighting functions are used to shape closed loop transfer functions. The so-called mixed-sensitivity approach is applied where the shape of the closedloop transfer functions of the sensitivity function S y and the complementary sensitivity function T y are optimized. d r e K G y - Figure 8-4: General closed-loop system. The general structure of the closed-loop system is shown in Figure 8-4. In the diagram G is the plant, K is the controller, d the disturbance input, r the command variable, y the system output, the sensor noise and e is the error signal to be controlled. The sensitivity function is the transfer function from r to e, the complementary sensitivity function is the transfer function from r to y and the disturbance rejection function (S y G) is the transfer function from d to y: 1 GK G S y, T y, SyG 1GK 1GK 1GK Doc. No: HYP-2-01 Page 55

61 It is easy to see that if the plant is unity then the sensitivity function is equal to the disturbance rejection function. So shaping the sensitivity function will also shape the disturbance rejection. The Simulink model used for the H -optimization is shown in Figure 8-5. Figure 8-5: Simulink model used for the H -optimization It has to be taken into account that the Power Spectral Density (PSD) of the sensor noise of the DFS is about two orders of magnitude higher for the less-sensitive axis (the x-axis) than for the other two ultrasensitive axes (see RD03). Therefore the closed-loop bandwidth of the drag-free controller for the x-axis cannot be as high as the bandwidth for the other two axes in order to have a better noise rejection. This has been verified during the optimization and performance analyses of the different controllers. The weighting functions W e and W y that are used for the design of the drag-free controller for the x-axis are shown in Figure 8-6 (left plot) along with the sensitivity function and the complementary sensitivity function of the closed-loop system (right plot). The weighting functions and the two sensitivity functions of the controller used for the drag-free control of the y and the z-axis are shown in Figure 8-7. It should be noted here that the point where the two sensitivity functions cross each other is approximately equal the closed-loop bandwidth of the controller. The individual controllers have been designed continuously and were discretized using a zero-order-hold method which yielded discrete H controllers that are sampled with 10 Hz. Doc. No: HYP-2-01 Page 56

62 Figure 8-6: Weighting and sensitivity functions of the drag-free controller for the x-axis Figure 8-7: Weighting and sensitivity functions of the drag-free controller for the y and z-axis 8.3 Preliminary Performance Analysis Nominal Performance The preliminary performance analysis is performed with one-dimensional systems, in line with the design assumptions. As an example the Simulink model used for the performance analysis of the attitude control around the y and the z-axis is shown in Figure 8-8. The integration method used during the simulation is a fifth order Runge-Kutta method with a fixed step size of 0.05 s (20 Hz). The simulation duration is four orbits, i.e. approximately s. Doc. No: HYP-2-01 Page 57