Dynamic modeling and analysis of micro-vibration jitter of a spacecraft with solar arrays drive mechanism for control purposes

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(14) . Sanfedino, Francesco and Alazard, Daniel and Pommier-Budinger, Valérie and Boquet, Fabrice and Falcoz, Alexandre Dynamic modeling and analysis of micro-vibration jitter of a spacecraft with solar arrays drive mechanism for control purposes. (2017) In: 10th International ESA Conference on Guidance, Navigation & Control Systems (GNC 2017), 29 May 2017 - 2 June 2017 (Salzburg, Austria).. .

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(24) DYNAMIC MODELING AND ANALYSIS OF MICRO-VIBRATION JITTER OF A SPACECRAFT WITH SOLAR ARRAYS DRIVE MECHANISM FOR CONTROL PURPOSES Francesco Sanfedino(1) , Daniel Alazard(2) , Valérie Pommier-Budinger(3) , Fabrice Boquet(4) , Alexandre Falcoz(5) (1),(2),(3). University of Toulouse, ISAE/DCAS, 10 Avenue Edouard Belin, BP-54032, 31055 Toulouse Cedex 4, France, +33 (0)695134718, francesco.sanfedino@isae.fr, +33 (0)561338094, daniel.alazard@isae.fr, +33 (0)561338420, valerie.budinger@isae.fr (4) ESA/ESTEC, Keplerlaan 1, 2201 AZ Noordwjik, The Netherlands, +31 715653780, fabrice.boquet@esa.int (5) Airbus Defence and Space, 31 Rue des Cosmonautes Z.I. du Palays, 31402 Toulouse Cedex, France, +33 (0)562198549, alexandre.falcoz@airbus.com. ABSTRACT Pointing performances are always more demanding for the scientific and observation Space missions of the future generation. A deep study of all the possible micro-vibration sources therefore results mandatory. This article presents several tools to analyze and control the micro-perturbations induced by a Solar Array Drive Mechanism (SADM) on a spacecraft. A first tool allows the user to build the dynamic model of a spacecraft composed of different flexible appendages attached to the main body by revolute joints. A second tool provides the SimScape model of a SADM that is interfaced with the spacecraft dynamics and allows analyzing the mission pointing performances. Time and frequency studies can be performed to deeply investigate pointing performances in presence of vibrations. An example that shows the interest of the proposed tools is provided. An observation spacecraft with two flexible solar arrays and an antenna is studied. The pointing metrics are computed for two complementary solutions to the micro-vibration issue: a microstepping technique for the SADM driver and a Fast Steering Mirror (FSM) for the active control of the Line of Sight (LOS).. 1. INTRODUCTION. Modern observation satellite missions always aim for better performances in high resolution real time imagery and video products. The needs are several: automated moving target identification, detection, increase of reconnaissance and identification capabilities, borders and assets surveillance, disaster monitoring, search and rescue. Line-of-Sight (LOS) satellite jitter caused by micro-vibrations deteriorates these resolution performances. The main sources of micro-disturbances are found in Attitude and Orbital Control System (AOCS) components (especially control wheels), cry-coolers (when present), Solar Array Drive Mechanism (SADM), antenna trimming mechanism or payload mechanisms. All these sources, together with the mission environment, have to be investigated in order to prevent coupling effects with the normal modes of the flexible structures, like solar arrays (SA) and ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 1.

(25) antennas. Several scientific missions in Space history, as the Hubble Telescope [1], demonstrated how this vibration mitigation can avoid performance degradation. Notice that in case of rotating flexible surfaces, like solar panels or antennas, the dynamic content of the spacecraft can change during the mission: a shift of the spacecraft normal mode can occur. In [2] a mixed µ−analysis helped to find the worst-case configurations for different solar panel tilts for the ESA MetOp mission. In missions, as BepiColombo, where highly-accurate pointing performances are demanded, the microvibrations induced by a SADM system represent a real concern [3]. Current solutions to solve the micro-disturbances problem rely frequently on fixed solar arrays configurations without any steerable capabilities. However this approach presents two main drawbacks: need to oversize the solar panels in order to guarantee full payload power generation and reduction of imaging time window caused by more complicated orbital maneuvers. This paper analyzes the impact of two different stepper-motor drive techniques: full-step and microstep. The micro-step drive method for orientable solar arrays is examined in order to potentially reverse the design trend of fixed solar arrays by leading to the same pointing performances while relaxing the mass penalties. Two design constraints are thus fulfilled: low level of micro-disturbances and avoidance of solar arrays over-sizing. The combination of micro-stepping solution with a Fast Steering Mirror (FSM) active control is addressed in order to produce a preliminary jitter budget for an observation Space mission. This paper provides tools to model the spacecraft dynamics coupled with a SADM model for the design of the micro-step mechanism and the analysis of the disturbances. The three main contributions are: the extension of the Satellite Dynamics Toolbox (SDT) [4] to take into account flexible appendages composed of kinematic chains of flexible bodies (like a solar array composed of several plates) and varying appendage tilt; the analysis of spacecraft dynamics with steerable solar arrays driven by micro-step mechanisms; the performance analysis with a FSM for jitter active control. 2. IMPROVEMENTS IN THE SATELLITE DYNAMICS TOOLBOX. The Satellite Dynamics Toolbox based on [5] is a tool able to extract the linear dynamic model of a spacecraft with several flexible appendages and on-board angular momentums. Let us consider the system in Fig. 1 composed of a rigid body B (the spacecraft central body), submitted to the external forces/torques Fext , Text,B and to forces/torques FB/A , TB/A,P , due to the interactions with a flexible appendage A (solar array, antenna, robotic arm). Let us consider that the connection between B and A is rigid in P . The main body dynamic model is obtained thanks to the Euler/Newton equations w.r.t. B and under the assumption that the angular rates are small: Fext − FB/A aB B = DB (1) Text,B − TB/A,B ω˙ where aB and ω˙ are the linear and angular accelerations of the point B and B m I3 03×3 B DB = 03×3 JBB. (2). with mB and JBB mass and inertia matrices of the body B. ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 2.

(26) The cantilever hybrid model MPA (s) of the flexible appendage A cantilevered on B gives the relationship between the 6 degrees of freedom (DOF) acceleration vector of the point P and the 6 DOF forces/torques vector applied by the main body in frame Ra (xa , ya , za ): " # " #  F a  P B/A  = DPA + LP η¨   T ω˙ B/A,P " # (3)  a  P 2 T   η¨ + diag(2ζi ωi )η˙ + diag(ωi ) η = −LP ω˙. where: DPA is the 6 × 6 mass/inertia model matrix of A at point P ; LP = [l1,P , ..., li,P , ..., lN,P ] is the matrix of the participation factors of the N flexible modes of A at point P ; ωi , ζi and li,P are the pulsation, the damping ratio and the 6 DOF modal participation vector of the i-th flexible mode. In the case where arevolute joint links A to B at the point P as in Fig. 1 in the direction ra , the 7 × 7 A augmented model PP Ra (s) (with s Laplace variable) at point P (expressed in frame Ra ) will be taken into account:     T aP FB/A A I6 I6  TB/A,P =  ω˙ Ra  (4) MP Ra (s) Ra 0 0 0 xra yra zra 0 0 0 xra yra zra Cm θ¨. where Cm is the revolute actuation torque (for example the SADM torque for a solar panel), θ¨ is the angular acceleration inside the revolute joint and xra , yra , zra are the coordinates of the vector ra in the appendage frame Ra . The final direct model PCA+B of the assembly base + appendage with the revolute joint is obtained by moving Eq. (3) from point P to point B thanks to the kinematic model τBP and writing it in Rb (using the rotation matrix Tba ) before being inserted in Eq. (1). This model can be represented in a block diagram as in Fig. 2. Note that this model has to be still moved to the center of the mass of the entire system for control purposes. In the previous version of the SDT the rotational matrix Tab was fixed for a determinate position of the revolute angle. Thanks to a smart parametrization of the tilt angle proposed in [6] and [7], a unique minimal Linear Fractional Transformation (LFT) model of a spacecraft with all possible revolute configurations can be obtained. If for simplicity a rotation is considered around ya -axis, parallel to yb -axis, Tab is written as:   cos θ 0 − sin θ 1 0  Tab =  0 (5) sin θ 0 cos θ By introducing a new parameter τ = tan(θ/4) varying from -1 to 1 the trigonometric functions can be rewritten as: (1 + τ 2 )2 − 8τ 2 cos θ = (1 + τ 2 )2. 4τ (1 − τ 2 ) sin θ = (1 + τ 2 )2. ∀τ ∈] − 1; 1]. (6). and the final Linear Fractional Realization (LFR) is obtained.. ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 3.

(27) 26. 2. Satellite Dynamics Toolbox: principles. Taking into account a revolute joint between the hub and an appendage with an embedded angular momentum then allows for the modelling of CMGs (Control Moment Gyros), the axis of the joint being the precession axis of the CMG.. θ¨. 2.5 Revolute joint between the appendage and the hub.  . → − aB → − ω˙. . . . . . B. . A. ~ra. ~ya. O B. . 3 + +  . 25. A ~xa P ~za.  x ra  yra  z ra. → − aP → − ω˙. Tba 03×3 03×3 Tba. MPA. i. Ra. . . Ra. θ¨.  .  . → − F ext → − T ext,B. .  . → − aB → − ω˙.  . → − F B/A → − T B/A,P . → − F B/A → − T B/A,B.  . Ra. Tba 03×3 03×3 Tba h. τPTB. . h. 6. . Rb. . Rb. +. i DBB R b. . i. +. Rb. Cm. [xra yra zra ]. . ~yb . 3 3. . . T . A PP Ra (s). (s). [τP B ]Rb. Cm. ~zb ~xb. → − F B/A → − T B/A,P. . 3. . h.  . → − F ext → − T ext,B.  . Rb. 6. Figure 2.5: Assembly of the base B and the appendage A linked with a revolute Figure 2.6: Direct dynamics model (7 × 7) block diagram of the assembly hub + joint along ~za appendage PBA+B (s) with revolute joint, expressed in frame Rb .. Figure 1: Assembly of the base B and the apFigure 2: Direct dynamics model (7×7) block pendage A linked with a revolute joint along diagram of the assembly base + appendage → − Because of the revolute joint, the projection of the torque T , exerted by the B/A,P For control laws synthesis, one can usejoint, the following inverse model: za P A+B with revolute expressed in frame base on the appendage at point P , along ~ra axis is either: null in case of a free B #  #   " →  " → − − Rb . revolute joint or equal to Cm in case of an actuated joint. aB F ext → − Cm = T B/A,P .~ra .. (2.12).  . → − ω˙ θ¨. Rb.  A+B  = PB. −1 Rb.  (s) . → − T ext,B Cm. Rb.  . (2.15). Expressing the direct dynamics model MPA (s) of the appendage at point P in frame that allows the user to introduce, between the seventh input to the seventh output, Ra enables us to write that: a local model of the joint mechanism or controller K(s) according to Figure 2.7. " → # " # − → − aP F B/A A = M (s) (2.13) → − → − P ¨ra ω˙ + θ~ T B/A,P 2.6 Generalisation. Thus thanks to the SDT it is possible to obtain the LFT representation of a satellite for any kind of uncertain parameters (inertia, resonance frequencies, damping ratios, appendage position) and for the entire rotational range of a revolute appendage. For the spacecraft in Fig. 3, if an incertitude of From (2.12) and (2.13), one can write the augmented direct model (7 × 7) of the One can generalise approach: 20% isPtaken into account for all the resonance frequencies of the theprevious two solar panels, the singular values A appendage P R (s) at point P and expressed in frame Ra : • tothe taketwo Na appendages (Ai , i = · · · , Na ) linked with 4 the(obtained hub at points Pi  diagram of the complete model for three 0positions of solar panels is1,plotted in Fig. into account, " # #    " →   the 0complete  → − by −Fthe Matlabfunction usubs from   a P uncertain LFT model). B/A  A → − I6 0  −    →    ISAE/DMIA/ADIS Page flexible 26/58 = I M (s)  .  Another  T B/A,P Rimportant 6 in the to analyze appendages with P R   is ω˙theR possibility 0 0 0 xrimprovement yr zr xrSDT  ¨ Cm θ   yr multiple attaching points, as the deployable solar panels of Fig. 3, thanks to the Two-Input Twozr {z [8]. Self-made } | Output Port (TITOP) approach super-element beams and bending plates elements are P (s) [ ] now available for the study of multibody structures. (2.14)A dedicated interface also allows integrating the The block diagram of Figure 2.6 represents operation. It directly also shows the conMSC/Nastran normal mode this analysis reading the f06 file results. A by a. a. a. a. a. a. a. a. a. a. A P Ra. nection of the first six inputs and outputs between PP Ra (s) and the hub’s direct B model DB in order to get the assembly model PBA+B (s), expressed in frame Rb . ISAE/DMIA/ADIS. Page 25/58. 𝑧 𝑧. ℛ𝐴𝑛𝑡. 𝑧. 𝑦 ℛ𝑆𝐴𝑆𝑜𝑢𝑡ℎ. 𝑥 𝑦. 𝑥 𝑧. ℛ𝑆𝐴𝑁𝑜𝑟𝑡ℎ. 𝑥. 𝑦. ℛ𝑆𝐶. 𝑦. 𝑥. Figure 3: Observation spacecraft with two flexible solar panels and an antenna. Figure 4: Spacecraft Singular Values plot for three different SA positions. ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 4.

(28) 3. MODEL FOR A SOLAR ARRAY DRIVE MECHANISM. The SADM is a complex spacecraft system used for steering the spacecraft solar arrays. It is generally composed of different elements. A stepper motor, an electronic driver and a reduction gearbox (when foreseen) are generally the most dimensioning ones. In this section a block-shaped model is presented for design purpose, where each of the previous elements is treated as an individual system. 3.1. Stepper motors for SADM. The stepper motors classically used in Space systems are bi-phase permanent magnet-like. This type of motors employs a permanent magnet joint to the rotor axis and bi-phased wires alternately disposed and regularly spaced on the stator. The rotor, composed of a p number of magnetic poles, is steered by the magnetic field produced by the pair of poles and the interaction with the rotor magnets. The equations describing the stepper motor are given as follows (see [9], [10] for a complete understanding of the equations derivation): diA = vA − RiA + pΨm ω sin(pθ) dt. (7). diB = vB − RiB − pΨm ω cos(pθ) dt. (8). L L. dω = −pΨm iA sin(pθ) + pΨm iB cos(pθ) − Bst |ω| − Bdyn ω − KD sin(4pθ) − τL (9) dt dθ =ω (10) dt where L and R are respectively the self-inductance and resistance of each phase winding; iA , iB and vA , vB are the currents and the voltages in phases A and B; Ψm is the motor maximum magnetic flux; J is the rotor inertia; Bst and Bdyn are the static and viscous friction coefficients; θ and ω are the rotor position and speed; τL is the load torque. The second member of Eq. (9) is the motor and loss torques budget: the first two terms represent the electromagnetic torque produced by the motor, the third and the forth terms are the friction torque (static and dynamic), the fifth one is the detent torque and τL has been already defined. The detent torque is a passive electromechanical torque caused by the interaction between the rotor permanent magnet and the stator wirings while no current passes throw them. The detent coefficient KD is strongly dependent on the motor physical characteristics (i.e. quantity of ferromagnetic material in the rotor). All the torques mentioned before constitute a perturbation source and can cause a pointing degradation. J. 3.2. Electronic driver. The electronic driver is the element which switches current between the motor phases. For a permanent magnet stepper motor, a bipolar drive is required to give bidirectional phase currents. The switching device chosen in this work is a MOSFET, which needs a small drive power. Fig. 5 shows. ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 5.

(29) the block diagram of the driver elements for one of the two motor phases. The phase current measured on the H-Bridge Driver, which is composed of four MOSFET transistors, is driven in closed-loop feedback and compared with the reference phase current. A current controller (generally a PI with a hysteresis comparator) guarantees the minimization of the error w.r.t the reference. Finally a PWM generator receives the duty cycle by the hysteresis comparator and feeds the H-Bridge circuit to obtain the phase voltage for the stepper motor. Current Reference Generator. 𝐼𝐴𝑟𝑒𝑓 + -. Current Controller. PWM Generator. Anti-aliasing filter. Current Sensor. 𝑉𝐴. H-Bridge Driver 𝐼𝐴. Figure 5: Driver block diagram The reference profiles used in this work for the phase currents are of two types: the classical full-step and the micro-step shape. The two phase currents obtained from the two solutions are shown in the example in Fig. 6.. 2. 0 -1. 1 0 -1. 1 0 -1. -2. -2. -2. -3. -3. -3. 0. 5. 10. =4 = 16 = 64. 2. Current (A). 1. Phase A. 3. 2. Current (A). Current (A). Phase B. 3. 0. Time (s). 5. Time (s). (a) Full-step. 10. Phase B. 3 2. Current (A). Phase A. 3. 1 0 -1 -2 -3. 0. 5. Time (s). 10. 0. 5. 10. Time (s). (b) Micro-step: 1/4,1/16,1/64. Figure 6: Full-step and micro-step phase currents In full-step mode the stator flux is rotated 90 electrical degrees every step of the motor, so only two current levels are possible (Fig. 6a): 0 and ±Ipeak . In micro-step the currents in the phases follow a sinusoidal law: IAref = Ipeak sin(θe ), IBref = Ipeak cos(θe ) (11) where θe is the full-step electrical angle after subdivision in µ micro-steps. According to Eq. 9 the effect of the micro-step approach is a smoother movement than in full-step by driving the rotor in a more continuous way. 3.3. Reduction gearbox. Solar arrays can be driven in two ways: - by the direct driving which is performed without reduction and allows limiting the problems introduced by an intermediate transmission (a further source of perturbations). However this solution can be considered only for small dimension mechanisms;. ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 6.

(30) - by a reduction gearbox for huge dimension mechanisms. In this case several solutions are available: Harmonic Drive for Low Earth applications or straight teeth for geostationary operations. However the gearbox is a great friction source to be faced. If Nred is the reduction ratio, the gearbox motion equation is given as follows: dωGear θ JGear = KGear − θGear − BstGear |ωGear | − BdynGear ωGear dt Nred. (12). where JGear is the gearbox inertia; θGear and ωGear are the outer shaft position and speed of the gearbox; KGear is the stiffness associated to the connection between the rotor and the outer shaft; BstGear and BdynGear are the static and viscous friction coefficients. In presence of a gearbox an additional reaction torque has to be considered in Eq. (9): θ 1 − θGear (13) TGearRot = −KGear Nred Nred 3.4. Complete model of the SADM. The complete model implemented in a dedicated Matlab/Simulink tool is composed of four blocks as shown in Fig. 7: a driver, a stepper motor, a gearbox and an inertia representing a rigid load. In the particular application of this work, two symmetric solar panels are employed and a sole driver unit feeds the two stepper motors. This prevents from any geometrical asymmetry (no differential tilt between the two solar arrays) by synchronizing the two stepper motors on the same fundamental frequency. Each block (except for the driver) communicates to the next one its state (angle of rotation and rate) and receives the reaction torque from it. The driver and the stepper motor blocks are implemented in a SimScape environment to take advantage of a realistic electronic circuit and analyze its impact on the performances in a multi-domain simulation. The solar arrays block contains the dynamic law of an inertia on the shaft axis representative of the particular application. The load is indeed considered as rigid for the first dimensioning of the SADM system: demanded electric power, step accuracy and required motor torque. Then this block can be easily replaced by a flexible element, more representative of a real solar panel, and the model can feed the entire spacecraft dynamics. In this second analysis the match between the SADM frequency content and the structural resonance frequency is investigated together with the potential coupling with the spacecraft axes, orthogonal to the SADM shaft axis. The model is discrete and a fixed time step size solver is selected to simulate a Hardware-in-theLoop (HITL) application. Nevertheless particular care has to be taken for the choice of the step size according to three constraints: integration problems caused by the fastest dynamics in the loop (i.e. non-linearites, electronic components, friction models); Matlab/Simulink storage capability; need to catch all the significant dynamics for Fast Fourier Transform (FFT) analysis while avoiding aliasing effects. Typically a step size frequency around 100 kHz can satisfy all these exigencies. 3.5. Case of study. Let us consider the observation mission in geostationary orbit of the Section 2. The two symmetric solar arrays have to make a complete rotation around their axis in a day. Let us consider a simple ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 7.

(31) Figure 7: SADM + Rigid Solar Arrays block diagram. rotation profile with a constant speed (0.0042 ◦ /s). Other profiles can be examined in case of particular pointing exigencies: stopping of rotation in imaging phase and further acceleration. The ideal situation is to keep the profile constant to maximize the Sun energy collection while preserving the pointing performances. Table 1: SADM parameters Parameter Full-step-angle Gearbox reduction factor Phase current Motor resistance Motor self-inductance Given outer shaft speed SA inertia on shaft axis. Symbol αf ull Nred Ipeak R L ωSA JSA. Value 1◦ 180 2.5 A 20 Ω 5 · 10−2 H 0.0042 ◦ /s 13.5 Kgm2. Let take as reference the SADM parameters of Table 1. The Fast Fourier Transform analysis of the total torque transmitted to one solar panel is shown in Fig. 8, where full-step and micro-step solutions are compared. The majority of the frequency content is concentrated at low frequencies where structural modes are typically located. These frequencies have to be cross-checked with the expected structural modes to prevent coupling effects. The full-step and the 1/2 micro-step show respectively an amplification of the 5th and the 3rd harmonic of their fundamental motor frequencies. For the other micro-stepping solutions the disturbing torque fundamental frequency grows up as expected and the amplitude progressively decreases. The temporal simulations in Fig. 9 of the motor driven in full-step and in 1/64 micro-step show the torques provided by the stepper motor as sum of the electromagnetic and the detent torques. The full-step motor torque presents a more discontinuous and energetic disturbance than the micro-step torque, where the same energy is more continuously distributed thanks to a smoother driving profile. This fact has consequences both on step accuracy (Fig. 10) and pointing performances (faced in the next section). ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 8.

(32) In particular according to the step resolution in Fig. 10 it can be argued that micro-stepping acts as an electronic damping element, which in certain case can substitute a mechanical gearbox. Amplitude Spectrum. Amplitude Spectrum. 0.012 Full step =2 =4 =8 = 16 = 32 = 64. 0.2. 0.1. Full step =2 =4 =8 = 16 = 32 = 64. 0.01. Torque [Nm]. Torque [Nm]. 0.3. 0.008 0.006 0.004 0.002. 0 0. 2. 4. 6. 8. 10. 12. 14. 16. 18. 0 20. 20. 22. 24. 26. 28. Frequency (Hz). (a) Emissivity (0 ÷ 20) Hz. Torque [Nm]. 6. 32. 34. 36. 38. 40. (b) Emissivity (20 ÷ 40) Hz. 10-3. Amplitude Spectrum Full step =2 =4 =8 = 16 = 32 = 64. 4. 2. 0 40. 30. Frequency (Hz). 42. 44. 46. 48. 50. 52. 54. 56. 58. 60. Frequency (Hz). (c) Emissivity (40 ÷ 60) Hz. Figure 8: FFT of the SADM total torque. 0.4. 0.8. Torque (N.m). Torque (N.m). Electromagnetic Torque Detent Torque Motor Torque. 0.6 0.4 0.2 0. Electromagnetic Torque Detent Torque Motor Torque. 0.2. 0. -0.2 -0.2 -0.4 10.576. -0.4 10.578. 10.58. 10.582. 10.584. Time (s). (a) Full-step. 10.586. 10.588. 10.59. 8. 8.5. 9. 9.5. 10. 10.5. 11. 11.5. 12. Time (s). (b) Micro-step: 1/64. Figure 9: Full-step and 1/64 micro-step stepper motor torques. 4 4.1. PRELIMINARY JITTER BUDGET FOR AN OBSERVATION MISSION AOCS control. The objective of this section is to highlight the contribution of the SADM to the pointing perturbations. For this reason all the other sources of micro-vibrations, like the control wheels, are supposed ideal. Let us consider the spacecraft platform of Section 2 and a simple proportional derivative (PD) AOCS control based on the main axis spacecraft inertia: a negative feedback with a strictly positive real controller ensures closed-loop stability. Since the three satellite axes are almost decoupled, the. ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 9.

(33) Position (deg). SA Position. 0.1 Full-step =8 = 64. 0.05 0. Rate (rad/s). 5. 10 10 -3. 2. 15. 20. Time (s) SA Rate. 𝜃𝑆𝐶𝑟𝑒𝑓. 0. +. 𝐾𝑝. +. −. 𝐾𝑣. 𝑇𝑅𝑊. −. -2 5. 10. 15. 20. Time (s). Figure 10: Solar array angular position and rate. 𝜔𝑆𝐶. 𝜃𝑆𝐶. Figure 11: PD AOCS law. AOCS law is based on three decoupled PD controllers tuned on the total spacecraft static inertia. According to Fig. 11 and assuming that spacecraft attitude θSC and rate ωSC are perfectly measured, the controlled reaction wheels torques result:. ω. TRWi = Kvi (Kpi (θSCi − θSCref ) − ωSCi ),. i = x, y, z. (14). i and Kvi = 2ζdesi ωdesi JSCi with ωdesi , ζdesi and JSCi respectively desired closedwhere Kpi = 2ζdes desi loop bandwidth, damping ratio and total spacecraft static inertia on the i-axis. For this work the values ωdes = 0.06 rad/s and ζdes = 0.7 are retained for the three axis. Notice that for the control wheels only saturation is modelled to limit their control torque to realistic values. A saturation value of 0.01 Nm is adopted.. 4.2. Pointing performances. The complete simulation environment for the jitter budget is shown in Fig. 12. The perturbations acting on the spacecraft are only the control torques and the SADM driving torques. Notice that the attitude sensors are considered perfect (no noise) with a unitary gain. The satellite is considered in nominal configuration with both the SA rotation angles at 0◦ , for design purposes. Once the SADM is dimensioned all the spacecraft parameters (solar arrays tilt angles, inertias, resonance frequencies, positioning, etc.) can be varied to perform Monte Carlo simulations for robustness verification. The pointing metrics used in this work are those mostly employed for the modern ESA missions [11]: the Absolute Performance Error (APE) as difference between the target (commanded) parameter (attitude, geolocation, etc.) and the actual parameter in a specified reference frame; the Relative Performance Error (RPE) as difference between the APE at a given time within a time interval ∆t, and the Mean Performance Error (MPE: mean value of the APE) over the same time interval. For a mission in geostationary orbit the APE requirement is quite relaxed: 5% of the nadir Field of View (FOV) in km is sufficient, which corresponds to 6.4 km equivalent to 180 µrad. What is demanding is the RPE objective: a stability requirement of 25 nrad over 100 ms has to be met to ensure a highly flexible imaging system.. ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 10.

(34) The pointing performances obtained with the stepper motor in Section 3.5 are shown in Fig. 13.. Figure 12: Spacecraft control loop 10-7. 8. 3. RPE (rad). 6. APE (rad). 10-8. 4. 4. x. 2. y z. x. 2. y. 1. RPE requirement. z. 0 Fu. ll. p ste. 1/2. 1/4. 1/8. /16. 1. 1. /32. Microstep Ratio. (a) APE performances. 4 1/6. 0 tep ll s. Fu. 1/2. 1/4. 1/8. 6 1/1. 2 1/3. 4 1/6. Microstep Ratio. (b) RPE performances. Figure 13: Pointing performances with different stepping solutions The value retained both for the APE and the RPE is the maximum value reached after a simulation of 1000 s, by considering that the tranquillization phase in which the errors reach a quasi-stable value lasts almost 100 s. The attitude angle mostly affected by the error is the angle around the spacecraft y−axis, where all the perturbations act, as expected. The APE requirement is well satisfied both by the full-step and the micro-stepping driving techniques, whereas the more constrained RPE performance is not met for the full-step and the 1/2 micro-step solutions. Although the RPE performance improves by increasing the micro-step number (theoretically null for an infinite sub-division), it has to be noticed that overcoming 64 µ−steps is complicated in practice and demands a great amount of power to keep the rotor teeth in a stable micro-step position between two successive full-step configurations. For the case faced here a 4 µ−steps solution can be retained, which corresponds to a 3.024 Hz motor fundamental frequency. Note one of the advantages of the proposed SDT model is the possibility to easily introduce any parameter uncertainty to perform a robust analysis of stability and investigate the amount of jitter ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 11.

(35) induced by different SADM configurations. This analysis can be used for co-designing the motor parameters (pairs of poles, micro-steps number, phase current and required holding torque constant) in order to assess their impact on the spacecraft LOS jitter. 4.3. Fast Steering Mirror in high pointing missions. In the example illustrated in the previous section the pointing performances are both met only by employing a micro-step driving technique. In other cases this solution cannot accomplish the pointing issue. If for instance the spacecraft and the SADM are those employed in the previous sections and the gearbox reduction factor is equal to 160, the RPE requirement is not met with 64 µ−steps, where it reaches the maximum value of 37.42 nrad for the spacecraft y−axis. In these cases, an active control system with an FSM can be foreseen to attain the required pointing performances. The device employed in this work for active vibration control is the commercial Physik Instrumente (PI) S-330.2SL Tip/Tilt platform. It is a two-axis tilt mirror platform, driven by four piezoelectric diagonally opposite actuators. They work in pair in push/pull mode in order to provide tilt of each of the two axes. Each tilt axis needs one controlled operating voltage in the range of 0 to +100 V and one constant voltage of +100 V. The S-330.2SL is a 2-inputs 2-outputs system (1 input for each rotational axis, 2 deflection angles as outputs). It should be controlled as a MIMO system. However, by considering that the two axes are completely decoupled and have the same dynamics (as demonstrated in identification phase), the system can be studied with a SISO model analysis (same dynamics for the two axes). The relationship between the input voltage and the tip angle established on one axis can be reduced to a third order transfer function as first approximation: Fpz (s) =. Gpz θpz (s) = 2 2 ) Vin (s) (τAmp s + 1)(s + 2ζpz ωpz s + ωpz. (15). where Gpz is the static gain, τAmp is the amplifier time constant proportional to the piezoelectric capacitance load, ζpz and ωpz are the damping factor and the pole location of the platform tilt resonance. The theoretical static gain Gpz can be computed thanks to the hypothesis that, with the maximum input voltage (10 V) in steady state, one of the piezo stacks of the pair is expanded to 100% (15 µm) of its maximum expansion and the other is not expanded. By knowing that at the zero position (tilt angle zero) both actuators of a pair are expanded to 50% of their maximum expansion [12] the maximum theoretical tip angle will be θpzmax = tan−1 (15 µm/Rpz ) = 2.7 mrad with Rpz distance of the piezo stack from the platform center. Thus Gpz = θpzmax /Vinmax = 2.7 · 10−1 mrad/V. The amplifier time constant τAmp can be found in PI E-505 amplifier specifications [13], where the amplifier first order transfer function has a cut-off frequency of approximately 300 Hz (for a total capacitance charge of 3 µF per axis). Finally the theoretical pole location of the platform resonance ωpz can be estimated as follows [12]: ωpz0 ωpz = p 1 + Im /Ipz0. (16). where ωpz0 and Ipz0 are the pole location of the tilt resonance and the inertia of the platform without the mirror (values provided by the supplier [12]) and Im is the customer mirror inertia. ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 12.

(36) Note that all the supplier parameters are provided with a huge uncertainty level (±20% on the first tilt resonance frequency), thus an identification operation results mandatory. A continuous-time identification technique based on the Instrumental-Variable (IV) Algorithm is performed by the System Identification Toolbox (SID) function tfest thanks to data recollected by a Polytec CLV 1000 vibrometer with the test bench in Fig. 14. The identified parameters are: Gpz = 1.99·10−1 mrad/V,. τAmp = 5.42·10−1 ms,. ζpz = 0.3044,. ωpz = 8.26·103 rad/s (17). The different value obtained for the static gain compared to the theoretical one suggests that the piezo stack are not completely extended at the maximum voltage. Decoder Output Module Module CLV-M030 CLV-M002. Laser. 𝑚𝑚 5 /𝑉 𝑠. 𝑧 𝑅. 𝑉𝑣𝑖𝑏. NI BNC2110. 𝐿𝑃 5 𝑘𝐻𝑧. 𝑉𝑆𝐺𝑆. 𝑉𝑖𝑛. NI PCI6251 Data Card. 𝑉𝑖𝑛. 𝑉𝑆𝐺𝑆 , 𝑉𝑣𝑖𝑏. FSM 𝑉𝑃𝑍𝑇 PZT-Servo 𝑉𝑆𝐺𝑆. Controller E509.X3. LVPZT Amplifier E505.00. Figure 14: Identification bench The identified device is coupled with a CCD camera for an active control loop of the two payload FOV-axes. If a Korsch Telescope is chosen for payload, an FSM can be used to correct the spacecraft jitter as shown in Fig. 15. If no aberration is taken into account for a preliminary performance budget, the FSM active control loop is reduced to the block diagram of Fig. 16, where the reflection coefficients link the mechanical angles to the optical angles. The CCD Camera is modelled as a first order filter with an imaging rate of 10 frame/s and a time delay. The controller chosen for the two axes is a classical Proportional-Integral (PI). The dimensioning element in this control loop is the CCD Camera both for its frequency bandwidth and temporal delay. In particular the integration time for imaging treatment can vary according to the mission exigencies. Pointing performances are consequently affected by this parameter. CCD. PRIMARY SECONDARY FSM. Jitter. TERTIARY. Reference + -. 1/2. FSM Control. Ampli + FSM. +. APE. +. 2 Reflection. Reflection. CCD Camera. Figure 15: Korsch Telescope. Figure 16: FSM active control. ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 13.

(37) Different PI control laws have been synthesized according to different CCD time delays in the loop in order to guarantee a smooth and fast response (damping factor near 0.7) and good stability margins (Phase Margin > 45◦ and Gain Margin > 6dB). Let us consider the jitter affecting the spacecraft with the SADM with Nred = 160 and 64 µ−steps. The resulting simulated pointing performances are shown in Fig. 17. Note that the FSM controller is activated after a simulation time of 100 s. The positive action of the FSM is degraded by an incrementing CCD integration delay as expected. In particular the RPE error on the y-axis, the most influenced by the SADM perturbation, does not meet the requirements for a CCD delay of 20 ms. This result constrains the choice of the camera performances. 10 -8. 2. 10 -7. 8 Nominal =20 ms CCD. 1. Nominal =20 ms CCD. 6. =10 ms CCD. =10 ms. CCD. =1 ms. =1 ms. APE (rad). APE (rad). CCD. 0. CCD. 4. 2. -1. 0. -2 0. 50. 100. 150. 200. -2. 250. 0. 50. 100. Time (s). (a) APE x-axis. 200. 250. (b) APE y-axis. 10 -8. 3. 150. Time (s). 2. 10 -8. 4. Nominal =20 ms CCD. 2. =10 ms. CCD. =1 ms. RPE requirement. 0. RPE (rad). RPE (rad). 0. CCD. 1. -2 -4. Nominal =20 ms CCD. -1 -6. =10 ms. CCD. -2. =1 ms. -8. CCD. RPE requirement. -3 0. 50. 100. 150. Time (s). (c) RPE x-axis. 200. 250. -10 0. 50. 100. 150. 200. 250. Time (s). (d) RPE y-axis. Figure 17: Pointing performances with different CCD delays. Piezo control starting at 100 s. 5. CONCLUSION. This article proposed a framework for a preliminary analysis and control of the micro-vibrations induced by a SADM system. The updated version of the SDT allows the user to request an LFT model for a spacecraft with rotating flexible appendages on the entire range of their rotation. A SimScape model has been developed for a SADM multi-physical analysis for design purposes. An interface with an SDT model is also available to analyze the impacts of the induced SADM torques on the spacecraft dynamics. A micro-step driving technique has been compared to the full-step approach to highlight its advantages in terms of pointing performances. An active control system with a piezoelectric FSM has finally been proposed to meet fine pointing requirements when the micro-step solution does not manage to completely satisfy the pointing specifications.. ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 14.

(38) ACKNOWLEDGMENT This work was founded by the European Space Agency (ESA) and Airbus Defence and Space under a Networking/Partnering Initiative (grant number: 4000116571/16/NL/MH). The view expressed herein can in no way be taken to reflect the official opinion of the European Space Agency. REFERENCES [1] C. L. Foster, M. L. Tinker, G. S. Nurre, and W. A. Till, “Solar-array-induced disturbance of the hubble space telescope pointing system,” Journal of Spacecraft and Rockets, vol. 32, no. 4, pp. 634–644, 1995. [2] C. Beugnon, B. Girouart, B. Frapard, and K. Lagadec, “Mixed µ-analysis applied to the attitude control of the metop spacecraft,” in European Control Conference (ECC), 2003. IEEE, 2003, pp. 2643–2649. [3] M. Vitelli, B. Specht, and F. Boquet, “A process to verify the microvibration and pointing stability requirements for the bepicolombo mission,” in International Workshop on Instrumentation for Planetary Missions, vol. 1683, 2012, p. 1023. [4] D. Alazard, “Satellite dynamics toolbox, version v1.3.” http://personnel.isae.fr/daniel-alazard/ matlab-packages/satellite-dynamics-toolbox.html, 2014 (Online; accessed April 20, 2017). [5] D. Alazard, C. Cumer, and K. Tantawi, “Linear dynamic modeling of spacecraft with various flexible appendages and on-board angular momentums,” 2008. [6] N. Guy, D. Alazard, C. Cumer, and C. Charbonnel, “Dynamic modeling and analysis of spacecraft with variable tilt of flexible appendages,” Journal of Dynamic Systems, Measurement, and Control, vol. 136, no. 2, p. 021020, 2014. [7] V. Dubanchet, “Modeling and control of a flexible space robot to capture a tumbling debris,” Ph.D. dissertation, Institut Supérieur de l’Aéronautique et de l’Espace, Polytechnique Montréal, 2016. [8] D. Alazard, J. A. Perez, C. Cumer, and T. Loquen, “Two-input two-output port model for mechanical systems,” in AIAA Guidance, Navigation, and Control Conference, 2015, p. 1778. [9] T. Kenjo, Stepping motors and their microprocessor controls. [10] P. P. Acarnley, Stepping motors: a guide to theory and practice.. Clarendon Press, 1984. Iet, 2002, no. 63.. [11] ESSB-HB-E-003, “Esa pointing error engineering handbook,” in ESSB-HB-E-003 Working Group, 2011. [12] Physik-Instrumente, “S-330 tip/tilt platform,” in PZ 149E User Manual, 2007. [13] PI, “E-500/e-501 modular piezo controller,” in Physik-Instrumente Manual, 2008.. ESA GNC 2017 – F. Sanfedino, V. Pommier-Budinger, D. Alazard, F. Boquet, A. Falcoz. 15.

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