Extremely Accurate Lne of Sght Control between a Pontng and a Target Spacecraft Pergovann Magnan a, b, Amala Ercol-Fnz b, Franco Bernell-Zazzera b a Galleo Avonca, Mlano, Italy b Dpartmento d Ingegnera Aerospazale, Poltecnco d Mlano, Italy Abstract Extremely accurate Lne Of Sght (LOS) control of a Pontng Spacecraft (S/C) wth respect a Target S/C has been exploted n order to devse very advanced n-orbt scentfc and technologcal experments. The pontng performances are acheved by the utlsaton of accurate sensors, low dsturbance actuators and the development of a control scheme that takes advantage of state of the art technology. In ths paper the smulaton approach s presented, as well as the acheved results, when a relatve algnment of two S/Cs nomnally co-orbtng n GEO s requred to be better than some mllarcseconds. Sutable dynamc and measurement models and control methods are developed and proved that the algnment can be obtaned whle oscllaton jtters are mantaned extremely low n the frequency band above a reference value of one Hz. Introducton The n orbt scenaro taken as reference for the analyss ams at a tentatve approach to detect possble non symmetres of the fundamental space-tme tensor caused by electro magnetc felds, e.g. unfed felds theores [1]. The experment conssts of three co-orbtng Spacecrafts (S/C): the Pontng S/C, the Target S/C and the Dstorter S/C. The Pontng S/C, an extremely stable platform, needs be optcally algned to a large detector sensor placed at the Target S/C so that a hghly collmated laser beam can form a spot on such sensor. Upon actvaton of the electromagnetc felds at the Dstorter S/C the spot may be subject to very small transversal dsplacements, whch can be determned by centrodng technques, provdng an ndcaton on the searched non symmetres. Gven the nature of the scentfc nvestgaton to be performed, the operatonal approach consdered can be based on the followng ponts [2]: 1. The Pontng S/C needs to be optcally algned onto the sensor surface array of the Target S/C for sequences of Basc Tme Interval (BTI), each allowng the executon of a porton of scentfc nvestgaton. Durng each BTI a free drftng phase has to be guaranteed, that s the vbraton on the Pontng S/C and Target S/C need be really mnmum n order not to dsturb the scentfc measurement; 2. Durng BTI the Pontng S/C LOS tends to drft away from the target because of the effects of non zero ntal relatve angular speed rate and the dsturbance torques due to the radaton pressure and gravty gradent. Furthermore the Pontng S/C poston, as well as the Target S/C poston, tend to drft due to non
zero ntal speed rate, radaton pressure, Sun/Moon perturbatons, Earth potental dstorton terms; 3. After each generc BTI the actuators can be fred wthn a sutable Frng Tme (FT), as schematzed n Fgure 1, n order to generate the approprate startng condtons for the Pontng S/C to properly ntate and allow the development of the next BTI by re-centerng the Pontng S/C LOS to the Target S/C, and recover the drftng atttude for the Target S/C, only wth a coarse accuracy. 4. Durng the repetton of several combned BTI followed by FT sequences, the S/Cs abandon ther nomnal locatons. Once the dsplacements exceed some allowed lmts the S/Cs poston recovery can be performed. However such a recovery needs be performed by thrusters n the class 2 (mn) and n tme ntervals not overlappng the scentfc measurements. Free Drft (BTI) Intal Poston Motor Frng (FT) Postonng Manoeuver (Motor Frng) Target sensor Array Fgure 1: Relatve pontng control strategy The Pontng S/C s the key element nvolved n the target acquston process and t nstalls the hgh accuracy sensors [2] and the ultra-fne thrustng actuator system [3][4] necessary to the purpose. The Target S/C acts, n ths respect, as a free drftng passve platform but for the laser beam castng so to mplement the gude star functons. The Dstorter S/C, whch s near to the Pontng S/C, wll be controlled relatve to the Pontng S/C n a coarse manner and does not play a key role n the fne algnment process. The feasblty of such a msson wll depend, prmarly, on the performances attanable durng three specfc phases: target acquston wth a very hgh accuracy, scentfc phase that asks for a sutable tme of permanence on target wth very low jtter level, poston recovery wth the utlsaton of very low thrust actuators. In the followng such phases are evaluated n detal n order to assess ther feasblty. Dynamc and measurement model The Pontng S/C structural elements utlsed for the smulaton nclude an nner optcal platform carryng the hgh accuracy optcal payload and gyroscopes, an outer structural frame carryng all remanng typcal S/C sub systems, and two short appendages representng e.g. solar panels folded and locked or antennas. The
connecton between the nner optcal platform and the outer structure s modelled by means of 6 sprngs (3 lnear, 3 angular) and 6 dampers (3 lnear, 3 angular). The connecton between each of the two appendages and the outer structural frame s modelled by means of 2 sprngs (1 lnear, 1 angular) and 2 dampers (1 lnear and 1 angular). The Pontng S/C state vector s descrbed by 38 components (32 needed to descrbe postons and speeds of the bodes and 6 needed as extensons, to descrbe the non whte nose of the solar radaton pressure fluctuatons). The pontng spacecraft measurement vector gather the followng types of measurements: relatve yaw and relatve ptch (from the hgh accuracy nterferometrc relatve atttude sensor [2]), absolute bank (from the hgh accuracy star tracker), poston dsplacements from the nomnal algnment frame (from the poston determnaton system based on SLR satellte laser rangng, transpondng technques, GPS) and absolute angular rates (from rate gyros). The thrusters are assumed connected to the outer spacecraft structure, wth a confguraton layout arranged to adapt to the fnal S/C outer body structural arrangements. For the objectves of the present analyss the system composed by the Pontng S/C and the Target S/C has been lnearzed and modelled around the nomnal orbtal confguraton. The followng set of equatons (1) have been consdered: p = A p B1 p1 q = E q F1 q1 = 4 y = C p D g = 1 = 7 = 2 = 4 = 2 ( q) B p F q Pontng S/C dynamcs Target S/C dynamcs Measurements (1) The Pontng S/C dynamcs parameters are represented by p, 38-element state vector, p1, 6-element nomnal thrust vector, p2, 6-element resoluton thrust vector, p3, 3-element Moon/Sun perturbaton vector, p4, 2-element Earth perturbaton vector, p5, Sun flux ntensty, p6, 2-element gravty gradent torque vector, p7, 12- element nose vector (thrust and radaton flux). The Target S/C dynamcs parameters are represented by q, 6-element state vector, q1, 6-element nomnal thrust vector, q2, 3-element Moon/Sun perturbaton vector, q3, 2-element Earth perturbaton vector, q4, Sun flux ntensty. The measurements parameters are represented by y, 9-element measurement vector, g1, 6-element atttude sensors offset vector, g2, 3-element forcng vector, g3, 3-element poston sensors offset vector and g4, 9-element measurement nose vector. As we can see the Target S/C has been assumed as an ndependently controlled, non cooperatve platform whose coordnates q enter the measurement equatons of the Pontng S/C. In turn the control loop of the Pontng S/C njects the nformaton on the Target S/C poston nto the Pontng S/C dynamcs va the regulator equatons.
The objectve s to keep the Pontng S/C as much as possble relatvely algned to the Target S/C and wth the mnmum possble relatve jtter. Control schemes In all schemes the control s acheved by mplementng loops based on sx ndependent PID regulators. In partcular, also wth reference to Fgure 2, two PIDs are dedcated to the control of the relatve atttudes (relatve α and relatve γ angles), one PID s dedcated to the control of one absolute atttude (β angle relatve to the local algnment frame), and three PIDs are dedcated to the control of the poston coordnates (xp, yp, zp relatve to the local algnment frame). Target S/C Relatve α β Pontng S/C xp Relatve γ zp Target S/C nomnal algnment frame 1 km separaton yp Pontng S/C nomnal algnment frame Fgure 2: Schematcs of the sx controlled parameters (wdely not to scale) The control analyss has been carred out by consderng two types of thruster systems utlsed n dfferent perods of the msson: a very accurate mcronewton class thruster, lmted to 1 µn and 3 µnm, dedcated to the relatve target acquston (and keepng) phase and a mllnewton class thrusters, lmted to 2 mn and 6 mnm, dedcated to the poston recovery phase. Durng the relatve target acquston phase only the two relatve atttudes and the absolute atttude are controlled by the mcronewton system whle the poston control s not actve. In fact not only s t not needed by the msson, but also the utlsaton of hgh thrust necessary for poston recovery would nduce strong dsturbances n the relatve pontng/algnment. Durng the poston recovery phase nstead, the mcronewton thrusters are not actve snce they would not be of any effect whle the control s performed n all sx coordnates. Control scheme for relatve target acquston (and keepng) The basc control scheme utlsed s shown n Fgure 3, wth the poston control not actve, and t s based on Kalman Flter (KF) mplementaton. Two tmng are consdered n the smulaton: an nterval DT for the smulaton of the dynamcs and KF propagaton and an nterval REG for the smulaton of the control loops, the atttude/poston sensors acquston (relatve atttude nterferometer, hgh accuracy star tracker and poston measurement system even f not used n ths scheme) and KF update.
Fgure 3: Control scheme for target acquston and keepng xˆ( k 1) z( k 1) = M x( k 1) G W xˆ( k 1) = xˆ( k 1) Kk 1 z( k 1) [ M xˆ( k 1) G] 14 ) 44 24443 ( 1) = z k ) I( k 1) = I( k) ( t t ) xˆ C ( k 1) = Φ x( k) = xˆ( k 1) F { z( k 1) z( k 1) } GI I( k 1) k 1 k (State propagaton) (Measurement) (2) (State update) (Added ntegraton acton) (Correcton of State update) Concernng the KF mplementaton, the followng consderatons are noted: a) The system dstrbuton matrces, the radaton pressure average dsturbances, the measurement matrx and some forcng parameter n the measurement equaton have been mplemented wth naccuraces; ndeed 21 perturbng parameters are present drectly affectng the atttude behavour. b) The system dynamc matrx have been assumed strongly perturbed n two prmary stffness parameters to smulate partal structural falure durng launch. Above perturbatons have two types of effects at the level of performances consdered: the perturbatons of type a) create some relatve atttude offsets (n the order of.1 arcsec) n the KF state reconstructon showng mnor senstvty whle the perturbatons of type b) are more mportant and tend to make the KF unstable showng an mportant senstvty. To cope wth above stuatons the followng approach has been used: a) The mnor offsets have been controlled by mplementng an ntegraton acton wthn the KF structure (see also eq. 2).
b) The strong senstvty to uncertantes n the dynamc matrx have been controlled by extendng the KF to the uncertan parameters; the KF becomes then non lnear so t has been lnearzed [5]. The approach utlsed allows a robust estmator wth no need of an excessve extenson of the flter. Other methods lke memory loss can ndeed control the nstabltes due to dynamc matrx uncertantes but the relatve atttude errors ncrease to approxmately.2-.3 arcsec and the jtter n the band >1 Hz ncreases by a factor of 5 whch s to be avoded for the specfc applcaton consdered. The gans of the regulators have been selected based on two approaches: tunng of gans wth tral and test (e.g. placement and regulaton of proportonal gans followed by placement and regulaton of dervatve gans and ntegral gans wth teratons) and utlsaton of determnstc optmum control technques [6] to derve proportonal and dervatve gans of comparable regulators. Control scheme for poston recovery phase Durng the recovery phase relatve large amount of poston dsplacements are to be managed, typcally tens or hundreds of meters n the case consdered. A control scheme whch utlses the fnal commanded coordnates drectly n the control loop can requre large tme of stablsaton f the control has very lmted acton capabltes (e.g. non lnear control caused by small saturaton thresholds lke n our case). In such crcumstances speed lmtatons may be needed otherwse the regulaton acton of the PID could requre several oscllatons near the fnal reference coordnates to damp the moton. An alternatve and more determnstc approach can be devsed takng as examples the trajectory generaton technques also used n other branches of engneerng (lke robotcs). Wth such an approach smooth reference trajectores can be planned wth polynomal profles compatble wth the characterstcs of the actuators and the desred qualty of moton. For the specfc case a quadratc generaton has been consdered for each of the sx controlled co ordnates. The recovery manoeuvres are mplemented n two steps: durng the frst step a speed slow down (tll zero) s acheved, n the second step the poston s recovered to the nomnal zero coordnate (and zero speed). The selecton of the parameters used for the trajectory profle generaton affect prmarly the tme of manoeuvres and the electrc power requrements, to a lesser extent the fuel consumpton. The gans used n the control loop, related to the allowed errors durng manoeuvres, affect the fuel consumpton; optmum determned gans from lnear theory are consdered. However t has to be sad that the closed loop system very easly becomes non lnear (due to the forces saturaton) as soon as the manoeuvre s tred to be faster. The approach consdered allows to perform stable, relable and fuel lmted recovery manoeuvres. The control on all axes (postons and atttudes) has been smulated by consderng a relatve atttude sensor wth characterstcs typcal of an absolute hgh accuracy star tracker and by consderng actuators of Xenon type capable of thrust level compatble wth ths type of msson. The obtaned
performances are representatve of an absolute control n the nomnal algnment frame. Numercal smulatons and results Several smulatons have been carred out for the three key msson phases: target acquston and keepng, scentfc phase (both repeated acqustons and free drfts) and poston recovery. Concernng the target acquston and keepng phase, the man objectves were to evaluate the pontng accuracy and jtter. Durng the analyss, large ntal msalgnments and Target S/C hgh drftng speeds have been assumed n order to assess the capablty and the necessary tme for correct Target acquston and trackng. Furthermore partcular mportance has been gven to robustness of the closed loop system aganst naccuraces and perturbatons for the dstrbuton, measurement and system matrxes. The evaluatons performed on the scentfc phase had the man objectves to estmate the tme of permanence of the LOS onto the Target S/C (ndeed on ts large sensor array) and the resdual jtters for dfferent levels of external dsturbances. The robustness of the closed loop system aganst Target S/C strong perturbatons durng repeated target acquston steps have been assessed. The evaluatons performed on the poston recovery phase amed at the overall verfcaton of the recovery strategy based on trajectory plannng technques. On top of the basc feasblty aspects, two major ndcators have been explored: fuel consumpton and power consumpton. Target Acquston and Keepng phase Fgure 4 presents a sample of performances concernng the relatve atttude angles: ptch (relatve γ) and yaw (relatve α). 3.6 x 1-3 -9 x 1 4 relatve gamma (arcsec) 3.4 3.2 3 2.8 2.6 2.4 2.2 2 3.5 rela 3 tve ga mm a fft 2.5 (ar cse c) 2 rms 1.5 1.5 1.8.5 1 1.5 tme (s*1) 2 2.5 3 x 1 4 1 2 3 4 5 6 7 8 9 1 freq (Hz) shown range 1-1 Hz Fgure 4: Some results on the Target Acquston and Keepng phase The target acquston manoeuvre starts from an ntal combned relatve atttude of 2.5 arcsec and a relatve atttude speed of.6 arcsec/s snce a fast movng target wth ntal speed of.3 m/s has been assumed. Furthermore a strong target
stepwse poston perturbaton (7 m combned) has been njected to check the trackng capablty of the system. The results evdence that the acquston and trackng capablty s farly effectve and robust, stablsng n about 2 s. Furthermore the pontng error, over a stablsed observaton wndow, s approxmately.5 arcsec of whch about.25 arcsec are the basc offset of the relatve atttude sensor. Ths correspond to approxmately.25 m at the target plane. The combned jtter n the band > 1 Hz s about 2 arcsec rms, whch corresponds to approxmately 1E-6 m rms at the target plane. Scentfc phase Fgure 5 summarzes some key performances durng the repeated acquston sequences durng Scentfc phase. For ths smulaton the allowed drft dstance from the centre of the target was set to 12 m above whch the target acquston was restarted. The overall phase s therefore an alternate sequence of target acquston and free drfts. The control system have been proved very robust aganst senstvty verfcatons (S/C optcal propertes unbalance, radaton pressure fluctuatons, target geometry sze, poston perturbatons on the target S/C) and the followng consderatons are noted: - One complete cycle s performed n approxmately 2 s. - The free drft phase last for approxmately 5-6 s durng whch the core of the scentfc expermentaton can be done. - The total fuel consumpton n 1.5 years, assumng a duty cycle of scentfc phases versus poston recovery phases of 1:2 turns out to be of.7 kg of propellant (Caesum or Indum for the FEEP technology). 3.5 x 1-7 3 fuel cumulatve consumpton (kg) 2.5 2 1.5 1.5 1 2 3 4 5 6 7 x 1 4 tme (s*1) Fgure 5: Example of Scentfc Phase: repeated acquston The jtter behavour above 1 Hz durng the free drft phases s summarsed n Fgure 6. It s noted that jtters on target plane better than 1E-1 m =.1 µm are achevable and even assumng uncertantes of 1-1 on the PSD value of the radaton pressure fluctuatons, the jtter would reman more than one order of magntude smaller than a value of.1 µm whch could represent a reference resoluton on the laser spot centrodng determnaton.
Poston Recovery phase Fgure 7 summarzes some representatve results related to a recovery phase. The behavour of the system s tme varant snce t depends on the relatve Earth-S/C- Moon-Sun postons. In the case shown a knd of worst case stuaton has been consdered snce the startng condton has been assumed around the full algnment of the bodes nvolved. relatve gamma fft (arcsec) rms 4 x 1-13 3.5 3 2.5 2 1.5 1.5 relatve Xs fft (m) rms 2 x 1-11 1.8 1.6 1.4 1.2 1.8.6.4.2 1 2 3 4 5 6 7 8 9 1 freq (Hz) shown range 1-1 Hz RMS= 1.82e-12 (arcsec) from 1 to 5 Hz 1 2 3 4 5 6 7 8 9 1 freq (Hz) shown range 1-1 Hz RMS= 5.2454e-11 (m) from 1 to 5 Hz Fgure 6: Example of Scentfc Phase: free drft (scentfc measurement) The recovery starts from an ntal poston offset of about 33 m and ntal speeds whch would accumulate from a prevous Scentfc phase lastng n the order of three hours. The frst step of the recovery manoeuvre takes about 8113 s and slows down the S/C moton to zero speed whle the poston error would grow to about 6 m; the second step lasts about 146 s and would brng the S/C to ts zero poston wth zero speed ready to start a new scentfc phase. 6.15 5 X opt (m) 4 3 2 1 cumulatve fuel consumpton (kg).1.5-1 1 2 3 4 tme (s*5) 5 6 7 8 x 1 5 1 2 3 4 tme (s*5) 5 6 7 8 x 1 5 Fgure 7: Example of Poston Recovery Phase (second step) The total fuel consumpton for the recovery, n the assumed condtons s approxmately.21 kg whch would lead, for a 1.5 years msson, to approxmately 3 kg of fuel (e,g, Xenon). Concernng the electrc power
consumpton present an average value of 92 W wth sustaned perod of 14 W whch wll pose due requrements on the Power and Dstrbuton System. Whle the tme s evolvng the perturbaton effects from the Sun and Earth are contnuously changng but we can assume the case reported representatve for the feasblty of the manoeuvre. Conclusons The extraordnary possbltes offered by the space envronment and the progressve mprovements n space related technologes open to new potental scenar for n orbt expermentaton related to basc physcs ssues. The results presented n ths paper, based on approprate dynamc modellng and control system, proved that LOS relatve control of a Pontng S/C to a dstant Target S/C can be acheved wth very hgh performances. In fact, for two reference S/Cs n GEO at a nomnal dstance n the order of 1 km, a Target S/C ntercepton error of.25 m (relatve pontng error of.5 arcsec) wth Target S/C ntercepton jtter of 2 µm appears feasble durng sustaned target acquston. When the algnment s reached and stablzed the Lne of Sght relatve jtter wll drop to somethng n the order of.1 µm f the relatve control s swtched off leavng the S/Cs n free drft mode. A very mportant ssue n order to acheve above performances s the on board avalablty of drect relatve atttude measurements wth performances far below the arcsec level and ultra low nose thrusters. On lne dynamc flterng appears furthermore to be qute effectve and can be brought robust enough wth respect to the modellng uncertantes (dstrbuton matrxes, dynamc matrx, measurement matrx). Wth respect to the n orbt operaton scenaro taken as reference for the analyss, the key ssue of extremely accurate and stable relatve algnment of the Pontng S/C on to the Target S/C appears to be achevable. References [1] Magnan PG., Fnz A.E., A Possble In Orbt Relatvstc Experment Msson wth Very Accurate S/C Relatve Control, accepted for publcaton on AIDAA journal Aerotecnca Mssl e Spazo, 24 [2] Magnan PG., Methodologes for accurate spacecraft relatve control n support to very demandng scentfc mssons, PhD Thess, Poltecnco d Mlano, 24 [3] Tajmar M., Genovese A., Steger W., Indum feld emsson electrc propulson mcrothruster expermental characterzaton, Journal of Propulson And Power, vol. 2, n.2, pp. 211-218 mar.-apr. 24 [4] Marcucco S., Genovese A., Andrenucc M., Expermental performance of feld emsson mcrothrusters, Journal of Propulson And Power, vol. 14, n. 5, pp. 774-781, sep.-oct. 1998 [5] Gelb A., Kasper J.F., Nash R.A., Prce C.F., Sutherland A.A., Appled Optmal Estmaton, the M.I.T. Press, 1986 [6] Fredland B., Control System Desgn: an Introducton to State Space Methods, McGraw Hll, 1986.