A New Approach for Small Satellite Gyroscope and Star Tracker Fusion

Transcription

1 Indian Jornal of Science and echnology, Vol 9(7), DOI: /ijst/06/v9i7/9363, May 06 ISSN (Print) : ISSN (Online) : A New Approach for Small Satellite Gyroscope and Star racer Fsion Ngo Dy an, hai Qang Vinh and Bi rong yen Space echnology Institte, Vietnam Academy of Science and echnology, Hanoi, Viet Nam; ndtan@sti.vast.vn, bttyen@sti.vast.vn Institte of Information echnology, Vietnam Academy of Science and echnology, Hanoi, Viet Nam; tvinh@ioit.ac.vn Abstract his paper discsses the se of hybridization of measrements from different sensors (namely the gyroscope and star tracer) in order to estimate the attitde of a satellite. In this stdy, an adaptive fzzy logic algorithm is developed in order to add robstness to the eisting etended Kalmanfiltering method of sensor fsion. Models of the sensor are developed and a simlation is programmed in Matlab in order to eamine the effectiveness of the new algorithm. Reslts from the simlations show that the fzzy logic algorithm allows for better pointing accracy dring the period of Star tracer navailability. his method sed the EKF filter effectively and optimally when the system parameters and noises are nown. However in practice some of these parameters are ncertain, leading to the ineffectiveness when applied to system with high accracy reirements or comple system. In order to overcome this constraint, fzzy algorithm can be sed to evalate the reliability of the each component then estimate the appropriate tning parameters. his approach will facilitate small satellite attitde falttolerance estimator. Keywords: Fsion, Fzzy Logic Estimator, Satellite Gyroscope, Star racer. Introdction Sensor fsion generally refers to the process of combining measrements from several different sensors in order to obtain a better reslt than considering each sensor separately. Sensor fsion is especially sefl for sch system eipped with different types of sensors which are differently characterized by otpt types, sampling freencies and accracies. Common attitde sensors on board an Earthobservation small satellite are groped into two types: Inertial sensors: sch as accelerometer and velocity sensors. Reference sensors: sch as Sn sensor, Earth sensor, magnetometer and star tracer (or stellar sensor). Each type of the above sensors has its own advantages and disadvantages. Particlarly, inertial sensors are of high accracy (typically to the one hndredth of a degree) bt sffering from drift over time and may contain moving parts; reference sensors do not provide highly accrate otpt (accracy of arond one to several degrees) and depend on the reference frame, bt are low cost, robst and containing no moving parts. Moreover, an inorbit satellite is considered as a nonlinear system de to inflences by nwanted factors inclding: Eternal Forces: Pertrbation forces, solar pressre, gravity gradient. Internal Forces: Vibration effects by solar panel mechanism, momentm wheels, distrbance de to the actation of proplsion valves or pyrotechnie Sensors inaccracies. he diagram in Figre details the components of an AOCS sbsystem onboard a small satellite. his paper focs on the attitde estimator by fsing gyroscope and stellar sensor. Etended Kalman filter (EKF) is one of the most common algorithms becase of its simplicity and effectiveness for lowearth orbit satellites with light reirements in attitde accracy. However, this method is impacted by ncertain parameters sch as the system noise, measrement noise and especially in * Athor for correspondence

2 A New Approach for Small Satellite Gyroscope and Star racer Fsion the case of failre or navailability of one sensor. his fact cases the estimator delivering nreliable measrements, i.e. when the star tracer is blinded by the Moon or the Sn, or when the gyroscope sffer from overdrift or scale factor. As a reslt, it is necessary to apply adaptive algorithms, fzzy algorithm in this paper, to develop a falttolerance estimator. Figre. Small satellite attitde and orbit control sbsystem.. Qaternion Representation for Satellite Attitde Satellite attitde can be represented by: Eler angles, directional cosine matrices, and aternions. Among them, aternions are preferable thans to its efficiency and stability in comptation, and are widely sed in simlation programs and attitde estimation software. A aternion is a 4dimension vector, as follows: = [ ] º [ ] () Where: = [ 3] = esin( ) () 4 = cos( ) e is the Eler ais nit matri and is the angle of rotation. As this is a 4dimension vector and sed to represent a 3dimension rotation, the components of aternion cannot be independent and have to satisfy the following condition (normalization condition): 3 4 = = (3) Vol 9 (7) May 06 he corresponding directional cosine matri can be compted as: A( ) = ( ) I ( ) Y ( ) (4) Where: I 33 is the nit matri é ù I 4 33 X ( ) = é ù I 4 33 Y ( ) = é 0 3 ù = (5) he dynamic eation shall tae the form:. = X ( ) w= W ( w) (6) Where ω is the 3dimensional velocity vector and: éw wù W ( w) = w 0 (7) herefore, the ey advantage of sing aternions is that the inetic eations are linear and nonsinglar. Additionally, sccessive rotations can be obtained by aternion mltiplication. For eample: AA ( ) ( ) = A ( Ä ) (8) he prodct of two aternions is bilinear: ' Ä =Y [ ( ) ] =X [ ( ) ] (9) he inverse of a aternion matri is given as: é ù = 4 Where A( ) = A ( ) It is noticeable that: Ä = [000] is the nit aternion. 3. Modelling of Attitde Sensors (0) A. Modelling of Gyroscope Gyroscope is considered as an orientation measring device with relatively good accracy. Mechanical sensors operating on the principle of conservation of anglar momentm, sally consists of a rotating wheel or dis Indian Jornal of Science and echnology

3 Ngo Dy an, hai Qang Vinh and Bi rong yen arond its shaft. One of the main disadvantages of these sensors is its moving parts, leading to the declining in their sage. hey are being replaced by optical fiber or laser gyroscope. Both of them operate on the Sagnac effect. hey are of high accracy; containing no moving parts and high reliability. However they are sffering from drifts and noises. In particlar, the commonly sed optic gyroscopes drifts are highly ssceptible to the Earth s magnetic field. his needs to be seriosly considered when sing gyroscopes for attitde determination. For a 3ais attitde stabilized satellite, each ais will be eipped with a gyroscope for its attitde prediction. As sch, there will be error reslted from the mechanical misalignment of the monting of the gyroscopes on the satellite platform. Mathematical model of a gyroscope can be represented in Eation (). w= ( I S) w b h b= h s = h s 3X3 = h = h L L æ s U ö U º ç L U3 S s ç çèl L3 s 3 ø V () Where: w as the measred velocity, β as the drift, S as the scale factor matri s, U and L as the misalignment error; h, hv, hu and hlas the Gassian white noises with the zero mean vale and: { hv() hv ()} = sd v (t) 33 { h() h()} = sd (t) 33 { hu() hu()} = sd U (t) 33 { hl() hl()} = sd L (t) 33 () Where E {} denotes the epectation and d( t t) denotes the Dirac plse fnction. B. Modelling of Star racer A star tracer is high accracy sensor. he sensor image the stars and compares the captred pictres with a star map on board, the differences then be sed to calclate the satellite orientation. he sensor s accracy depends on several factors sch as the nmber of stars it can trac, its star map and the ality of its optical components. Its accracy is sally in the range of several arc seconds. Star tracer is modeled as: = dä (3) Vol 9 (7) May 06 Indian Jornal of Science and echnology 3 s Where: s is the aternion measred attitde, s is the aternion real attitde, δ is the sensor noise. 4. Gyroscope Star racer Fsion sing Etended Kalman Filter (EKF) he general strctre of an EKF filter is given in. However, mltiple sensors of different types are sed on board; there mst be some modifications to the measrement matri H and the error vector comptation ε for each sensor. he state vector of the estimator is given as: = [ b b b ] (4) y z Where: = [ 3 4 as the satellite attitde aternion. his vector is defined by (), () and (3): he error aternion is given as: d= Ä (5) Where d= [ d d4] and the inverse aternion = [ ]. aing the difference of (5) leads to: 4 (6) d = Ä Ä Eventally, the inematic eation is calclated as: = X ( ) w= W( w) (7) In order to find the aternion propagation eation, the above inematic eation will be epanded sing a series of eponential fnction as: e ìï ü ïë û ë û ï ý ïî ïþ é ù é ù W ( w) t W( w) t W ( w) t ï ê I ú ê ú = 4 4åí ( w) W( w) ï å = 0 ( )! = 0 ( )! ( w t) sin ( ) æ ö = I 4 4cos ç ( w) t W ( w) è ø ( w) (8)

4 A New Approach for Small Satellite Gyroscope and Star racer Fsion he propagation eation is given simply as: =W ( w ) (9) With w and as prediction vectors after pdate and W( w ) is defined as: æ ö æ ö I3 3cos w t Y ( ç D w ) W =ç ç è ø æ cos w ö D t è ç Y èç ø ø ( ) cos w Dt w Y (0) w w = w b b b () t is the sampling period. he covariance matri is calclated as follows: P =F P P GQG æf F ö F =ç ç ç çf è F ø sin( w t) { cos( w t)} X X F = I33 éw ù é w ù ê 3 ë ê w ë w F = 03X 3 F = I3X 3 æi 3X3 03X3ö G = ç çè 03X3 I 3X3ø æ 3 ö ( svt st ) I3X3 ( st ) I3X3 3 Q ç ( st) I3X3 ( svti ) 3X3 çè ø () With s and s v as the standard deviations. In order to determine the pdate fnction, the state prediction error is given as: X º [ b ] (3) For small angles it is possible to approimate d d» and 4». As sch, the 4dimension vector can be replaced by the 3dimension Eler angle vector. he net step is to determine the measrement matri H. he measrement model is given as: y º h ( X ) V (4) With y as the measred vector, X as the state, h as the measrement matri and v as the noise. For the star tracer: = d Ä =X [ ( ) ] d (5) Its measrement matri is given as: dh ( X ) é H ( X º = X( )0 X ê ë 3X3 X ù (6) Lastly, the pdate fnction for the state error vector is given as: (7) = K [ y h ( With K as the filter gain, h ( X ) as the measrement estimation, for star tracer this estimation is the aternion. Update fnction for the gyroscope: b = b b (8) Covariance matri and filter gains are calclated as followed: P = [ IKH( )] P (9) K = P H ( )[ H( ) P H ( ) R] Smming p, the gyrostellar fsion sing EKF is given in able. able. EKF Filter for GyroStellar Fsion Initialize ( 0) = 0 b( 0) = b0 P ( ) = P 0 0 Propagation w = wb b = b =W( w ) P =FP F GQG Gain comptation K = P H ( )[ H ) P H ( ) R] H( é ù ) = X( )033 Update = K [ y h ( )] b = b b ( ) X = b d P = [ IKH( )] P he simlation reslts of this estimator will be illstrated in the session VI. 4 Vol 9 (7) May 06 Indian Jornal of Science and echnology

5 Ngo Dy an, hai Qang Vinh and Bi rong yen 5. Proposed Fzzy Algorithm for nning the EKF GyroStellar Estimator Dring the operation on a Snsynchronos orbit, the attitde determination sbsystem has to handle the following constraints: Star tracer: Despite of its high accracy, it is not always possible to deliver measrements with the same accracy. In particlar if there are too few stars in the field of view of the sensor the accracy will be degraded. In addition, when there are bright objects (lie the Sn or the Moon) in its field of view, its measrements will be nreliable de to its error. Gyroscope: the drift over time has a large impact on the accracy of the anglar measrements. And, its scale factor error and calibration error mst be careflly considered. On the other hand, Kalman filter only wors well when the system noise model w (with covariance matri Q ) and the measrement noise v (with covariance matri R ) are white noise processes with zero mean. However, in practice all the parameters of the system cannot be nown eactly and these noise processes are not white noises. his fact leads to the srpls error of the EFK ( yh( )) not being white noise any more. If this error gets too large de to ncertain elements the EFK filter will not be able to converge. hese are only two factors that affects the convergence of the EFK filter, there may be many more other sch as the drift or scale factor of the gyroscope, the misalignment of the mechanical monting process of the sensors. Based on the above analysis, in order to maintain the accracy, stability and falttolerance of the attitde estimator there has to be a mechanism to monitor the evoltion of this srpls error, and a mechanism for compensating the gains of the EFK filter. Fzzy logic algorithm is proposed to monitor and compensate the parameters Q and R of the filter. his is proposed as follows: Q R = Q a _ new 0 = R a _ new 0 ( ) ( ) (30) With α as the tning coefficient, if α = then this is the normal EFK filter,q 0, R 0 are constant m atrices. he fzzy algorithm wors on the following principle: eep monitoring the mean and standard deviation of the srpls error of the EKF filter, then assess and calclate the calibration vale α according to this flctation. In practice this process aimsat assessing the sensors reliability, if a sensor s reliability is high then the filter will increase the confidence weight and prioritize the se its measrement and vice versa. his is also the bacgrond to develop evalation and raning algorithm, creating the fondation for attitde determination system with faltresistance capability. he fzzy algorithm taes the mean and standard deviation of the srpls error as inpts and the otpt is the tning coefficient α. he bloc diagram of the algorithm is given in Figre. Figre. Fzzy based tning bloc. Net step, the Membership fnction (MF) is developed. he MF fnction of the inpts is divided into 3 levels: zero, small and large corresponding to the vale of the inpts. he MF fnction for the mean vale inpt is given in Figre 3. Figre 3. MF fnction for inpt mean vale. For the otpt α, 4 levels are the range of [;.] as: zero, small, average and large. Its MF fnction is given in Figre 4. he fzzy referring is proposed in able. able. Fzzy referring for GyroStella estimator Vale of α Zero Inpt : vale of the mean Small Large Inpt : Zero Small Small Large standard Small Zero Large Average deviation Large Large Average Zero Vol 9 (7) May 06 Indian Jornal of Science and echnology 5

6 A New Approach for Small Satellite Gyroscope and Star racer Fsion Figre 4. MF fnction for otpt. 6. Simlation he simlation is implemented in MALAB with the following parameters: Satellite specifications: Inertial matri: æ ö J = 0 0 ç çè 0 0 8ø g. m Sampling freency: 0 Hz Simlation time: 0 s Star tracer: 4 radian (3σ) error Gyroscope: 6 o /h drift, anglar rate wal ARW=0.5 o / srt(h). Pertrbation tore: t = [ ] et Initial condition: = [ ] he reslt of the simlation is depicted by Figre 5. Sppose that the star tracer sffers a loss of some measrements (de to noise) dring the operation then the reslt of the simlation is illstrated in Figre 6. Figre 6. Simlation reslts by normal EKF filter (impacted by noise). It can be seen that when the measrement data of a star tracer loses reliability then the EKF can only se data from the gyroscopes to estimate the attitde. However de to the drifts of the gyroscopes the longer the dration of sensor loss the more the error of the filter will accmlate, decreasing the reliability of the filter, and even leads to divergence. As a compensation method, fzzy algorithm is applied. his algorithm calclates the reliability of the sensors sing the srpls error, then estimate the calibration coefficient. he reslt after applying fzzy algorithm is given in Figre 7. Comparing the reslts, the fzzy algorithm has improved greatly the effectiveness of the filter, especially when measrement anomaly appears in the star tracer. In the scope of this article, fzzy algorithm is applied only to EKF filter with star tracer and gyroscopes as two inpts. However, this technie can also be applied to other sensors fsion with more than two types of sensors in order to improve their accracy and stability. his is also the fondation to develop adaptive sensor fsion with falttolerance capability. Figre 5. Simlation reslts by normal EKF filter. Figre 7. Simlation reslts by fzzy tned EKF filter (impacted by noise). 6 Vol 9 (7) May 06 Indian Jornal of Science and echnology

7 Ngo Dy an, hai Qang Vinh and Bi rong yen 7. Conclsions his article describes the method of sensorfsion sing the etended Kalman filter to estimate the attitde of a small satellite. his method sed the EKF filter effectively and optimally when the system parameters and noises are nown. However in practice some of these parameters are ncertain, leading to the ineffectiveness when applied to system with high accracy reirements or comple system. In order to overcome this constraint, fzzy algorithm can be sed to evalate the reliability of the each component then estimate the appropriate tning parameter. his approach will facilitate small satellite attitde falttolerance estimator. 8. References. John L Crassidis, John L Jnins. Chapman and Hall: Optimal estimation of Dynamic Systems. nd edn Shepperd SW. Qaternion from Rotation matri. Jornal of Gidance and Control. 978; (3): Crassidis JL. SigmaPoint Kalman Filtering for Integrated GPS and Inertial Navigation. San Francisco, CA: AIAA Gidance, Navigation and Control Conference Wahba GA. LeastSares Estimate of Satellite Attitde. SIAM Review. 965; 7(3). 5. Lerner GM. Qmethod. Netherlands: Klwer Academic Pblishers: Spacecraft Attitde Determination and Control, Wertz JR Shster MD, Oh SD. hreeais Attitde Determination from Vector Observation. Jornal of Gidance and Control. 98; 4(). 7. Vinh hai Qang, Hirota Kaor: Decentralized robst fzzy sliding mode control design of interconnected ncertain system. Jornal of Advanced Comptational Intelligence Van der Merwe Wan E. John Wiley & Sons: he Unscented Kalman Filter. Kalman Filtering and Neral Networs Crassidis JL, Marley FL. Unscented Filtering for Spacecraft Attitde Estimation. Jornal of Gidance, Control and Dynamics. 003; 6(4). 0. Landis Marey F, John L Crassidi, Cheng Yang. NASA Goddard Space Flight Center: Nonlinear Attitde Filtering Methods an ran Mạnh. Congnghevetinh. Nhaatban KH&K Passino KM, Yrovich S. AddisonWesley: Fzzy Control soalas LH, Uhrig RE. Fzzy and Neral Approaches in Engineering. WileyInterscience Lin C, Lee CS. Prentice Hall: A NeroFzzy Synergism to Intelligent Systems Qang M. Lam, John L. Crassidis. Precision Attitde Determination Using a Mltiple Model Adaptive Estimation Scheme. IEEE Aerospace Conference Jason thill. Design and Simlation of a nanosatellite attitde determination system Vol 9 (7) May 06 Indian Jornal of Science and echnology 7