Adaptive Fusion of Inertial Navigation System and Tracking Radar Data

Transcription

1 Amirkabir Universiy of Technology (Tehran Polyechnic) Amirkabir Inernaional Jounrnal of Science & Research Elecrical & Elecronics Engineering (AIJ-EEE) Vol. 48, No. 2, Fall 2016, pp Adapive Fusion of Inerial Navigaion Sysem and Tracking Radar Daa M. Fahi 1,*, N. Ghahramani 2, M. A. S. Ashiani 3, A. Mohammadi 4, M. Fallah 5 1- Ph.D. Candidae, Deparmen of Aerospace Engineering-Fligh Mechanics and Conrol Group, Malek Ashar Universiy of Technology 2- Associae Professor, Deparmen of Elecrical Engineering-Conrol Group, Malek Ashar Universiy of Technology 3- Associae Professor, Deparmen of Aerospace Engineering-Fligh Mechanics and Conrol Group, Malek Ashar Universiy of Technology 4- Assisan Professor, Deparmen of Elecrical Engineering-Conrol Group, Malek Ashar Universiy of Technology 5- Assisan Professor, Deparmen of Elecrical Engineering-Communicaion Group, Malek Ashar Universiy of Technology ABSTRACT (Received 10 Ocober 2015, Acceped 29 May 2016) Agains he range-dependen accuracy of he racking radar measuremens including range, elevaion and bearing angles, a new hybrid adapive Kalman filer is proposed o enhance he performance of he radar aided srapdown inerial navigaion sysem (INS/Radar). This filer involves he concep of residual-based adapive esimaion and adapive fading Kalman filer, and unes dynamically he filer parameers, including he fading facors and he measuremen and process noises scaling facors based on he raio of he acual residual covariance o he heoreical one. In fac, due o he unknown and fasvarying saisical parameers of he radar measuremen noises and heir nonlinear characerisics, applying a convenional Kalman filer o INS/Radar fusion yields a low-performance navigaion and in-fligh alignmen. The Mone Carlo simulaion resuls of he inegraed navigaion sysem on an inercepor missile rajecory indicae he new algorihm has an effecive performance in he face of nonlineariies and uncerainies of he racking radar measuremens. These resuls allow knowing wheher he fine infligh alignmen and high-performance navigaion can be possible for he long-range air defense missile using he low-cos INS/Radar sysem wihou aiding global navigaion saellie sysem signals. KEYWORDS: Adapive Kalman Filer, Inerial Navigaion, In-fligh Alignmen, Radar Please cie his aricle using: Fahi, M., Ghahramani, N., Ashiani, M. A. S., Mohammadi, A., and Fallah, M., Adapive Fusion of Inerial Navigaion Sysem and Tracking Radar Daa. Amirkabir Inernaional Journal of Elecrical and Elecronics Engineering, 48(2), pp DOI: /eej URL: hp://eej.au.ac.ir/aricle_818.hml * Corresponding Auhor, mahd_fahi@mu.ac.ir AIJ - Elecrical & Elecronics Engineering, Vol 48, No. 2, Fall

2 Adapive Fusion of Inerial Navigaion Sysem and Tracking Radar Daa 1- Inroducion I is widely acceped ha he modern long-range air defense missiles will require a wo-phase guidance sraegy employing a mid-course phase followed by erminal homing. These acical missiles will require an on-board srapdown inerial navigaion sysem, INS, o perform he funcions, such as he provision of he daa for a navigaion of he missile prior o a period of erminal homing, provision of homing head poining commands and provision of auopilo feedback signals [1]. Due o INS inrinsic defec and special mission of an ani-air missile, four problems can appear. Firs, INS navigaion errors propagae as a resul of insrumen errors, iniial alignmen error, and imperfecions in he srapdown compuing algorihms. Second, many air defense missiles require an accurae alignmen of he onboard INS in order o achieve a high probabiliy of killing agains heir inended arges [2]. Third, air defense sysems require rapid reacion ime, hus significan pre-launch delays for alignmen are no olerable [2]. Fourh, ypically, such acical sysems require medium grade inerial qualiy insrumens (high cos) o carry ou heir funcions and o ensure an accurae ransiion o he erminal phase of fligh. As a soluion o overcoming hese problems, he racking radar (radio deecion and ranging) aided INS was presened in [1,3]. In his paricular applicaion, he launch plaform can provide only a very coarse iniializaion for he missile s INS hen he missile is fired o inercep an inended arge. The missile and he arge are racked by a ground-based radar sysem. The radar provides measuremens of missile range, elevaion and bearing wih respec o he radar se. These measuremens may be passed o he missile via an uplink ransmier and used o aid he on-board INS. The missile on-board inerial navigaion sysem implemens a Kalman filer o deermine in-fligh correcions o he navigaion errors. Several sudies have been conduced on INS/ Radar o gain he benefis of in-fligh alignmen of INS, in-fligh calibraion of INS, improvemen of INS performance and in-fligh regisraion of radar frame wih navigaion frame. Boh he in-fligh alignmen by INS/Radar inegraion and he pre-launched alignmen by shipboard measuremens for a ship launched missile s INS are developed and heir relaive meris are compared in [1]. The subjec of [2-8] is relaed o he developmen of a radar and GPS (global posiioning sysem) aided INS for a navy acical ballisic missile defense inercepor missile (Sandard Missile SM- 3). An overview and a preliminary performance assessmen of an INS/Radar/GPS sysem based on a direc Kalman filer wih 20 saes are provided in [3]. The research in [2,4] proposes a new meric o deermine he degree of observabiliy of aiude errors in fligh alignmen of INS/Radar/GPS over specified missile fligh profiles. In-fligh alignmen of an onboard INS using he exernal radar and GPS daa and a convenional Kalman filer for daa fusion have been explained in deail in [5,6]. Sure enough, a cornersone o he in-fligh alignmen was laid ou in [7-9] which addressed he quesion: wha kind of horizonal maneuver is preferable in he in-fligh INS alignmen; he ineresed reader is direced o hese references for a more deailed discussion of in-fligh alignmen. The references [10,11] discuss he successful fligh es resuls of he INS/Radar/ GPS sysem for he missile SM-3. A srapdown INS augmenaion scheme comprising asronavigaion sysem and secure radio posiioning sysem is proposed in [12]. Also, he issue of he Mars enry navigaion using inegraion of informaion sensed from radio beacons and informaion derived from inerial measuremen uni is addressed in [13]. All he above-menioned researches relevan o he inegraion of INS and radar daa assume ha he measuremen noise covariance marix (R) in he Kalman filer is fixed. In oher words, i is supposed ha he racking radar measuremens have consan sandard deviaions. However, i is well known ha he radar measuremens accuracies are very range depended [14, chaper 11], [15, chaper 4]. Indeed, he accuracies of range and angles (elevaion and bearing) measured by he radar are he funcions of he range (see secion 2 for he deails). I means he marix R should be a varying marix, no a fixed one. Insufficien a priori informaion abou his varying marix R affecs he accuracy of he INS/ Radar inegraed sysem. In fac, insufficienly known a priori filer saisics will reduce he precision of he esimaed filer saes or inroduce biases o heir esimaes [16]. In addiion, wrong a priori informaion will lead o he pracical divergence of he filer. For example, if R is oo small a he beginning of he esimaion process, he uncerainy ube around he rue value in a probabilisic sense will ighen and a biased soluion will resul. If R is oo large, a filer divergence, in he saisical sense could be resuled. 82 AIJ - Elecrical & Elecronics Engineering, Vol. 48, No. 2, Fall 2016

3 M. Fahi, N. Ghahramani, A. Mohammadi, M. A. S. Ashiani, M. Fallah Besides, i will resul in a longer esimaion ransiion for he filer whose oal operaion ime is very shor (abou 1 o 3 minues for a long-range air defense missile fligh). These minue poins imply ha using a fixed filer designed by convenional mehods for an INS/Radar sysem wih he changing radar measuremens accuracies is a major drawback. From his poin of view, his sudy proposes ha he fixed esimaion formulaion for he INS/Radar inegraed sysem has resuled in a poor esimaion performance and should be replaced by an adapive esimaion filer wih a varying measuremen noise covariance marix. I can be expeced ha wih he adapive fusion scheme, a beer navigaion performance can be achieved. The paper is organized as follows. In secion II, he racking radar is inroduced and he measuremen errors of a specific racking radar are analyzed. Secion III oulines he residual-based adapive esimaion / he adapive fading Kalman filer hybrid mehod (RAE/AFKF) which is he main conribuion of his paper. In secion IV, he process and measuremen model for INS/Radar inegraion sysem are provided. In secion V, he performance of RAE/AFKF is evaluaed in Mone Carlo simulaions compared wih an exended Kalman filer, EKF. Finally, secion VI provides concluding remarks. 2- An assessmen of racking radar measuremen errors A ypical racking radar has a pencil beam o receive echoes from a single arge and racks he arge in angle, range, and/or doppler. Discussion in [15, chaper 4] and [17, chapers 8, 9] shows ha he several major effecs on he accuracy of he racking radar measuremens are as follows: Glin, or angle noise which affecs all he racking radars especially a a shor range. The error due o glin varies inversely wih he range. Receiver noise which also affecs all he radar ypes, and mainly deermines o rack accuracy a he long range. The receiver noise causes he error o vary as he square of he range. Ampliude flucuaions of he arge echo ha boher he conical-scan and sequenial lobing ype of radars, bu no he monopulse ype. The ampliude flucuaions effec is independen of he range. Cluer and mulipah effecs; Servo noise and servo lag of racking mechanism; he servo noise is independen of he range. The conribuion of glin, receiver noise and ampliude flucuaions o he accuracy of a racking radar as a funcion of range is illusraed in Fig. 1. Fig. 1 is a very qualiaive plo showing he general naure of each of hese facors, while precise invesigaions of racking errors associaed wih a paricular monopulse phased-array radar sysem and wih he radar AN/FPS-16 are discussed and presened in [18] and [14, page 11-12], [19, pages 48-94], respecively. Bu he specific operaing condiions assumed in [14,18,19], such as he arge radar cross secion, RCS, and aliude are no compaible wih wha his research needs. Thus, based on he free-space performance analysis and he missile arge RCS, sandard deviaion (STD) of he oal angle and range errors of he racking radar An/ FPS-16 as a funcion of range are roughly esimaed as illusraed in Figs. 2 and 3 (see appendix A for he radar equaions). Alhough his radar is one of he early monopulse radars, i remains in a wide use and represens one of he mos accurae racking devices employed in he es-range insrumenaion. Is major parameers are lised in [17, pages ]. 3- INS/Radar adapive inegraion sysem As menioned in secion I, due o he unknown range-varying saisical parameers of he radar measuremen noises, applying he EKF o INS/Radar inegraion resuls in a low-performance esimaion. In order o enhance is performance, his unknown and variable saisical parameer needs o be esimaed and adaped wih he sysem sae and he error covariance. Two popular ypes of he adapive Kalman filer algorihms include he innovaionbased adapive esimaion, IAE, approach and he Fig. 1. Qualiaive plo showing relaive conribuions o he angle racking error wihou unis [15, page 235] AIJ - Elecrical & Elecronics Engineering, Vol. 48, No. 2, Fall

4 Adapive Fusion of Inerial Navigaion Sysem and Tracking Radar Daa Fig. 2. A rough esimaion of he racking angle error STD for he AN/FPS-16 in he free-space analysis he hybrid approach IAE/AFKF developed in [23] were applied o INS/Radar inegraion problem as none of hem solely was able o saisfy he desired performance. Therefore, a new hybrid approach involving he concep of residual-based adapive esimaion, RAE, and AFKF is presened. The raio of he acual residual covariance based on he sampled sequence o he heoreical residual covariance will be employed for dynamically uning hree filer parameers, including fading facor (λ P ), measuremen noise scaling facor (λ R ) and process noise scaling facor (λ Q ). To provide hese facors, he Kalman filer approach is coupled wih he adapive uning sysem, ATS. In he ATS mechanism, he adapaions on he error saes covariance marix (P), on process covariance marix (Q) and on he measuremen noise covariance are involved. The idea is based on he concep ha when he filer achieves esimaion opimaliy, he acual residual covariance based on he sampled sequence and he heoreical residual covariance should be equal. Fig. 3. A rough esimaion of he racking range error STD for he AN/FPS-16 in free-space analysis adapive fading Kalman filer, AFKF, approach [20]. The innovaion sequences in IAE mehod have been uilized by he correlaion and covariance-maching echniques o esimae he noise covariance. The basic idea behind he covariance-maching approach is o make he acual value of he covariance of he innovaion consisen wih is heoreical value. The implemenaion of IAE-based adapive Kalman filer, IAKF, o navigaion designs has been widely explored in [16,21]. The idea of AFKF is o incorporae subopimal fading facors as a muliplier ino he prediced covariance marix o deliberaely increase he variance of he prediced sae vecor o enhance he influence of innovaion informaion and improve he racking capabiliy in high dynamic maneuvering [22]. In his sudy, he mehods IAKF, AFKF and 3-1- Inegraion algorihm As shown in Fig. 4, he posiion saes of INS, i.e. x, y, z, may be compared wih he same quaniies obained from a radar model. This model convers he radar measuremens of he radar Polar coordinae sysem ino he radar Caresian one using x=rcosθsinψ, y=rcosθsinψ and z= RsinΘ where R, Θ and Ψ are range, elevaion, and bearing, respecively. The differences beween he acual and prediced measuremens are he filer measuremens innovaions. These quaniies are muliplied by he Kalman gains o provide esimaes of he errors in he INS indicaed posiion, velociy, and aiude. Also, because of insufficien knowledge of he radar measuremen noise saisics, a residual-based adapive uning sysem effecively adaps R, Q and Fig. 4. INS/Radar Sysem Block Diagram 84 AIJ - Elecrical & Elecronics Engineering, Vol. 48, No. 2, Fall 2016

5 M. Fahi, N. Ghahramani, A. Mohammadi, M. A. S. Ashiani, M. Fallah P o compensae such lack of informaion. Finally, he esimaes of he errors are subsequenly used o correc he INS soluions over ime, and updae akes place following he arrival of each radar measuremen of he missile posiions ARAE/AFKF hybrid algorihm The process model and measuremen model are represened [24, chaper 4] as: (1) (2) where he sae vecor x k R n, process noise vecor w k R n, measuremen vecor z k R m, and measuremen noise vecor v k R m. In Eqs. (1) and (2), boh he vecors w k and v k are zero mean Gaussian whie sequences having zero cross-correlaion wih each oher: (3) (4) (5) where Q k is he process noise covariance marix, R k is he measuremen noise covariance marix, F is he sae ransiion marix, E [.] represens expecaion and superscrip T denoes marix ranspose. From he measured z k and he prediced measuremen z k+ =H k x k + based on he updaed filer saes x k+, he residuals sequence is defined as v k =z k z k+. Also, he heoreical covariance marix of he residual sequence, developed in [16], is given by: (6) Defining C vk as he saisical sample variance esimae of C vk, marix C vk can be compued by averaging inside a moving esimaion window of size N: (7) where N is he number of samples (usually referred o he window size); j 0 =k N+1 is he firs sample inside he esimaion window. The window size N is chosen empirically o give some saisical smoohing. The block diagram of RAE/AFKF hybrid mehod and furher deails regarding he residual-based adapive uning sysem loop are illusraed as a flow char in Fig. 5. where P k is he error covariance marix defined by E[(x k x ) (x x k k k )T ], in which x is an esimaion of he k Fig. 5. Block Diagram of RAE/AFKF for an Adapive Esimaion sysem sae vecor x k, and he weighing marix K k is generally referred o as he Kalman gain marix. Also, he value 0<α 1 is deermined empirically hrough compuer simulaion. 4- Mahemaical modeling In his secion, linear differenial equaions of he INS are derived and are presened as he process model. Also, he measuremen model based on he difference beween he radar and he INS measuremens is obained Process modeling The angen coordinae frame (-frame) is defined o be an earh-fixed which is aligned wih a geographic frame a he fixed locaion missile launcher on he earh. In his sysem, he navigaion equaions are [25, chaper 3]: (8) (9) (10) where ω i is he urn rae of he earh expressed in he AIJ - Elecrical & Elecronics Engineering, Vol. 48, No. 2, Fall

6 Adapive Fusion of Inerial Navigaion Sysem and Tracking Radar Daa -frame: (11) L f, Ω and f b represen laiude of he missile launcher locaion, earh s rae, and he specific force in he missile body frame, respecively. Also ω b b is he urn rae of he body wih respec o he -frame; Ω b b is he skew-symmeric form of ω bb, C b which is a direcion cosine marix o ransform he measured specific force vecor o navigaion axes (-frame) and g l represens he graviy field vecor. The linearized differenial Eqs. (8) o (10) are: Fig. 6. Radar, navigaion and body frames, background image couresy Rayheon company copyrigh 2002 (12) Kalman filer algorihm. The difference beween he radar and INS measuremens in he navigaion frame is expressed as: (17) (18) (13) (14) (15) where, he radar provides measuremens in he polar coordinae sysem, i.e. he measuremens of range, R, elevaion, Θ, and bearing, Ψ. Caresian quaniies may be expressed in erms of he polar coordinaes as follows: (16) where G and g n are he graviy gradien marix and he graviy vecor referenced in he local level geographic frame (n-frame), respecively. Alhough G may be negligible for a low cos or acical grade INS, in Appendix B a new model for G is derived. δr and δv are he INS posiion and velociy errors, and ϕ is he body-o-navigaion aiude error. ε is defined as he error vecor in he ransformaion beween he radar face and navigaion coordinae frames. The componens of ε are he misalignmen errors beween hem as shown in Fig. 6. Superscrips or subscrips, R and b denoe he angen, radar and body frames, respecively. w a and w g are acceleromeers and gyros noise vecor. Eq. (12) in discree-ime domain akes he form Eq. (1) using he error sae vecor expressed in componen form as x=[δr δv ϕ ε ] T Measuremen modeling The radar oupus are combined wih INS posiions o form a measuremen for processing in an onboard (19) Expanding Eq. (17) abou a nominal posiion r and neglecing producs of error quaniies yields: where (20) (21) (22) (23) v represens he error in he radar measuremens. v is assumed o be a zero-mean, Gaussian whie noise process wih he covariance marix R. C R is a known 86 AIJ - Elecrical & Elecronics Engineering, Vol. 48, No. 2, Fall 2016

7 M. Fahi, N. Ghahramani, A. Mohammadi, M. A. S. Ashiani, M. Fallah direcion cosine marix beween he radar frame and he navigaion axes (-frame). In his research, he marix C R is assumed ime-invarian due o he fixed ground base of he radar saion and he missile launcher. 5- Simulaion sudies In his secion, firsly he fligh rajecory needed for navigaion simulaion is inroduced. Then, he specificaions of inerial sensors and he racking radar are presened. Finally, he simulaion resuls of INS/Radar adapive inegraion navigaion sysem are displayed and discussed Fligh rajecory Fig. 7 illusraes he represenaive scenario used in he simulaion analyses. Over he period of fligh, he missile under proporional guidance laws follows a boos phase and inerceps a maneuvering arge. The missile s inegraion inerial navigaion sysem before second 4 is in he INS only mode and afer receiving he radar daa in he second 4 swiches o INS/Radar mode. Table 1. Inerial sensor and radar measuremen specificaions Specificaion Value (1σ) Acceleromeer In-Run Bias, mg 1 Acceleromeer Noise Densiy, µg / Hz 110 Gyro In-Run Bias, deg/hr 12 Gyro Noise Densiy, ( / s ) / Hz Radar Azimuh Measuremen Noise, deg Radar Elevaion Measuremen Noise, deg Radar Range Measuremen Noise, m Radar Face o Nav. Frame Misalignmen, deg ~ 0 Radar daa uplink rae, Hz Sensor specificaions The low- cos inerial sensors specificaions of INS are presened in Table 1. For he radar measuremens simulaion, we use he informaion given in Figs. 2 and 3. Bu, he design of an exended Kalman filer for he INS/Radar inegraion needs he fixed radar measuremens specificaions which are lised in Table 1. Also, owing o preliminary alignmen beween he ground-fixed radar frame and navigaion frame, i is supposed ha he misalignmen ε is negligible. Fig. 7. The missile rajecory 5-3- Simulaion resuls Time domain Mone Carlo, MC, and he simulaions of he INS/Radar inegraed navigaion sysem are carried ou o compare he performance of RAE/AFKF, EKF and a new definiion filer under he ile of ideal EKF, IEKF. In fac, he EKF and IEKF display more or less wo boundaries in he simulaion resuls for evaluaing he performance of RAE/AFKF. In he IEKF, unlike RAE/AFKF and EKF, i is supposed ha he rue radar measuremens accuracy in every range conforming o Figs. 2 and 3 is known and he measuremen noise covariance marix of he filer is improved in every second based on his exising and virual informaion. In oher words, he ideal EKF can be equal o an adapive EKF whose marix R is only adaped and is esimaion abou R is definiely precise, accurae and similar o he real radar measuremens accuracy, i.e. Figs. 2 and 3. Hence, he IEKF represens how good he navigaion performance can be if such excellen informaion abou he accuracies of he radar measuremens is esimaed and provided for he EKF disregard of he esimaion mehod. Therefore, his virual filer highlighs he capaciy of adapive filers in he performance enhancemen of INS/Radar inegraion AIJ - Elecrical & Elecronics Engineering, Vol. 48, No. 2, Fall

8 Adapive Fusion of Inerial Navigaion Sysem and Tracking Radar Daa navigaion sysem. The simulaion resuls described below, show he sandard deviaions (STD) of he aiude and posiion errors. Each se of resuls has been obained from a bach of 100 MC simulaions. Figs. 8 o 10 illusrae he sandard deviaion of in-fligh alignmen errors. These resuls explain ha he RAE/AFKF has he superior performance in comparison wih he EKF in accuracy and convergence rae. If here is he aiude error requiremen of 0.2 degree (1σ), RAE/AFKF relaively saisfies i, bu EKF fails o mee his supposed requiremen. As shown in Figs. 8 o 10, he convergence speed of he heading and pich alignmen error is faser han hose of bank alignmen error. These differences are due o he dissimilar observabiliy of hese errors in daa fusion process. As shown in Figs. 10, near he end of fligh ime, he error curves approach nearly o he same error value. This approach is due o he acceleraions profile of he inercepor missile. Fig. 8. STD of bank aiude error based on MC simulaion Fig. 9. STD of pich aiude error based on MC simulaion: a) all daa and b) zoomed porion Fig. 10. STD of heading error based on MC simulaion: a) all daa and b) zoomed porion 88 AIJ - Elecrical & Elecronics Engineering, Vol. 48, No. 2, Fall 2016

9 M. Fahi, N. Ghahramani, A. Mohammadi, M. A. S. Ashiani, M. Fallah The sandard deviaion of posiion errors is presened in Figs. 11 o 13. These resuls show ha he RAE/AFKF in comparison wih he EKF considerably provides more accurae navigaion. Figs. 8 o 13 indicae ha unil nearly second 50, he performance of IEKF in accuracy and convergence rae is beer han ha of RAE/AFKF. I means ha he accuracy of R-adapaion in he RAE/AFKF is no sufficien. However, afer second 50, he navigaion and alignmen accuracy relevan o RAE/ AFKF represen a noiceable preference over IEKF. I implies ha he R-adapaion in an adapive INS/ Radar inegraed navigaion sysem is necessary for he navigaion and alignmen performance improvemen agains he radar measuremens uncerainy, i bu is no sufficien. From he comparison of EKF and ideal EKF resuls in Figs. 8 o 13, i can be found ha a proper R adapaion in an adapive INS/Radar inegraed Fig. 11. STD of X posiion error based on MC simulaion Fig. 13. STD of Z posiion error based on MC simulaion navigaion sysem offers an effecive ool o compensae for our insufficien prior informaion abou he radar measuremens accuracies and enhance he navigaion and in-fligh alignmen performances as a resul of he compensaion. Neverheless, in spie of he rue R adapaion in IEKF, he posiion error of navigaion sysem using IEKF in final seconds is more han ha of EKF as seen in Fig. 12. I denoes ha he navigaion error can be improved by he addiion of he oher adapion like P adapion or Q adapion. 6- Conclusions In his paper, a new hybrid adapive filer based on wo conceps of residual-based adapive esimaion and adapive fading Kalman filer is proposed for he inerial navigaion sysem/radio deecion and ranging (INS/Radar) inegraed navigaion sysem. The Mone Carlo simulaion resuls show ha he presened algorihm is effecive enough o improve he in-fligh alignmen and navigaion performance in he face of he unknown and highly range-dependen accuracy of he racking radar measuremens. The proposed scheme is superior compared o he exended Kalman Filer and offers a possibiliy ha he in-fligh alignmen may be achieved wihin a relaively shor period of ime. Also, his new adapive filer pus forward a capaciy ha he lowcos INS/Radar inegraion sysem wihou aiding global navigaion saellie sysem signals (e.g. Global Posiioning Sysem or GPS) can be used hroughou he inercepor missile s mission. Fig. 12. STD of Y posiion error based on MC simulaion 7- Appendix A Thermal noise errors of he radar angle and range AIJ - Elecrical & Elecronics Engineering, Vol. 48, No. 2, Fall

10 Adapive Fusion of Inerial Navigaion Sysem and Tracking Radar Daa measuremens are random variables wih he sandard deviaions given by respecively in degrees and meers [14, chaper 10], [15, chapers 4, 6]: (A.1) (A.2) where he free-space, and single-pulse signal-o-noise raios (S/N) are deermined using he radar range equaions [14, chaper 3], [17, chaper 1]: (A.3) The basic and equivalen glin error sandard deviaions in angle and range for uniform scaered over a arge span L are given by σ θg =L x /3R and σ RG =L r /3, respecively in degrees and meers [17, pages ]. 8- Appendix B The graviy vecor in -frame in erms of he graviy vecor in he local level geographic frame, n-frame, is: (B.1) where (B.2) C n is a direcion cosine marix from n-frame o -frame. The marix C n has been explained in deail in [26, page 37]. By differeniaing (B.1) respec o r, he graviy gradien marix G is obained as where (B.3) (B.4) and for he acical missiles, i.e. regional range, i can be supposed ha Applying (B.4) o (B.3) gives: (B.5) (B.6) 9- References [1] Tieron, D. H. and Weson, J. L.; The Alignmen of Ship Launched Missile IN Sysems, Inerial Navigaion Sensor Developmen, IEE Colloquium on, London, [2] Johnson, C.; Ohlmeyer, E. J. and Pepione, T. R.; Aiude Diluion of Precision A New Meric for Observabiliy of in Fligh Alignmen Errors, Guidance, Navigaion, and Conrol Conference and Exhibi, Denver, pp , [3] Ornedo, R. S.; Farnsworh, K. A. and Sandhoo, G. S.; GPS and Radar Aided Inerial Navigaion Sysem for Missile Sysem Applicaions, IEEE Posiion Locaion and Navigaion Symposium, Palm Spring, CA, [4] Ohlmeyer, E. J.; Pepione, T. R. and Hanger, D. B.; Effec of Trajecory Shaping on Observabiliy of NTW Inercepor In-Fligh Alignmen Errors, Dahlgren VA, Naval Surface Warfare Cener, [5] Ohlmeyer, E. J.; Hanger, D. B. and Pepione, T. R.; In-Fligh Alignmen Techniques for Navy Theaer Wide Missiles, Guidance, Navigaion and Conrol Conference, Monreal, [6] Bezick, S. M.; Pue, A. J. and Pazel, C. M.; Inerial Navigaion for Guided Missile Sysems, Johns Hopkins APL Technical Diges, Applied Physics Laboraory, Vol. 28, pp , [7] Bar-Izhack, I. Y. and Pora, B.; Azimuh Observabiliy Enhancemen During Inerial Navigaion Sysem In-Fligh Alignmen, Journal of Guidance, Conrol and Dynamics, Vol. 3, No. 4, pp , [8] Pora, B. and Bar-Izhack, I. Y.; Effec of Acceleraion Swiching During INS In-Fligh Alignmen, Journal of Guidance, Conrol and Dynamics, Vol. 4, No. 4, pp , [9] Bar-Izhack, I. Y.; Minimal Order Time Sharing Filers for INS In-Fligh Alignmen, Journal of Guidance, Conrol and Dynamics, Vol. 5, No. 4, pp , [10] Anders, J. L.; Johnson, C.; Luckau, A. M.; Moore, T. A. and Ornedo, R. S.; Successful Fligh Tes of a GPS and Radar Aided Inerial Navigaion Sysem, Proceedings of he Naional Technical Meeing of The Insiue of Navigaion, San Diego, CA, AIJ - Elecrical & Elecronics Engineering, Vol. 48, No. 2, Fall 2016

11 M. Fahi, N. Ghahramani, A. Mohammadi, M. A. S. Ashiani, M. Fallah [11] Anders, J. L.; Buhar, C.; Esrada, V.; Johnson, C. and Ornedo, R. S.; New Generaion GPS and Radar Aided Inerial Navigaion Sysem for Ballisic Missile Inercepor, Procedings of he 60 h Annual Meeing of The Insiue of Navigaion, Dayon OH, [12] Gul, F.; Fang, J. and Khan, S.; SINS Augmenaion by ANS and Secure Radio Posiioning Sysem, 2 nd Inernaional Conference on Emerging Technologies (ICET), pp , [13] Li, S. and Peng, Y.; Radio Beacons/IMU Inegraed Navigaion for Mars Enry, Advances in Space Research, Vol. 47, No. 7, pp , [14] Baron, D. K.; Radar Evaluaion Handbook, Arech House, Norwood MA, [15] Skolnik, M. I.; Inroducion o Radar Sysems, McGraw Hill, New York, 3 rd Ediion, [16] Mohamed, A. H. and Schwarz, K. P.; Adapive Kalman Filering for INS/GPS, J. Geodesy, Vol. 73, No. 4, pp [17] Baron, D. K.; Modern Radar Sysem Analysis, Arech House, Norwood, MA, [18] Ewell, G. W.; Alexander, N. T. and Tomberlin, E. L.; Invesigaion of Targe Tracking Errors in Monopulse Radars, Georgia Insiue of Technology, Alana, Georgia, [19] Baron, D. K. and Baron, W. F.; Modern Radar Sysem Analysis Sofware and User s Manual Version 2.0, Arech House, Norwood MA, [20] Jwo, D. J.; Chung, F. C. and Weng, T. P.; Adapive Kalman Filer for Navigaion Sensor Fusion, Sensor Fusion and is Applicaions, Shanghai, InTech, pp , [21] Hide, C.; Moore, T. and Smih, M.; Adapive Kalman Filering for Low-Cos INS/GPS, The Journal of Navigaion, Vol. 56, Vol. 1, pp , [22] Xia, Q.; Rao, M.; Ying, Y. and Shen, X.; Adapive Fading Kalman Filer wih an Applicaion, Auomaica, Vol. 30, No. 8, pp , [23] Jwo, D. J. and Weng, T. P.; An Adapive Sensor Fusion Mehod wih Applicaions in Inegraed Navigaion, The Journal of Navigaion, Vol. 61, No. 4, pp , [24] Simon, D.l Opimal Sae Esimaion, John Wiley and Sons, Hoboken, New Jersey, [25] Tieron, D. H. and Weson, J. L.; Srapdown Inerial Navigaion Technology, Peer Peregrinus, London, [26] Briing, K. R.; Inerial Navigaion Sysems Analysis, John Wiley and Sons Inc., USA, AIJ - Elecrical & Elecronics Engineering, Vol. 48, No. 2, Fall