Attitude Estimation of A Rocking Ship with The Angle of Arrival Measurements Using Beacons

Transcription

1 IOSR Journal of VLSI and Signal Processing (IOSR-JVSP) Volume 6, Issue 5, Ver. I (Sep. - Oc. 2016), PP e-issn: , p-issn No. : Aiude Esimaion of A Rocing Ship wih he Angle of Arrival Measuremens Using Beacons Kyungwoo Yoo 1, Joohwan hun 1 and Seungoh Yoo 2 1 (Deparmen of Elecrical Engineering, KAIS, Korea) 2 (Agency for Defense Developmen, Korea) Asrac: o improve he accuracy of navigaion soluions such as posiion, velociy and aiude, various navigaion echniques have een developed. For example, he gloal posiioning sysem (GPS) and inerial navigaion sysem (INS) have een used in general navigaion sysems. Nowadays, o provide more accurae navigaion soluion, exended Kalman filer (EKF) ased GPS/INS coupled sysems, which calirae he navigaion soluion of INS y using he GPS, are also widely used for miliary sysems. However, wih he ordinary GPS/INS coupled sysems he aiude canno e caliraed, since he GPS is ale o calirae posiion and velociy only, and his feaure can e vulnerale o cerain applicaions aiude informaion is needed. o overcome his drawac, an angle-ased aiude esimaion echnique using eacons is proposed in his paper. he proposed echnique uses he angle of arrival (AOA) measuremens of he received signals using an array anenna, and esimaes he own-ship aiude. he proposed echnique is also coupled wih INS and he performance of he proposed echnique is demonsraed hrough simulaion. Keywords - aiude esimaion, eacon, exended Kalman filer (EKF), INS/GPS coupled sysem I. Inroducion he inerial navigaion sysem (INS) and gloal posiioning sysem (GPS) are widely used independenly as general posiion esimaion echniques in navigaion. However, coupled navigaion sysems such as GPS/INS coupled sysems have een also used o compensae he drawac of each; INS diverges maredly as ime goes on and GPS only canno improve he precision over cerain level [1], [2]. While GPS is ale o deermine he posiion and velociy of he own-ship, aiude canno e deermined direcly using GPS. his limiaion canno e a prolem when he posiion and velociy esimaion is our final need. However, if we need susequen conrol acions such as firing hruser or weapons, accurae own-ship aiude informaion is essenial. If we rely only on gyroscopes and acceleromeers, hen he resuling aiude esimae would e unreliale especially for rocing small ships. In his paper, a eacon-ased posiion and aiude algorihm is proposed. We assume ha he array anenna is insalled a he sea carriage and using his array anenna, Angle of Arrival (AOA) of he signals ransmied y he several eacons and disance eween array anenna and several eacons can e measured [3]. Using he disance measuremen, he posiion esimae can e oained hrough he riangulaion mehod and in urn he aiude ecomes calculaed y using AOA of he signals ased on he esimaed posiion. he echnique is descried in 3D, in addiion, he INS/eacon coupled sysem ased on exended Kalman filer (EKF) is also suggesed. o evaluae he performance of he proposed echnique, Mone-arlo simulaion is performed in erms of accuracy of AOA informaion, he numer of eacons, disance eween sea carriage and eacons and so on. Simulaion resuls show he proposed echnique esimaes he posiion and aiude well. Also, he performance of he INS/eacon coupled sysem is also compared wih convenional echniques. he res of his paper has four secions. In secion II, an AOA-ased aiude esimaion echnique using eacons is proposed in 3D. An INS/eacon coupled sysem is descried in secion III. o evaluae he performance of he proposed echnique, several simulaion resuls are given in secion IV. Finally, secion V concludes his paper. II. Angle-Based Aiude Esimaion o define he own-ship aiude, defions of coordinae frames should e preceded. ale 1 descries four coordinae frames ha are widely used in navigaion applicaions [4]. ale 1: oordinae Frames Frame Axis origin x-axis y-axis z-axis Inerial frame Earh cener Vernal equinox Righ-hand rule Axis of earh roaion DOI: / Page

2 Aiude Esimaion of A Rocing Ship wih he Angle of Arrival Measuremens Using Beacons (i-frame) Earh frame (e-frame) Navigaion frame (n-frame) Body frame (-frame) Earh cener Inersecion eween equaorial plane Righ-hand rule Axis of earh roaion and Greenwich meridian plane user Norhward direcion Easward direcion Down direcion ener of graviy of he user Forward direcion ransverse direcion Righ-hand rule In his paper, we assume ha he origin of he n-frame and ha of he -frame are he same and define he angles roll ( ), pich ( ) and yaw ( ) which are he Euler angles wih respec o he x-axis, y-axis and z- axis from he n-frame o he -frame as shown Fig. 1. he direcion cosine marix (DM) Euler angles: (1) n x y z Fig.1 Defions of roll, pich and yaw. n which ransforms n-frame o -frame can e wrien in erms of cos 0 sin cos sin 0 0 cos sin sin cos 0 0 sin cos sin 0 cos a is he roaion marix which roaes counerclocwise y angle wih respec o he a - axis. As descried aove, our aiude parameers are relaed o he n-frame and -frame. herefore, he localizaion of he own-ship is essenial o oain he aiude parameers, since he n-frame and he -frame can e decided afer posiioning. In his regards, we may calculae he own-ship posiion y rilaeraion [5], using he range measuremens eween he own-ship and eacons. Such rilaeraion echnique is well-nown and is omied in his paper. Afer he posiion of he own-ship is deermined, he aiude esimaion may e carried ou. Noe ha he AOAs of he eacons are measured in he -frame. herefore, in he sequel i, X i, X u, El,i and Az, i will denoe respecively, he index of eacon ( i 1, 2,..., N ), he posiion of he ih eacon, he own-ship (user) posiion, he elevaion angle of he ih eacon and he azimuh angle of he ih eacon. Since he own-ship is locaed a he origin of he -frame, he coordinae of he own-ship is se o 0, 0, 0. If we denoe he disance eween ih eacon and he own-ship r i, he posiion of ih eacon can e expressed as (2). i i El. i Az. i i El. i Az. i i El. i X r cos cos, r cos sin, r sin (2) Since he n-frame is fixed when he own-ship posiion is fixed, he coordinae of he ih eacon in he n- frame is also fixed. hen, using he DM n, he relaionship eween X i and X ni, which is he coordinae of he ih eacon in he n-frame can e expressed as follows: X X (3) i n ni In equaion (3), we wan o deermine he Euler angles, and using measuremen priori informaion). Susiuing (1) for n in (3), we can rewrie (3) as (4). X ( X is a- i ni DOI: / Page

3 X Aiude Esimaion of A Rocing Ship wih he Angle of Arrival Measuremens Using Beacons x x z z i ni i yi n Xni yni i ni 11 xni 12 yni 13 zni x y z g 31xni 32 yni 33z ni coscos ni 22 ni 23 ni i cossin sin sin sin cos cos sin,, sin sin sin cos cos. (5) sincos cos sin cos sin sin cos sin sin sin cos coscos For N eacons, equaion (4) can e exended simply as: X1 g1,, (6) X N gn,, g,, is nonlinear funcion wih respec o, and. i o solve he aove nonlinear equaion using he Newon-Raphson mehod, we may rewrie (6) as a zero-prolem (ZP): f1,, X1 g1,, 0 (7) fn,, X N gn,, f1,, In (7), he soluion of 0 ecomes he esimaed aiude. If we se he ial value of, and fn,,, which are used in Newon-Raphson mehod as,,, respecively, he linearized (7) ecomes (8) f 0 f 1,,,, N f1,, A fn,, f1,, f1,, f1,, fn,, fn,, fn,,,, (4) (8) DOI: / Page

4 A is A Aiude Esimaion of A Rocing Ship wih he Angle of Arrival Measuremens Using Beacons f1,, f1,, f1,, (9) fn,, fn,, fn,, herefore, he updaed, and ecome,, f1,, 1 A A A (10) fn,, Rewrie he ime index as n 1,...,, 1,... K, 1 f1,, 1 1 A A A (11) 1 fn,, As he recursion progresses, he aiude esimae ecomes more accurae. III. INS/Beacon oupled Sysem In general, pure INS updaes he aiude, velociy and posiion of he own-ship y using acceleraion and angular velociy measured wih acceleromeers and gyroscopes, respecively [4]. However, due o characerisics of INS, navigaion soluion oained from pure INS ends o diverge wih he passage of ime. o overcome he weaness of pure INS, EKF-ased coupled sysems, which calirae navigaion soluion such as he INS/GPS navigaion sysem has een used [1], [2]. In his paper, an INS/eacon coupled sysem is proposed. Since he difference eween INS/GPS and INS/Beacon coupled sysem is measuremen equaion, we shall focus on he measuremen equaion only, mosly neglecing he dynamic equaion. 3.1 Dynamic Equaion ypically, he dynamic equaion of he EKF is ased on INS error model derived from pure INS equaion [6]. x F x G w (12) F, x, x and w are, respecively he dynamic marix, error sae variales, differeniaion of error sae variales and whie Gaussian noise which follows w ~ N 0, Q in ime. he error sae variale is expressed as: x Llh v n (13) Llh, vn,, and indicae respecively, he errors of laiude, longiude, heigh, he velociy errors wih respec o n-frame, he aiude errors, and he ias errors of acceleromeer and gyroscope, wih respec o he -frame. he whie Gaussian w models he random ias errors and. he marices F and G are given in [4] Measuremen Equaion he measuremen equaion can e expressed as follows [6]: z Hx (14) represens whie Gaussian measuremen noise disriued as ~ N 0, R DOI: / Page. In secion II, o calirae aiude, he AOA measuremens from eacon signals are used. However, as deriving equaions o oain aiude, he measuremen equaion a ime can e wrien as follows. f1,, z A (15) fn,,

5 Aiude Esimaion of A Rocing Ship wih he Angle of Arrival Measuremens Using Beacons If we rewrie (15) in erms of he original full error sae variales, hen 033 Llh f1,, 0 33 vn z A fn,, H x 33 (16) Now, y carrying ou he ime-updae and he measuremen-updae in (17), he errors of he pure INS can e correced [6]: xˆ f xˆ P P G QG K P H H P H R xˆ xˆ K z H xˆ P I K H P f is he nonlinear dynamic equaion and mean and covariance marix of he error sae variales, and covariance marix. F 1 1. Also, xˆ e xˆ and P P and (17) are ime-updaed are measuremen-updaed mean and IV. Simulaion resuls his secion illusraes he performance of he AOA-ased aiude esimaion algorihm and he INS/eacon coupled sysem via Mone arlo simulaion. 4.1.AOA-ased aiude esimaion algorihm he performance of he AOA-ased aiude esimaion algorihm is evaluaed wih respec o he error of measured AOA (azimuh and elevaion) and he numer of eacons. he disance eween user and each of he eacons is se o 1000m. he errors of measured azimuh and elevaion range from o 10 and he numer of eacons ranges from 2 o 10. he ial aiude (roll, pich, yaw) used Newon Raphson mehod is uniformly o o disriued in [0,10 ]. he simulaion is carried ou using 1000 rials for each AOA errors and he numer of eacons. Fig. 2. shows he sandard deviaion of aiude errors wih respec o he AOA error when he numer of eacons is wo. his resul indicaes ha aiude errors is proporional o AOA error. Also, he errors of roll, pich and yaw have similar paerns. Fig. 2. Aiude error wih respec o AOA error (Numer of eacons = 2) DOI: / Page

6 Aiude Esimaion of A Rocing Ship wih he Angle of Arrival Measuremens Using Beacons he sandard deviaion of he aiude errors wih respec o he numer of eacons is illusraed in he case of roll in Fig. 3. Noe ha he performance of he AOA-ased aiude esimaion algorihm is proporional o he numer of eacons, as he measuremens increase when he numer of eacons increases. Also, he improvemen of performance ecomes converge wih respec o increase of he numer of eacons. 1 of aiude error [deg] AoA error : 10deg AoA error : 1deg AoA error : 0.1deg AoA error : 0.01deg AoA error : 0.001deg he numer of eacons Fig. 3. Aiude errors wih respec o he numer of eacons INS/Beacon coupled sysem he aiude esimaion performances of he INS/GPS coupled sysem and he INS/Beacon coupled sysem are compared wih a simple scenario. he sandard deviaions of acceleraion, angular velociy measured from inerial measuremen uni (IMU), GPS posiion error and AOA measured from eacon signals are se o 50 g, 1/hour, 1m and 1, respecively. All sensors have he same daa rae, 100Hz. he scenario lased 100 seconds wih no flucuaion and movemen. he ial alignmen which aligns ial aiude efore saring navigaion coninues for 10 seconds, and INS/Beacon and INS/GPS coupled sysem are carried ou for 90 seconds. he Fig. 4 shows he simulaion resuls of oh coupled sysems. Fig. 4. Aiude esimaion performance of INS/GPS and INS/eacon coupled sysems DOI: / Page

7 Aiude Esimaion of A Rocing Ship wih he Angle of Arrival Measuremens Using Beacons Fig. 4 illusraes he simulaion resuls for roll, pich and yaw. Noe ha he performance of he INS/eacon coupled sysem is superior o ha of he INS/GPS coupled sysem in all cases. o verify he performance of he INS/eacon coupled sysem in deail, Fig. 5 illusraes simulaion resuls relaed o INS/eacon coupled sysem. As depiced in Fig. 5, esimaed yaw has more errors compared o roll and pich. Also he esimaed aiude has flucuaion even if hese values are exremely small. hese may e caused y he measuremen equaion which has correlaion eween measuremen and sae variales when he measuremen is oained and could ecome worse as ime progresses. Fig. 5. Aiude esimaion performance of he INS/eacon coupled sysem. V. onclusion In his wor, an AOA-ased aiude esimaion echnique using eacon and INS/eacon coupled sysem are proposed o overcome he drawac of he convenional navigaion sysems such as INS or INS/GPS coupled sysem. Firs, an AOA-ased aiude esimaion echnique using he Newon Raphson mehod is derived. hen, ased on his mehod, an INS/eacon coupled sysem is proposed. he performance of he proposed echniques are assessed y Mone arlo simulaion for a simple scenario, comparing hem wih ha of he convenional sysem. Alhough he proposed echnique gives eer performances compared o he convenional sysem, furher refinemen of he EKF model would e needed o increase he reliailiy of proposed echnique. References [1] J. H. Kim, An Implemenaion of a Loosely-coupled GPS/INS Inegraion Sysem, M.S. hesis, EE., hungnam naional universiy, Daejeon, Korea, [2] L. J. Wang, H. h. Zhao, and X. N. Yang, he modeling and simulaion for GPS/INS inegraed navigaion sysem, Inernaional conf. microwave and millimeer wave echnology, Apr. 2008, pp [3] E. Mosca, Angle esimaion in ampliude comparison monopulse sysem, IEEE rans. Aerosp. Elecron. Sys., vol. AES-5, pp , Mar [4] D. H. ieron and J. L. Weson, Srapdown Inerial Navigaion echnology. Sevenage, U.K.: Peregrinus, [5] J.B. sui, Fundamenals of Gloal Posiioning Sysem Receivers: A Sofware Approach. NY: John Wiley & Sons, [6] D. Simon, Opimal Sae Esimaion, 1s ed. NY: Wiley Inerscience, DOI: / Page