Parameter Estimation for Multipath Error in GPS Dual Frequency Carrier Phase Measurements Using Unscented Kalman Filters

Transcription

1 388 Eunsung Internatonal Lee Sebum Journal Chun of Young Control Jae utomaton Lee easam and Kang Systems Gyu-In vol. Jee 5 no. and 4 Jeongrae pp Km ugust 27 Parameter Estmaton for Multpath Error n GPS Dual Frequency Carrer Phase Measurements Usng Unscented Kalman Flters Eunsung Lee Sebum Chun Young Jae Lee* easam Kang Gyu-In Jee and Jeongrae Km bstract: hs paper descrbes a multpath estmaton method for Global Postonng System (GPS) dual frequency carrer phase measurements. Multpath s a major error source n hgh precson GPS applcatons.e. carrer phase measurements for precse postonng and atttude determnatons. In order to estmate and remove multpath at carrer phase measurements an array GPS antenna system has been used. he known geometry between the antennas s used to estmate multpath parameters. Dual frequency carrer phase measurements ncrease the redundancy of measurements so t can reduce the number of antennas. he unscented Kalman flter (UKF) s recently appled to many areas to overcome some of the lmtatons of the extended Kalman flter (EKF) such as weakness to severe nonlnearty. hs paper uses the UKF for estmatng multpath parameters. seres of smulatons were performed wth GPS antenna arrays located on a straght lne wth one reflector. he geometry nformaton of the antenna array reduces the number of estmated multpath parameters from four to three. Both the EKF and the UKF are used as estmaton algorthms and the results of the EKF and the UKF are compared. When the ntal parameters are far from true parameters the UKF shows better performance than the EKF. Keywords: rray antenna carrer phase measurement GPS multpath error unscented Kalman flter.. INRODUCION Multpath means that one or more reflected sgnals reach the antenna n addton to the drect sgnal. Under partcular crcumstances only the reflected sgnal may reach the antenna. Multpath s the major error source n hgh precson Global Postonng System (GPS) statc and knematc postonng. hs paper descrbes the development of a multpath mtgaton method for GPS carrer phase Manuscrpt receved January 2 27; accepted May Recommended by Edtor Jae Weon Cho. hs work was supported by the Korea Research Foundaton Grant (KRF D248) funded by the Korea Government (MOEHRD). Eunsung Lee s wth the Satellte Navgaton System Department the Korea erospace Research Insttute Korea (e-mal: koreagnss@kar.re.kr). Sebum Chun Young Jae Lee and easam Kang are wth the Department of erospace Engneerng Konkuk Unversty Korea (e-mals: tucano@tucanokorea.com {younglee Gyu-In Jee s wth the Department of Electronc Engneerng Konkuk Unversty Korea (e-mal: gjee@ konkuk.ac.kr). Jeongrae Km s wth the School of erospace and Mechancal Engneerng Hankuk vaton Unversty Korea (e-mal: jrkm@hau.ac.kr). * Correspondng author. measurements. Most mtgaton methods are very effectve for the multpath of code measurements but t s dffcult for the carrer phase measurements. Varous methods are supposed to mtgate the multpath of carrer phase measurements but there are no dstngushed methods. he multpath mtgaton methods for GPS measurements are to be dvded nto three types. () Improvng antenna gans (2) New GPS recever technology and (3) Post data processng. () In order to mprove antenna gan pattern several methods exsts. here s the usage of specal antennas spatal processng wth mult-antenna arrays proper antenna locaton strateges and long-term sgnal observaton. (2) seres of new GPS recever technology help to mtgate the multpath effect. Narrow Correlator has a. chp spacng and a larger bandwdth at the IF (Intermedate Frequency) and provdes good long delay multpath mtgaton []. ME (Multpath Elmnaton echnque) s an mprovement of Narrow Correlator [2]. MEDLL (Multpath Estmaton Delay Lock Loop) utlzes multple narrow-spaced correlators to estmate the multpath and remove t from the correlaton functon to provde a more pure sgnal correlaton functon [3]. ll the technologes can not be put together because the GPS recever technologes

2 Parameter Estmaton for Multpath Error n GPS Dual Frequency Carrer Phase Measurements Usng Unscented 389 belong to an ndependent GPS recever s company and the hardware and software of GPS recevers cannot be changed by users. (3) he post data processng ncludes the smoothng algorthm that combnes GPS code measurements wth carrer phase measurements to mtgate the multpath error n the code measurements [4]. nother method s usng Sgnal-to-Nose Rato (SNR) to correct the multpath error n dfferental phase measurements. hs method nvolves the antenna gan pattern. Modelng reflector parameters of the multpath [5] wth SNR s also used. Multple references are used to estmate the multpath error of the code and carrer phase measurements [6]. Wth the known baselne length the multpath effect can be estmated and can be dvded nto three types. he frst one s to use the baselne-length constrant to reduce the search volume of GPS carrer phase nteger ambgutes [6]. he second one s to smplfy relatve equatons wth the baselne geometry [7]. he last one s to reduce the number of the possble ambguty soluton canddates wth the relatonshp of each ambguty [8]. he frst and second methods are usually used for atttude determnaton whle the thrd one s used for postonng. If the known baselne lengths are used the mnmum number of satelltes should be three when conventonal GPS recevers are used for atttude determnaton [3]. In order to estmate and remove the multpath n carrer phase measurements a specal array GPS antenna system has been proposed n ths paper. he known geometry between the antennas s used to estmate multpath parameters. he array antenna system conssts of 3 antennas and recevers whch can receve dual frequency carrer phase measurements. ll recevers are derved by common external clock. Dual frequency carrer phase measurements ncrease the redundancy of measurements so t can reduce the number of antennas. he extended Kalman flter (EKF) and the unscented Kalman flter (UKF) are used to estmate multpath parameters and the results from two flters are compared. Recently developed the UKF overcomes some weak ponts of the EKF and mproves the performance of estmatng multpath parameters. In the secton Carrer Phase Multpath Modelng a bref revew of carrer phase multpath modelng and the mathematcal model of GPS are dscussed. In the secton rray ntenna System the method of reducng the multpath parameters s explaned. In the secton Unscented Kalman Flter the drawback of the EKF and the advantage of the UKF are represented. In the secton Smulaton Results the results of a smulaton are explaned. he fnal conclusons are summarzed. 2. CRRIER PHSE MULIPH MODELING Multpath s a phenomenon whch affects most of the rado navgaton systems and GPS s not an excepton. Due to the reflecton of nearby objects such as buldngs vehcles the ground and water surfaces the sgnal that s receved by the a GPS antenna s composed of the lne of sght sgnal drectly from the satellte and sgnals reflected by nearby objects called reflectors. he magntude and the phase of reflected sgnals depend on the surface property of the reflectors and on the geometry between the GPS satellte recever antenna and reflectors. hus the multpath affects the carrer phase observaton as well as envronment condtons. fter the GPS code sgnal has been removed n the phase-lock loop of GPS recevers the GPS L sgnal can be reconstructed as follows: where 2 f ( ) ( ω ) S t = cos t+θ () π p p s the ampltude of a drect sgnal ω s f s the frequency of L carrer phase measurements and Θ s the phase delay of a drect sgnal [9]. When the reflected sgnals are receved by a GPS antenna together wth the drect sgnal the composte sgnal can be wrtten as ( ) = p cos( ω +Θ) m + α ( ωt+θ+ γ ) S t t cos p k k k= (2) where αk s the reflecton rato over the drect sgnal ampltude at k th reflected sgnal k s a reflector ndex γ k s the phase delay of a reflect sgnal and m s the total number of reflectors. Equaton (2) can be smplfed as ( ) β ( ω ) S t = cos t +Θ+Ψ (3) p where β s the factor sgnal ampltude change Ψ s the phase error due to multpath. β and Ψ can be wrtten as m 2 m 2 k k k k k= k= m αk snγk k= m + αk cosγk k= β = + α cosγ + α sn γ Ψ= arctan.

3 39 Eunsung Lee Sebum Chun Young Jae Lee easam Kang Gyu-In Jee and Jeongrae Km he equaton of GPS carrer phase measurements s descrbed as follows: R don dtrop N δ Φ = δ + Ψ + w (4) where Φ s the carrer phase measurement s a satellte number (superscrpt) s a recever number (subscrpt) R s the true range from satellte to recever don s a onospherc error d trop s a tropospherc error s a carrer phase wave length N s a nteger ambguty δ δ s a satellte clock Ψ s phase error error s a recever clock error due to multpath w s a measurements nose []. he phase error n (4) can be wrtten for a sngle reflector case as αsnγ Ψ = arctan αcosγ + (5) where γ s the phase delay of a reflected sgnal also a t can be wrtten as γ = and a s the dstance delay of a reflected sgnal []. 3. RRY NENN SYSEM In order to estmate the multpath error parameters n carrer phase measurements an array GPS antenna system s used. he known geometry between the antennas s the useful nformaton to estmate the multpath error parameters. If the dual frequency recevers are used the array antenna system usng 3 antennas wth the same clock helps estmate the multpath error parameters effectvely. If the dstance between antennas s a half wavelength and the antennas are located on a straght lne t s easy to fx the ambguty nteger and the multpath error parameters of GPS carrer measurements can be estmated drectly. Let us assume two GPS antennas are fxed as Fg.. he dstance between the antenna and B s half of the GPS carrer phase sgnal wavelength. he dfferenced measurements equaton between the antenna and B s descrbed as B B Φ =Φ Φ RB + don + dtrop + N B = + δ + δb + Ψ B + wb (6) Fg.. Confguraton of two antennas. R + d + d + N + δ + δ + Ψ + w on trop Snce the dstance between the two antennas s short the common errors.e. onospherc tropospherc satellte ephemers and satellte clock from the same satellte can be cancelled out. Equaton (6) can be rearranged as follows: Φ B = RB + NB + δb + Ψ B + w B (7) where R B s the dfference of ranges N B s the dfference of nteger ambgutes δ B s the dfference of recever clock errors Ψ B s the dfference of phase errors w B s the dfference of measurements noses. When the two GPS recevers are drven by a common external clock the term δ B can be zero. Equaton (7) becomes Φ B = RB + NB + Ψ B + w B (8) where Ψ B s descrbed as B B Ψ =Ψ Ψ. αsnγb αsnγ 2 + α sn ( γb γ) = arctan. + αcosγb αcosγ 2 + α cos( γb γ) (9) Snce the relatve dstance between the two antennas s constant t s possble to calculate N B. Equaton (8) can be arranged as follows: ( B) B wb Φ = Ψ + () MP

4 Parameter Estmaton for Multpath Error n GPS Dual Frequency Carrer Phase Measurements Usng Unscented 39 Fg. 2. Relatonshp of the phase delay at antennas and B. where ( B ) MP B B B Φ s Φ R N. he left hand sde of () s a known quantty whle the rght hand sde s an unknown quantty or a quantty to be estmated. he subscrpt (*) MP means Multpath. he phase delay of the reflected sgnal at antenna has a correcton wth the phase delay of the reflected sgnal at antenna B. Fg. 2 shows the relatonshp and t s descrbed as γ B = γ + d Bcos( ϕ φb) cos θ () where γ s the phase delay of the reflected sgnal observed at antenna γ B s the phase delay of the reflected sgnal observed at antenna B db s the dstance between and B ϕ s the reflected sgnal azmuth φ B s the azmuth of the vector B θ s the reflected sgnal elevaton [9]. If an array antenna system s bult wth three antennas n a straght lne as n Fg. 3 the phase delay relatonshp becomes γ B = γ + d Bcos( ϕ φb) cos θ (2) γ C = γ + d C cos( ϕ φc) cos θ. (3) Fg. 3. Confguraton of an array antenna system. If three GPS antennas are fxed on the surface of the vehcle then they have a relatonshp wth a reflector as Fg. 3. Snce the dstances between each antennas are /2. Because of ths constrant φb = φc s obtaned. Equatons (2) and (3) can be rewrtten as follows: γb = γ + dbk (4) γc = γ + dck (5) where K s ( ) cos ϕ φ cos θ. B Lke () the dfferenced equaton between antenna B and antenna C s descrbed as ( BC) BC wbc Φ = Ψ MP + (6) where Ψ BC =ΨC ΨB αsnγc αsnγ B 2 + α sn ( γc γb) = arctan. + αcosγc αcosγ B 2 + α cos( γc γb) Snce all three antennas are separated wth equal dstance and snce they are on a straght lne the phase delay equatons become B S C S γ = γ + (7) γ = γ + 2 (8) 2 S = B = ab a = dbk. he multpath parameters are α γ θ and ϕ [9] but only three parameters α γ and S are requred to estmate multpath. he multpath parameters θ and ϕ are replaced by S n ths paper. he geometrc nformaton of the antenna array reduces the number of estmated multpath parameters reduced. If there s no geometrc nformaton the four parameters have to be estmated. However n ths paper the three parameters have to be estmated by usng the geometrc nformaton. When the sngle frequency carrer phase measurements are used at least four antennas are needed to estmate the multpath error parameters []. When the dual frequency carrer phase measurements are used the number of antennas can be reduced. L frequency carrer phase measurements and L2 frequency carrer phase measurements have a π where γ γ { }

5 392 Eunsung Lee Sebum Chun Young Jae Lee easam Kang Gyu-In Jee and Jeongrae Km gamma (radan) a (m) Phase (radan) me (sec) Multphath Delay (radan) 5 relatonshp lke Fg. 4. In Fg. 4 the sold lne shows L carrer phase measurements errors and the dashed lne shows L2 carrer phase measurements errors. If the dual frequency carrer phase measurements B MP L are used ( Φ ) ( Φ ) ( Φ ) 2 BC and ( ) MP L 2 BC MP L B MP L Φ can be obtaned for each satellte. Subscrpt L and L2 represent L and L2 frequences. he multpath parameters also have the relatonshps as follow: S L L2 = SL L2 γ Multphath Delay (m) me (sec) Multphath Error (radan) me (sec) Fg. 4. Multpath errors on carrer phase measurement at antenna. = (9) L L2 γ L L2 where the subscrpt L stands for L carrer phase measurements the subscrpt L2 stands for L carrer phase measurements. 4. UNSCENED KLMN FILER For multpath mtgaton the EKF has been a popular estmator whch uses a combnaton of the avalable nformaton from an array antenna system to estmate varous multpath parameters []. lthough the EKF nherts the fancy feature of a lnear Kalman flter under process and measurement noses the EKF has several drawbacks due to ts lnearzaton scheme. When estmaton ntervals are not suffcently small lnearzaton process may degrade the flter performance. he dervatons of the Jacoban matrces are nontrval n most applcatons and often lead to sgnfcant mplementaton dffcultes. On the other hand suffcently small tme step ntervals usually lead to hgh computatonal overhead as the number of calculatons for generatng state estmates and covarance become large. o overcome the weak ponts of the EKF the UKF has been recently developed as a state estmator. When a fxed number of parameters are used t s easer to approxmate a Gaussan dstrbuton than an arbtrary nonlnear functon. he UKF uses a parameterzaton whch captures the mean and covarance nformaton whle permttng the drect propagaton of the nformaton through an arbtrary set of nonlnear equatons. It wll be shown that ths can be accomplshed by generatng a dstrbuton from the mnmum number of ponts whch have the same frst and second moments where each pont n the dscrete approxmaton can be drectly transformed. he mean and covarance of the transformed ensemble can be computed as the estmate of a nonlnear transformaton of the orgnal dstrbuton. On the other hand the EKF uses lnearzed system dynamc equatons for the state propagaton. he UKF does not use the crude lnearzed scheme n the predcton procedure. he UKF employs the equvalent formaton of flterng procedure to the EKF [2]. Fg. 5 shows the concept of the unscented transform of the UKF. Generally t s easer to estmate ts statstcal value lke mean or varance than ts realzed value. By ths reason unscented transformaton s proposed. Unscented transformaton s the method to perform nonlnear transformaton on the assumpton that the transformed varable has a Gaussan probablty dstrbuton. Unscented transformaton has characterstcs as follows: - Standard mplementaton s avalable because t uses lmted sgma ponts. - It has an equvalent calculaton amount wth a lnearzed transformaton whch s used n the extended Kalman flter f two flters have the same nonlnear functon. - Unscented transformatons do not need to dfferentate nonlnear equatons for lnearzaton. - Unscented transformatons can be appled nto dscontnuty functons. he Kalman flter whch exchanges nonlnear transformatons to unscented transformatons s the UKF. Constructon of the UKF s as follow. Fg. 5. Unscented transform.

6 Parameter Estmaton for Multpath Error n GPS Dual Frequency Carrer Phase Measurements Usng Unscented Intal condton and sgma pont Equaton (2) s an augmented state vector of the UKF and (2) s ts sgma pont matrx. [ ] ( )( ) xˆ = E x P = E x xˆ x xˆ a a xˆ = E x = xˆ a a a a a P = E ( x xˆ )( x xˆ ) P = Q R χk a = xˆk a xˆ k a ± ( L+ ) Pk a (2) (2) where x s the state vector P s the covarance of the state vector x a s the mean of the augmented a state vector P s the covarance of the augmented state vector Q s the covarance matrx of the process nose R s the covarance matrx of the measurement a nose χk s a sgma pont matrx L s the dmenson of the state vector s the sgma pont scalng parameter me propagaton Equaton (22) s a tme propagaton procedure usng sgma ponts generated from the prevous step. ( k k) x x v kk = f χ χ χ ( m) x k = W χkk = xˆ ( χ )( χ ) ( c) x x k = kk k kk k = P W xˆ xˆ (22) Y = h x n ( χ χ ) kk- kk kk prevous step. ( c) yk y = k kk k kk k = ( c) xk y = k χkk k kk k = = Px k y P k yk yk k = k + κ ( k k ) k = k κ y. k y κ k P W Y yˆ Y yˆ P W xˆ Y yˆ κ xˆ xˆ y yˆ P P P 5. SIMULION RESULS (23) Numercally smulated data wth tme varant parameters have been used to test the performance of the proposed method. he followng fgures show the crcumstance of the experment. Fgs. 6 and 7 show the number of vsble satelltes and PDOP. In ths case the number of observable satelltes must be more than four or fve. In ths case the vsble satellte number s 8. he PDOP s about.72. hs smulaton supposes that the sgnal of satellte s contamnated by the multpath wth the tme varant parameters. ssumng the GPS array antenna system s fxed on the floor of the top of the car as n Fg. 3 Fg. 8 shows the sngle dfference multpath Fg. 6. Number of vsble satelltes. ( m) k = kk = yˆ W Y where x f ( x v ) = expresses system models k k k ( ) y = h x n expresses measurement models k k k ( c) ( m) W s a weght for the mean W s a weght for the covarance Measurement update Equaton (23) s the measurement update procedure. It uses sgma ponts whch propagated from the Fg. 7. Poston DOP.

7 394 Eunsung Lee Sebum Chun Young Jae Lee easam Kang Gyu-In Jee and Jeongrae Km Estmate Fg. 8. Sngle dfference multpath errors. able. Intal parameters. Intal Parameters Gven α L.3.8 α L2.2.7 γ L S L errors. Because the three antennas exst four sngle dfference multpath errors are obtaned. able expresses the ntal parameters whch are the nput to the EKF and the UKF. wo of those parameters are far from true ntal parameters. Fgs. 9 through 2 show the results of the EKF. he sold lnes are estmates and the stars are the true parameters. Fgs. 9 and show the estmated reflecton ratos (α ) of the L and L2 carrer phase measurements respectvely. Fg. shows the estmated phase delay ( γ L). Fg. 2 shows the estmated S L. he results of the EKF show that the EKF does not work properly for estmatng the parameters and that the parameters are dvergent. On the other hand Fgs. 3 to 7 show the results of the UKF. Fgs. 3 and 4 show the estmated reflecton ratos of L and L2 carrer phase measurements respectvely. Fg. 5 shows the estmated phase delay. Fg. 6 shows the estmated S L. he results show that the UKF works well and the estmated multpath parameters follow the true parameter correctly. ccordng to those results the convergent tme s less than about 25sec. 6. CONCLUSIONS Multpath s the major error source n hgh precson GPS applcatons.e. multpath accounts for most of the total error budget especally n carrer phase measurements. hs paper descrbes the procedure to estmate the multpath of GPS carrer measurements. n array antenna system wth antennas Fg. 9. Multpath parameter α L (EKF). Estmate Fg.. Multpath parameter α L2 (EKF). Fg.. Multpath parameter γ L (EKF). Estmated Estmated Fg. 2. Multpath parameter S L (EKF).

8 Parameter Estmaton for Multpath Error n GPS Dual Frequency Carrer Phase Measurements Usng Unscented 395 Estmated Fg. 3. Multpath parameter α L (UKF). Estmated Fg. 4. Multpath parameter α L2 (UKF). Estmated Fg. 5. Multpath parameter γ L (UKF). Estmated Fg. 6. Multpath parameter S L (UKF). was used wth a common external clock. Dual frequency carrer phase measurements ncrease the redundancy of measurements so t can reduce the number of antennas. In ths paper the EKF and the UKF are used to estmate multpath parameters and the results by the two flters are compared. Usng numercal smulaton data t has been shown that the proposed method can estmate the multpath parameters n GPS carrer phase measurements effectvely and the UKF shows an advantage over the EKF. hs proposed algorthm can be used for real tme precse postonng as well as atttude determnaton usng GPS carrer phase measurements. REFERENCES [] P. Fenton W. H. Falkenberg. Ford K. Ng and. J. Van Derendonck Novatel s GPS recever; he hgh performance OEM sensor of the future Proc. of ION GPS-9 pp September 99. [2] B. ownsend and P. Fenton practcal approach to the reducton of pseudorange multpath error n L GPS recever Proc. of ION GPS-94 pp September 994. [3] V. Nee Multple and Mult-ransmtter Interference n Spread-Spectrum Communcaton and Navgaton Systems Ph.D. dssertaton Delft Unversty 995. [4] B. Hofmann-Wellenhof H. Lchtenegger and J. Collns Global Postonng System: heory and Practce Sprnger-Verlag 997. [5] P. xelrad C. J. Comp and P. F. Macdoran SNR based multpath error correcton for GPS dfferental phase IEEE rans. on erospace and Electronc Systems vol. 32 no. 2 pp prl 996. [6] G. Lu Development of a GPS Mult-ntenna System for tttude Determnaton Ph.D dssertaton Unversty of Calgary December 994. [7] C. E. Cohen tttude Determnaton usng GPS Ph.D. dssertaton Stanford Unversty 996. [8] G. Lachapelle H. Sun M. E. Canmon and G. Lu Precse arcraft-to-arcraft postonng usng a multple recever confguraton Canadan eronautcs and Space Journal vol. 4 no. 2 pp [9] J. K. Ray M. E. Cannon and P. Fenton GPS code and carrer multpath mtgaton usng a mult-antenna system IEEE rans. on erospace and Electronc Systems vol. 37 no. pp January 2. [] E. Lee S. Chun Y. J. Lee. S. Kang G. I. Jee and M. F. bdel-hafez Performance mprovement of Wald test for resolvng GPS nteger ambguty usng a baselne-length constrant Internatonal Journal of Control

9 396 Eunsung Lee Sebum Chun Young Jae Lee easam Kang Gyu-In Jee and Jeongrae Km utomaton and Systems vol. 4 no. 3 pp June 26. [] E. Lee S. Chun Y. J. Lee. S. Kang and G. I. Jee Multpath mtgaton of GPS carrer phase measurements usng unscented Kalman flter Proc. of ION NM 25 pp January 25. [2] S. J. Juler and J. K. Uhlmann new extenson of the Kalman flter to nonlnear systems Proc. of the th Int. Symp. on erospace/defence Sensng Smulaton and Controls pp prl 997. Eunsung Lee s a Senor Researcher n the Satellte Navgaton System Department at the Korea erospace Research Insttute. He receved the Ph.D. degree n erospace Engneerng from Konkuk Unversty n 25. Hs research areas nclude GPS RK orbt determnaton and fault detecton of GPS systems. Councl. space systems. Gyu-In Jee s a Professor n the Department of Electroncs Engneerng at Konkuk Unversty Korea. He receved the Ph.D. degree n Systems Engneerng from Case Western Reserve Unversty. Hs research has been focused on GPS and ts applcatons. He s a Commttee Member of the GNSS echnology Jeongrae Km s an ssstant Professor n the School of erospace and Mechancal Engneerng at Hankuk vaton Unversty. He receved the Ph.D. degree n erospace Engneerng from the Unversty of exas at ustn n 2. Hs research topcs nclude satellte nstrument orbt determnaton and GPS applcatons to Sebum Chun s a Ph.D. canddate n the Department of erospace Engneerng at Konkuk Unversty Korea. He receved the M.S. degree n erospace Engneerng from Konkuk Unversty n 22. Hs research areas nclude GPS RK system ntegraton of mult-sensors. Young Jae Lee s a Professor n the Department of erospace Engneerng at Konkuk Unversty Korea. He receved the Ph.D. degree n erospace Engneerng from the Unversty of exas at ustn n 99. Hs research nterests nclude applcaton of GPS to arcraft atttude determnaton and precse postonng development of arcraft landng system. easam Kang s an ssocate Professor of the Department of erospace Engneerng at Konkuk Unversty Korea. He receved the M.S. and Ph.D. degrees from the Seoul Natonal Unversty n 988 and 992 respectvely. Hs current research areas are robust control theores that nclude flght control development of mcro aeral vehcle and MEMS nertal sensor development.