Yaw Rate Estimation Using Two 1-Axis Accelerometers

Transcription

1 5 Ameicn Cntl Cneence June 8-, 5. Ptlnd, OR, USA WeA. Yw Rte Estimtin Using Tw -Ais Accelemetes Wnshik Chee, PhD. Abstct This ppe pesents new w estimting w te vehicle using tw -is ccelemetes. Mesuement w te is inevitble vehicle stbilit cntl sstems dpted in cntemp high pemnce sedns nd the utmtic steeing cntl sstems intelligent vehicles. Cmped with ccelemetes, the cst te gs used t mesue the w te is signiicntl highe. As n ecnmicl stwe slutin t eplce the high-cst te gs with miniml dditinl hdwe, n lgithm t estimte w te using tw -is ccelemetes is ppsed in this ppe. Y I. INTRODUCTION AW te is ne the ke inmtin vehicle diectinl stbilit cntl sstems s well s the vehicle cntl pplictins. Especill, w te mesuement cn signiicntl impve the pemnce the ltel mtin cntl sstem intelligent vehicles. Tpicll, te gs e used t mesue the w te. Hweve, the cst the te gs is me epensive cmped with the senss such s ccelemetes. As lw cst ltentive t the w te g, n lgithm t estimte the w te using tw -is ccelemetes is pesented hee. Thee hve been tw ttempts t estimte w te using sevel ccelemetes ([],[]). Shimd et l. [], nd the ttempts listed in [] ppsed lgithms bsed n vehicle kinemtics. Theee, these methds e sensitive mesuement nise. An lgithm bsed n line vehicle mdel ws pesented in []. Hweve, due t the sens lctins, the lgithm pesented in [] is sensitive t tilt the vehicle. The lgithm ppsed hee vecmes the eect the tilts b instlling the ccelemetes lngitudinll. Using the tw ltel cceletin mesuements, te chnge the w te cn be und. Then, the estimtin the w te nd the slip velcit cn be btined b designing stte estimtin lgithm. Ognitin this ppe is s llws. Fllwing the Wnshik Chee, PhD is n ssistnt pess The Univesit Tes, Sn Antni, 69 N. Lp 6 W., Sn Antni, TX 789. Phne: ; F: ; e-mil: wnshik.chee@uts.edu. intductin, Desciptin the Vehicle Mdel t be used the estimtin design is pesented. Then, the min ide the ppsed lgithm is discussed. Anlticl nlsis the eect the tilt spung mss the vehicle geneted eithe b lling mtin b supe elevtin llws the min ide desciptin. Finll, simultin esults nd cnclusins will llw. II. DESCRIPTION OF THE VEHICLE MODEL F the lgithm develpment, the biccle mdel [] is used. The biccle mdel epesents vehicle mtins in hintl plne. The mdel is btined b lumping the tw nt wheels int ne imgin nt wheel nd the tw e wheels int ne imgin e wheel. In dditin, suspensin dnmics e neglected. Thus, ll, pitch, nd heve mtins e nt included. Fig.. Vibles nd Cdintes Biccle Mdel Figue pesents ke vibles t descibe the dnmics the biccle mdel, nd the cdintes used deivtin the equtin mtin. I we ssume tht the lngitudinl speed V is cnstnt, the equtins mtin using the es ied t the vehicle cn be witten s llws. m V V cs () () I l cs l In these equtins, V dentes the speed in the diectin /5/$5. 5 AACC

2 is nd V the ne in the diectin is. The velcit vect vehicle, V, is btined m V nd V. dentes the w ngle with espect t the bslute cdintes, OXY, nd epesents the steeing ngle the vehicle. nd e the side ces the nt tie nd the e tie, espectivel. Othe smbls nd thei nminl vlues e listed in Tble. TABLE : VARIABLE AND PARAMETERS OF BICYCLE MODEL C b s m, Cs b l () I III. THE ALGORITHM The ppsed lgithm hs been develped bsed n the cnigutin the tw -is ccelemetes s shwn in Figue. Hee, the tw -is ccelemetes e ssumed t be instlled n the lngitudinl centeline the vehicle. One (Fnt Accelemete) is instlled in nt the C.G., nd the the (Re Accelemete) is in e the C.G. s shwn in Figue. The side ce geneted b ech tie is unctin tie sideslip ngle. The slip ngle is deined s the ngle between the diectin n bject nd the velcit vect the bject. I the bject is tie, we cn btin the tie sideslip ngle s llws. l V () V l V () V Hee, nd epesent the nt nd the e tie sideslip ngle, espectivel. I we ssume the tie sideslip ngles e smll, the side ces e witten s (5) C s C s (6) Using the simpliied tie mdel, the lineied vesin the biccle mdel cn be witten s llws. V V b (7) V b (8) Hee, dentes the w te the vehicle, i.e.. The ceicients these tw equtins e deined s llws. C l s l, Cs V, mv mv l l Cs, I V l l Cs (9) I V Fig.. Vibles nd Cdintes Biccle Mdel The distnce between Accelemete nd CG is dented s S, nd the ne between Accelemete nd CG s S. Nte tht the chice S nd S cn be bit, but shuld nt be e. The cceletin mesued b Accelemete cn be mdeled s llws cnsideing the kinemtics. V V S () Als, the cceletin mesued b Accelemete cn be mdeled s llws. V V S () Then, b subtcting equtin () m equtin (), the llwing equtin cn be btined. () S S Nw, b tking equtin () s the utput the sstem epesenting the ltel dnmics (equtins (7) nd (8)), ne cn design n bseve s llws. Als, nte tht even thugh S nd S cn be vied due t vitin the lctin C.G., S S is lws cnstnt unless the ccelemetes e eplced. THEOREM : Suppse tht the utput the sstem epesented b equtins (7) nd (8) is chsen s Y () S S

3 Then, the sstem is bsevble n nne vehicle speed nd n cneing stiness i the weight distibutin ti is nt 5:5. (P) Reclling the equtin () nd (8), equtin () becmes s llws. Y V b (5) Then, bsevbilit mti cn be und s Nw, the nge the velcit the vehicle nd the nge the cneing stiness vlue, deteminnt the bsevbilit mti becmes nne since det i the weight distibutin ti is nt 5:5. Nte tht, i the weight distibutin ti is 5:5, becmes. Theee, the sstem is bsevble. THEOREM : Suppse tht stte bseve the sstem epesented b equtins (7) nd (8) using the utput btined s equtin () is cnstucted s Vˆ Vˆ ˆ b L Y Yˆ (6) Y ˆ Vˆ ˆ b L Yˆ (7) whee Yˆ ˆ ˆ V b (8) Then, estimtin es, V Vˆ nd ˆ, cnvege t e with ppe chice L nd L. (P) Omitted. IV. EFFECT OF ROLL AND SUPERELEVATION The ppsed lgithm ws develped b igning ll nd supe-elevtin. Hweve, the eect the ll nd the supe-elevtin cn be nlticll pedicted. Eectivel, the ll ngle nd the supe-elevtin ngle cn be epesented s ltel tilting ngle the spung mss the vehicle s shwn in Figue nd Figue. Suppse tht the vehicle shwn in Figue epeiences lling mtin whse mgnitude is. Then, the ltel cceletin mkes the sme ngle with the diectin the ccelemete mesuement. Nw cnside the situtin shwn in Figue. Hee, the ngle the supe-elevtin is dented s. Agin, the diectin the ltel cceletin nd the diectin the ccelemete mke the sme ngle with the supeelevtin ngle. Theee, bth cses, the cceletin mesuement cn be pesented s llws. cs g sin (9) mesued Hee, is the ltel cceletin the vehicle, which is pesented s eithe equtin () (), nd is the ngle the tilt which geneted b eithe ll mtin Supe-Elevtin. is the mesued cceletin mesued b the ccelemete. Fig.. Tilt due t ll mtin Fig.. Tilt due t Supe-Elevtin THEOREM : I the vehicle tilts b (dins) nd, the mesued te chnge w te b equtin () ( ()) will hve eltive e in the de. (P) When the vehicle tilts, the mesued cceletins pesented s equtins () nd () becme s llws. V V S cs sin g mesued V V S cs sin mesued g Using these tw equtins, () becmes Y mesued Y cs Nw, deine the e Y due t the tilt s Y Y Y e mesued 5

4 Then, the e becmes Y e Y cs Since, Tl seies cs cn be btined s llw. 6 cs O Theee, the eltive e cn be Y e 6 cs O Y Y e Y O Nte: F the tilt, the eltive e is.5.5%. F the tilt, the 8 eltive e is.6 6.%. Theee, the e due t the tilt cn be negligible esnble diving cnditins. shws the estimtin e. Fm the lwe plt, the mimum e is.96 d / sec. The eltive e is.8%. Fig 6: Yw Rte Estimtin when thee is n tilt nd n sens nise. V. SIMULATION RESULTS Simultin mdel hs been develped using MATLAB/SIMULINK. F the simultin, discete time vesin the bseve ws deived b ssuming e de hld []. The smpling time ws selected msec. The speed the vehicle is selected s.m / sec, which is 7 MPH. The bndwidth the bseve is chsen s.5 H, nd the dmping ti s.8. Sens nise is mdeled s unimed distibuted ndm vible nged ve [.m / s,.m / s ]. The llwing steeing ngle, shwn in Figue 5, is selected t vlidte the ppsed lgithm. Fig 5: Steeing Angle Simultin IDEAL CONDITION SIMULATION Nw, cnside the cse when thee e n tilt nd n sens nise. Simultins this cse e shwn in Figue 6, Figue 7 nd Figue 8. Figue 6 shws the pemnce w te estimtin. The uppe plt shws the cmpisn between ctul w te nd the estimted w te. The lwe plt Fig 7: Acceletin the ccelemetes when thee is n tilt nd n sens nise. Figue 7 shws the mesued cceletins m the ccelemetes. The slid line epesents the cceletin mesued b the nt ccelemete shwn in Figue 6. The dshed line epesents the cceletin mesued b the e ccelemete. Nte tht the dieence between the tw ccelemetes is celted with the slpe the w te shwn in Figue 6. The behvi cnims equtin (). In the wds, the dieence between the nt ccelemete nd the e ccelemete is the slpe the w te. Figue 8 shws the estimtin the slip velcit. Nte tht slip velcit cnnt be mesued b n mens. Since the ppsed lgithm is stte estimtin lgithm the sttes the biccle mdel, the slip velcit is ls btined. The mimum estimtin e the slip velcit is.975 m / sec, the mimum eltive e.69%. 6

5 the tilt n the ppsed lgithm cn be negligible mst cses. Fig 8: Slip Velcit Estimtin when thee is n tilt nd n sens nise. Nte tht the estimtin es bth the w te nd the slip velcit e less thn.5%. Theee, the ppsed lgithm cn pvide suicient ccuc the cntl pplictins such s vehicle stbilit cntl ltel guidnce. EFFECT OF TILT Nw, investigte the eect tilting the spung mss the vehicle, geneted eithe b lling mtin supe elevtin. In de t simpli the simultin, it is ssumed tht the vehicle is mneuveed with speciied tilting ngle. Thus, the eect the tilt cn be mdeled s pesented in (9). Fig : Acceletin the ccelemetes when thee is tilt nd n sens nise. Figue shws the cceletins mesued b the ccelemetes. Nte tht, due t the tilt, the bse line the cceletins e nt e. The bse line cceletin is.55m / sec, which is equl t 9.8 sin. Fig 9: Yw Rte Estimtin when thee is tilt nd n sens nise. Figue 9 shws the w te estimtin when the tilting ngle is. Nte tht tilt ngle is etemel lge mgnitude, nd, usull, tilt due t lling mtin nd/ the supe-elevtin is less thn. Nw, the mimum e is d / sec,, eltive e is.%. Nte tht the estimtin e this cse is smlle thn the idel cse. This cnims tht the eect Fig : Slip Velcit Estimtin when thee is tilt nd n sens nise. Figue shws the slip velcit estimtin nd its estimtin e. The mimum e is.95 m / sec, the eltive e is.85%. Fm Figue 9, Figue nd Figue, the mimum eltive estimtin e is still less thn.5%. Theee, the ppsed lgithm cn be eective in spite the tilt the vehicle. EFFECT OF TILT AND SENSOR NOISE Nw, emine the eect the sens nise n the pemnce the estimtin when the vehicle is tilted. Agin, the tilt ngle is chsen s. The sens nise is mdeled s unim ndm vible whse nge is [.m / s,.m / s ]. 7

6 Yw Rte(d/sec) Estimted Actul Yw Rte Estimtin E (d/sec) Time (sec) Fig. : Yw Rte Estimtin when thee is tilt nd sens nise. dminnt distubnce suce the estimtin e V (m/sec) Estimted Actul V Estimtin E (m/sec) Time (sec) Fig. : Slip Velcit Estimtin when thee is tilt nd is sens nise. Acceletin(m/sec ) 8 6 Fnt Accelemete Re Accelemete Time (sec) Fig. : Acceletin the ccelemetes when thee is tilt nd sens nise. Figue shws the w te estimtin when the sens nise is implemented. Fm this igue, the mimum estimtin e cn be und s.55 d / sec, eltivel.755%. Nte tht this quntit munts t lmst.5 times the cse withut sens nise. Eect the sens nise cn be educed b lweing the bndwidth the bseve. I the lwe bndwidth is ttempted, the estimtin will shw signiicnt phse lg nd signiicnt e when thee is discepnc between the ctul sstem pmetes nd the mdel pmetes used the design the estimtin lgithm. Theee, ne cnnt bitil educe the bndwidth. The mesued cceletins e shwn in Figue. Figue shws the estimtin the slip velcit. In this cse, the mimum estimtin e is 5. m /sec,, eltivel.69%. Cmped with the cse withut the sens nise, the eltive estimtin e gws but.5 times. Bsed n the indings m Figue nd Figue, sens nise is the VI. CONCLUSION A new estimtin lgithm t estimte the w te the vehicle using ccelemetes tht e instlled lng the centeline the vehicle ws pesented. Accding t the nlsis, the eect the tilt, geneted eithe b lling mtin b supe-elevtin, n the ppsed lgithm is negligible. Fm the simultin studies, it ws und tht the sens nise is the dminnt suce the estimtin e. Even thugh, unde the dvese eect the nise, the estimtin e ws bunded b %. Theee, it cn be cncluded tht the ppsed lgithm pvides ccute nd smth estimtin the w te. In the utue, stud bustness t the mdel pmete uncetinties needs t be pemed since the ppsed lgithm is mdel-bsed estimtin lgithm. Als, epeimentl vlidtin needs t be pemed. REFERENCES [] G. F. Fnklin, J. D. Pwel nd M. Wkmn, Digitl Cntl Dnmic Sstems, d Ed., Addisn Wesle Lngmn, Inc., 998. [] K. Shimd, Y. Nkmu, S. Hikshi, H. Sugw nd T. Mnji, Angul Rte Detectin Apptus, Acceletin Detectin Apptus nd Mvement Cntl Apptus Mving Bd, US Ptent 5,7,66, 99. [] N. Sivshnk nd A. G. Uls, Yw Rte Estimtin Vehicle Cntl Applictins, Junl Dnmic Sstems, Mesuement, nd Cntl, Vl., pp. 67 ~ 7, 998 [] J. Y Wng, The Gund Vehicles, nd Ed., Jhn Wile & Sns, Inc., 99. 8