**Department of Electrical Engineering, Shiraz University, Shiraz, IRAN. *Department of Computer Engineering, Shariaty University, Tehran, IRAN.

Transcription

1 Evaluatio of Solid-State Accelerometer for Poitioig of Vehicle S. Nikbakht **,* Maood Mazlom * A. Khayatia ** ikbakht@hariaty.et m8a@yahoo.com khayatia@hirazu.ac.ir **Departmet of Electrical Egieerig, Shiraz Uiverity, Shiraz, IRAN. *Departmet of Computer Egieerig, Shariaty Uiverity, ehra, IRAN. Abtract- A evaluatio of a low-cot, mall ized olid tate accelerometer i the abece of abolute poitioig iformatio i performed i thi paper. Acceleratio igal outputted by the eor i doubly itegrated with time that yield the travel ditace. Due to bia ad hermal oie of the accelerometer which i accumulate i time, Kalma filter i eed to be deig to icreae the accuracy of poitio etimatio ytem. A modelig for bia of olid tate accelerometer ADXL5 QCG without aidig eor evaluated i thi paper. Durig the precedig time itat, the poitio iformatio become icreaigly utrutworthy ad the validity of the poitio update decay. So, we ued a moother algorithm baed Kalma filter for better etimatio accuracy i poitio etimatio. Reult how moothig algorithm icreae the accuracy of poitio etimate for hort time duratio. Keyword: INS, IMU, Bia, olid tate accelerometer, Kalma filter (KF), Smoothig. -itroductio Iertial avigatio ytem ha bee ued i aeropace vehicle, military applicatio, uch a hip, ubmarie, miile, ad to muh leer extet, i lad vehicle applicatio. Oly few year ago the applicatio of iertial eig wa limited to high performace-high cot aeropace ad military applicatio. However, everal cotributio i o-military applicatio have recetly bee publihed makig ue of low cot iertial ytem. A full iertial avigatio ytem (INS) coit of at leat three accelerometer ad three orthogoal gyrocope providig acceleratio i three dimeio ad rotatio rate about three axe. heoretically, igle ad double itegratio of the gyro ad accelerometer output will provide velocity ad poitio iformatio. I practice whe workig with tadard IMU uit, the o-liearity ad oie preet i the eor make the trajectory predictio valid for hort period of time. So Kalma filter eed to utilize for olvig thi problem ([], []). he mot relevat referece foud that develop modelig the bia of olid accelerometer ad evaluatig it for poitio etimatio. A evaluatio of a INS coitig of a olid-tate gyrocope ad a olid tate accelerometer for a mobile robot ca be foud i [3]. Burha ad Durrat Whyte ivetigated the ue of a olidtate gyrocope ad accelerometer for mobile robotic applicatio. hey paid particular attetio to the gyrocope error model, ad came up with a expoetial curve to fit the chage i bia a the gyrocope warm up. So, they do thi tet for olid tate accelerometer. hey the implemeted a Kalma filter for etimatig poitio ad how that the error of etimatio decreae at leat by factor of 5. K. Motov i referece [4] decribed ytematic ad radom error i a accelerometer. Method are propoed to reduce the effect of radom error which iclude proper modelig of the accelerometer ad uig a feedback ytem. Referece [5] decribe how to fid the bia ad cale factor error by uig iertial data ad abolute poitio data. G. l. ad at al. i referece [6] decribe a low-cot GPS/INS car avigatio ytem. GPS i ued to compeate for error uch a drift rate ad offet bia error for two gyrocope ad three accelerometer. I ([7], [8]), Motov dicu a accelerometerbaed low-cot gyro-free iertial device for automotive applicatio. So, S. Park ad Chi- a Woo ued thi ytem for log term duratio with GPS eor [9]. Hugh H.S. Liu ad Gratham K.H. Pag evaluate a low-cot mallized olid-tate accelerometer []. hey how that the accelerometer could be viable olutio a a hort-duratio ditace-meaurig device for a mobile robot or platform. Stergio I. Roumelioti i referece [] decribed a techique to accurately etimate the tate of a robot helicopter uig a combiatio of gyrocope, accelerometer, icliometer ad GPS. hey ued two Kalma filter, oe for the gyrocope data ad the other for the accelerometer data. he theory, implemetatio ad uage of the ADXL5 accelerometer give i referece []. he primary motivatio for the work reported i thi paper ha bee the eed to develop a ytem

2 that capable of providig low-cot, high preciio, hort time-duratio poitio iformatio. I particular, the iteret ha bee i obtaiig locatio iformatio for hort period whe the vehicle i ot i cotact with ay aidig eor for example beaco or ladmark iformatio. o make bet ue of low-cot iertial eig ytem, it i importat that a detailed udertadig of the mechaim cauig drift error are udertood ad a model for thee derived.he approach take i thi paper i to icorporate i the ytem a priori Iformatio about the error characteritic of the iertial eor ad to ue thi directly i a Exteded Kalma Filter (EKF) to etimate poitio before upplemetig the INS with abolute eig mechaim. I ectio the experimetal etup decribed. he bia modelig of olid tate accelerometer decribed i ectio 3. Applicatio of Kalma filter for reductio of bia effect for poitio etimatio evaluated i ectio 4. I cocluio, the uefule of low cot olid tate accelerometer for hort time duratio dicued - Experimetal Setup ad Bia modelig of olid-tate accelerometer he accelerometer that ha bee evaluated i the ADXL5 QCG produced by Aalog Device, Ic. It i a low-cot low-power two-axi micro machied accelerometer with a meauremet rage of ± 5 g. It ca meaure both dyamic acceleratio ad tatic acceleratio. he output of accelerometer are aalog ad digital igal with duty cycle proportioal to the acceleratio i each of two axe. he amplig rate of the accelerometer deped o the dyamic of avigatio ytem or robot. With high-peed, the accelerometer hould be et to a higher badwidth, which require a fater amplig rate. he badwidth of our evaluated accelerometer i from.hz to Hz be ued. However, wider badwidth would caue higher white oie i igal. But i thi reearch we ued a accelerometer box that wa illutrated i Figure. Accelerometer box cotai two olid tate accelerometer ADXL5 QGC that i a dual axi ad we ued the aalog output of accelerometer. Buildig Error model for iertial eor i motivated by a attempt to reduce the effect of ubouded poitio error Depedig how ucceful thee model are, iertial eor may poibly be ued i a uaided mode or for loger duratio o their ow. Figure : Cube olid tate accelerometer box Drift o the output of iertial eor i the mot importat cotributor to avigatio ytem error, ad i maily depedet upo the device techology. o develop a error model for the accelerometer, it output wa recorded over log period of time whe ubjected to zero iput. he reult of thi experimet over a period of 6 hour i how i Figure. I the followig, let ε (t) be the bia error aociated with meaurig the true value of a quality uig iertial eor. A oliear parametric model of the followig form wa fitted to the data from the accelerometer: t ε mod c ( e el = ) + c () Where ε model i the fitted model to the accelerometer output whe zero iput wa applied, with parameter c, c, Τ to be tued. I geeral, a model fitted to experimetal data i regarded a beig adequate if the reidual from the fitted model cotitute a white, zero-mea proce. Hece, oe ca tart with ay reaoable model baed o ipectig the origial data ad tet it reidual for whitee. If the tet fail, the model ca be further developed util the reidual pa the whitee tet. hi implie that the tet for the validity of ay model i baically reduced to a tet for whitee. he ufficiecy of the above model i Equatio we attai the olid tate accelerometer data i tatioary for 6 hour ad hour. Figure ad Figure 3 how the reult i time domai ad frequecy domai. he parameterized model of Equatio for the determiitic bia error ca be repreeted by the followig differetial equatio: ( c + c) & ε = ε () With iitial coditio ε ( ) = c ad ε& ( ) =. c

3 .3 weightig fuctio uch that variace of error miimized x 4 Figure : he output of accelerometer for oe hour i domai with cale (/g) m/^ Power Spectral Deity (db/hz) Welch PSD Etimate time Frequecy (Hz) Figure 3: he output of accelerometer for oe hour i frequecy domai After dicretio: ε ( k + ) = ε ( c + c ) (3) With ε ( ) = c. Due to it recurive ature, thi differece equatio i idepedet of tart-up time but relie o a good etimate of the iitial bia. Alo, the reidual bia of model that wa fitted o the raw data of accelerometer i o motio cae i white tatioary igal (equatio 3). So, total bia model ca be derived: ε ( k + ) = ε + ( c + c ) + ω ε + + (4) ω ε (t) i white Gauia oie with variace ε. 3- Reductio of Radom Bia with Kalma filter Kalma filter i method that ued for reductio radom oie ad data fuio of multieor i iertial avigatio ytem. I thi method tochatic characteritic of meauremet model i recurive form ued for etimate tate of avigatio. Etimatio of thi tate achieve with 3- Deig Kalma filter Baed o otal State Kalma filter A Kalma filter model for accelerometer i well decribed i []. However, we decribe a oedimeioal Kalma filter here. We ue Poitio-Velocity-Accelerometer (PVA) model, a tatioary proce uch a Gau-Markov proce, for reductio the effect of bia i the olid tate acceleratio igal for poitio etimatio. Figure 8 how the accelerometer bia that modeled a a radom walk proce. hi bia i modeled a itegrated white oie with power pectral deity (PSD) beigw. PSD of iput oie obtai from the variace of derivatio of raw igal accelerometer i o motio tate from ome experimet to provide better reult. So, if a (t) be acceleratio i timet, we have: a& = ω (5) ω (t) i white oie with variace that equal with power pectral deity (W). So, tate equatio for thi method drive to followig form: r& = u( t) u& = a( t) a& = ω ( c + c ) & ε = ε + ω ε (6) he reultig tate equatio ad tate traitio matrix F ad proce oie covariace matrix after dicretizatio i derive a follow: X ( k + ) = F X + u + w( k) Z = H X + r F = + (7) u =, H = [,,, ] ( c + c ) +

4 5 4 Q = ε Figure 8: Proce model of the accelerometer data for the Kalma filter We ue raw data of accelerometer i motio tate a meauremet data for Kalma filter to etimate bia of eor. Figure 9 how the Kalma filter algorithm that ued for KF Kalma filter. Figure illutrate acceleratio etimated with Kalma filter ad Figure how that the determiitic bia etimated with Kalma filter from raw data of olid tate accelerometer i motio tate. Reult i Figure how that Kalma filter ha a better etimatio for poitio from itegratio method (that attai from acceleratio data with aumptio decreae from c ) ad decreae the effect of expoetial ad radom bia i poitio etimatio [6]. 4- Smoothig Baed Kalma Filter I previou ectio we built a kalma filter that etimate the poitio. Durig the precedig time itat, a we ca ee from the poitio plot i Figure, after ecod, the poitio iformatio become icreaigly utrutworthy ad the validity of the poitio update decay. So we apply a moothig algorithm every time we receive a good etimate of the poitio. he dicrete Kalma filterig algorithm wa preeted i previou ectio. We will ow coider the recurive moothig problem. he algorithm to be preeted here i due to Rauch, ug, ad Striebel [3-4] ad it derivatio i give i Meditch [5]. I the iteret of brevity, the algorithm will be ubequetly referred to a the RS algorithm. Coider a fixed-legth iterval cotaiig N + meauremet. hee will be idexed i acedig order z, z,..., z N. he aumptio relative to the proce ad meauremet model are the ame a for the filter problem. he computatioal procedure for the RS algorithm coit of a forward recurive weep followed by a backward weep. We eter the algorithm a uual at k = with the iitial coditio ˆx ad P. Figure 9: Kalma filter algorithm Figure : otal bia etimated with KF Etimated with KF (Solid lie) ad etimated ad tatioary data of acceleratio with itegratio (dahed lie) We the weep forward uig the covetioal filter algorithm. With each tep of the forward weep, we mut ave the computed a priori ad a poteriori etimate ( xˆ k ad xˆ k ) ad their aociated P matrice ( Pˆ k ad Pˆ k ). hee are eeded for the backward weep. After completig the forward weep, we begi the backward weep with iitial coditio xˆ N

5 demotrate the improvemet i etimatio with moothig algorithm for hort term duratio [6]. Figure : Real poitio (dah-dote lie), poitio with itegratio (dahed lie) ad etimated with KF (olid lie) Figure: Real poitio (dah-dot lie), poitio with Forward KF (dah lie) ad with Backward (olid lie) ad Pˆ N obtaied a the fial computatio i the forward weep. With each tep of the backward weep, the old filter etimate i updated to yield a improved moothed etimate, which i baed o all the meauremet data. he recurive equatio for the backward weep are: xˆ = xˆ + A( k)* b f [ xˆ ( + ) xˆ b k f ( k + )] (8) whhere the moothig gai A (k) i give by: ˆ A( k) = Pf F Pf ( k + ) (9) ad k = N, N,...,. he error covariace matrix for the moothed etimate i give by the recurive equatio: Pˆ = Pˆ + A( k) * b f ˆ ˆ [ Pb ( k + ) Pb ( k + )] A () Poitio etimated with forward ad backward filter i illutrated i Figure. Reult 5- Cocluio he experimetal reult have provided a ueful evaluatio of a low-cot olid tate accelerometer. he performace of the accelerometer i how to be acceptable a a hort-duratio ditace-meaurig device for mobile platform or robot [7]. So, we applied a moothig algorithm i the abece of abolute poitioig iformatio that cotructed the bet etimate of the poitio over a time period uig all the meauremet i that time iterval. Further reearch would focu o the better model for tochatic bia of olid tate accelerometer i order to reduce the effect of bia ad thermal oie ad fue the olid tate accelerometer data with other abolute eor that meaure poitio. 6- Referece [] Lawrece A., Moder Iertial techology, 993, priger verlag. [] R.G. Brow ad P.Y.C. Hwag, Itroductio to Radom igal ad Applied Kalma filterig. [3] B. Barha ad H.F. Durrat-Whyte, A ietial avigatio ytem for a mobile robot, i proc. It. cot. Itelliget Robot ad Sytem, yokohama, Japa, 993 [4] K. Motov, Iertial eor documetatio, Web Page of PAH, Uiv. Califoria, Berkeley. er.html [5] K. Motov, Method for correctio of ytematic iertial eor error, Web Page of PAH, Uiv. Califoria, Berkeley. er.html [6] J.-S. Gil, Y.-D. Cho, ad S.-H. Kim, Deig of a low-cot iertial avigatio ytem with GPS for car avigatio ytem, i proc. 5 th World Cogr. IS, 998 [7] K.S. Kotov, A.A. Soloviev ad.-k.j. Koo, Accelerometer baed gyro-free multi-eor geeric iertial device for automotive applicatio, i proc. IEEE Itelliget raportatio Sytem Cof., 997, pp [8] Chi-Woo a, Kirill Motov, Pravi Varaiya, Feaibility of a gyrocope-free iertial avigatio ytem for trackig rigid body

6 motio, Web Page of PAH, Uiv. Califoria, Berkeley. [9] Chi-Woo a ad Sugu Park, GPS-Aided gyrocope-free iertial avigatio ytem, Web Page of PAH, Uiverity of Califoria at Berkeley. [] Hugh H. S. Liu ad Gratham K. H. Pak Member IEEE, Accelerometer for mobile robot poitioig, IEEE raactio o Idutry Applicatio, Vol.37, No.3, May/Jue. [] Myugoo Ju, Stergio I. Roumelioti, Graurav S. Sukhatme, State etimatio of a autoomou helicopter uig kalma filterig, Proceedig of the 999 IEEE/RSJ, Iteratioal Coferece o Itelliget Robot ad Sytem. [] Aalog Device, Ic., Sata Clara, CA, ADXL5 data heet. [3] H. E. Rauch, Solutio to the Liear Smoothig Problem, IEEE ra. Auto. Cotrol, Ac-8:37, (996). [4] H.E. Rauch, F. ug ad C.. Striebel, Maximum Likelihood Etimatio of Liear Dyamic Sytem, AIAA J., 3:445, 965. [5] S. Meditch, Stochatic Optimal Liear Etimatio ad Cotrol, New York; McGraw- Hill, 969. [6] S. Nikbakht, " Evaluatio of Solid State Accelerometer Mouted o the Vertical Gyro for Platform Poitioig", MS hei, Dep. of Electrical eg., Shiraz Uiverity, Ira, 3. [7] S. Nikbakht, A. Khayatia,, Evaluatio of Solid-State Accelerometer Mouted O the Vertical Gyro for Platform Poitioig, Iraia Coferece o Electrical Egieerig, Mahhad, ira, May 4.