Method to Estimate the Position and Orientation of a Triaxial Accelerometer Mounted to an Industrial Manipulator

Transcription

1 Technicl epot fom Automtic Contol t Linköpings univesitet Method to Estimte the Position nd Oienttion of Tiil Acceleomete Mounted to n Industil Mnipulto Ptik Aelsson, Mikel Nolöf Division of Automtic Contol E-mil: elsson@isy.liu.se, mino@isy.liu.se 19th Septeme 2011 Repot no.: LiTH-ISY-R-3025 Sumitted to the IEEE Intentionl Confeence on Rootics nd Automtion 2012 Addess: Deptment of Electicl Engineeing Linköpings univesitet SE Linköping, Seden WWW: AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET Technicl epots fom the Automtic Contol goup in Linköping e ville fom

2 Astct A novel method to nd the oienttion nd position of tiil cceleomete mounted on i degees-of-feedom industil oot is poposed nd evluted on epeimentl dt. The method consists of to consecutive steps, hee the st is to estimte the oienttion of the senso dt fom sttic epeiments. In the second step the senso position eltive to the oot se is identied using senso edings hen the senso moves in cicul pth nd hee the senso oienttion is kept constnt in pth ed coodinte system. Once the cceleomete position nd oienttion e identied it is possile to use the senso in oot model pmete identi- ction nd in dvnced contol solutions. Comped to pevious methods, the senso position estimtion is completely ne, hees the oienttion is found using n nlyticl solution to the optimistion polem. Pevious methods use pmeteistion hee the optimistion uses n itetive solve. Keyods: Rootics, Acceleomete, Estimtion

3 Method to Estimte the Position nd Oienttion of Tiil Acceleomete Mounted to n Industil Mnipulto Ptik Aelsson, Mikel Nolöf Astct A novel method to find the oienttion nd position of tiil cceleomete mounted on i degeesof-feedom industil oot is poposed nd evluted on epeimentl dt. The method consists of to consecutive steps, hee the fist is to estimte the oienttion of the senso dt fom sttic epeiments. In the second step the senso position eltive to the oot se is identified using senso edings hen the senso moves in cicul pth nd hee the senso oienttion is kept constnt in pth fied coodinte system. Once the cceleomete position nd oienttion e identified it is possile to use the senso in oot model pmete identifiction nd in dvnced contol solutions. Comped to pevious methods, the senso position estimtion is completely ne, hees the oienttion is found using n nlyticl solution to the optimistion polem. Pevious methods use pmeteistion hee the optimistion uses n itetive solve. I. INTRODUCTION A novel method to estimte the position nd oienttion of tiil cceleomete mounted on n industil oot is pesented. The estimtion method uses to step pocedue hee the fist step is to identify the oienttion of the senso using nume of sttic epeiments. It is ssumed tht the senso is mounted in such y tht it cn e itily oiented using the si degees-of-feedom DOF oot m. The desied oienttion of the senso is hence knon hile the ctul oienttion is unknon. In [1] nd [2] the cceleomete clition is consideed nd intenl pmetes of the cceleomete, such s sensitivity nd is, ut lso lignment of ech one of the thee cceleomete mesuement chnnels, e identified. The min diffeences eteen the ppoch pesented in the pesent ppe, nd [1], [2], e tht the oienttion, sensitivity, nd is e found using n itetive optimistion ppoch in [1], [2] hile in the ppoch pesented in this ppe the solution cn e found in closed fom. In ddition, the pesent method lso uses the dynmics of the pocess to identify the position of the cceleomete. In [1], [2] it is ssumed tht the cceleomete is moved in such y tht only gvity ffects the mesuements. In contst, to identify the position it is necessy to ecite the dynmic cceletion, nd it is pesented ho this cn e chieved y doing nume of mesuements using the motion cpilities of the oot hile keeping the cceleomete in diffeent oienttions ith espect to the pth coodinte system. Finlly, the poposed method is evluted on epeimentl dt. All uthos e ith the Deptment of Electicl Engineeing, Linköping Univesity, SE Linköping, Seden {elsson, mino}@isy.liu.se. The estimtion polem is fomulted in Section II. In Section III, the method to find the oienttion of the senso is descied, nd the method to estimte the mounting position is descied in Section IV. The oienttion nd position estimtion is evluted on epeimentl dt in Section V nd Section VI concludes the esults. II. PROBLEM FORMULATION Assume tht the cceleomete is mounted on the oot ccoding to Figue 1 hee the senso is ssumed to e igidly ttched to the oot tool. Given definition of the tool coodinte system the estimtion method pesented in this ppe finds the eltive oienttion nd position of the tiil senso. The oienttion of the desied coodinte system cn e seen in Figue 1. Let ρ e n cceleomete mesuement vecto in the senso coodinte system O y of the cceleomete nd ρ s n cceletion vecto in the desied coodinte system O, desciing the cceletion in m/s 2. The eltion eteen ρ nd ρ s is given y, ρ s = κrρ + ρ 0, 1 hee R is the ottion mti R fom O y to O, κ is the cceleomete sensitivity nd ρ 0 the is. It is ssumed tht the sme sensitivity vlue κ cn e used fo ll thee sensos in the tiil cceleomete. The sensitivity nd is is chosen such tht the units in O e m/s 2. When the unknon pmetes in 1 hve een found the position of the cceleomete is identified, epessed eltive to the tool coodinte system. To solve fo the unknon pmetes ρ is mesued hile ρ s is computed fom model. In the sttic cse ρ s is simply the gvity vecto, hile in the dynmic cse hen the senso is moved the cceletion ill depend on the speed nd oienttion of the senso. To e le to divide the estimtion polem in to distinct polems the oienttion is estimted using sttic mesuements only hile the position of the senso is found y moving the cceleomete long knon pth ith knon speed. Using the knon oienttion of the cceleomete it is possile to numeiclly cncel the effect of gvity nd only mesue the dynmic cceletion, ith constnt speed in cicul pth, pependicul to the gvity field. The oienttion of the cceleomete is kept fied ith espect to the pth coodintes duing the motion. This mens tht the cceletion oiginting fom the movement cn e isolted fom the gvity component. A specil cse is hen O is otted such tht the coodinte system of the cceleomete is diected to give gvity mesuements long one

4 y The cceleomete nd its ctul coodinte system O y. The cceleomete nd the desied coodinte system O. Fig. 1. The cceleomete mounted on the oot. The yello ectngle epesents the tool o eight nd the lck sque on the yello ectngle is the cceleomete. The se coodinte system O of the oot is lso shon. coodinte is only. The to othe es of the cceleomete diectly gives the dynmic cceletion component hich cn e used to estimte the position. III. IDENTIFICATION OF ORIENTATION, SENSITIVITY AND BIAS To solve fo the pmetes R, κ nd ρ 0 in 1, fist define the esidul e k = ρ s,k κrρ,k ρ 0, 2 hee k indictes the smple nume. Net, minimise the sum of the squed nom of the esiduls, N minimise k=1 e k 2 suject to detr = 1 3 R T = R 1 hee the constins guntee tht R is n othonoml mti. Thee eists closed fom solution to this optimistion polem [3], / κ = N N ρ s,k 2 ρ,k 2, 4 hee k=1 R = M M T M 1/2, ρ 0 = ρ s κr ρ, ρ s = 1 N ρ = 1 N k=1 N ρ s,k, k=1 N ρ,k, k=1 4 4c 5 5 e the centoids fo the mesuements in O y nd O. ρ s,i = ρ s,i ρ s, ρ,i = ρ,i ρ, 6 6 denote ne coodintes nd N M = ρ s,kρ,k T. 7 k=1 N is the totl nume of mesuements nd it hs to e ssumed tht N 3. In ddition condition of sufficient ecitement hs to e fulfilled, such tht M T M hs full nk. As n ltentive to the fomultion ove hee the ottion is pmeteised y the othonoml mti R it is lso possile to find closed-fom solution to 1 using unit qutenions, see e.g. [4]. Consideing the nume of opetions the mti fomultion is, hoeve, computtionlly moe efficient. As indicted in Section II the oienttion nd the senso pmetes e found using sttic mesuements, i.e., moving the tool into nume, N C, of diffeent configutions. The gvity vecto is mesued y the cceleomete in ech of the N C configutions, hich gives N M,j, j = 1,..., N C mesuements fo ech configution. Let { } {ρ } = {ρ 1,i} N M,1 i=1,..., {ρn C,i }N M,N C i=1 8 denote the set of ll the N = N C j=1 N M,j mesuements in ll N C configutions, nd let {ρ s } = { {ρ 1 s} N M,1 i=1,..., {ρn C s } } N M,N C i=1 e the gvity vecto fom the model in the desied coodinte system O fo ech configution, hee ρ j s, j = 1,..., N C is constnt. Using the mesued cceletions nd the model vlues to solve the optimistion polem in 4 to 7 the tnsfomtion pmetes cn e computed. The N C diffeent configutions cn e chosen ity ut hee e suggest si diffeent configutions ccoding to Figue 2, hich give ρ 1 s = 0 0 g T ρ 3 s = 0 0 g T ρ 5 s = g 0 0 T 9, ρ 2 s = 0 g 0 T,, ρ 4 s = 0 g 0 T,, ρ 6 s = g T, hee g = 9.81 m/s 2. The sign of g in 10 is opposite the gvity vecto in Figue 2. The eplntion fo this is tht n cceleomete mesues the noml foce hich is opposite the gvity vecto. The si configutions in Figue 2 e stightfod to otin fo i degee of feedom industil mnipulto [5]. The pocedue to estimte the tiil cceleomete senso pmetes is summised in Algoithm 1. Algoithm 1 Estimtion of the senso pmetes 1 Mesue the cceletion fo the diffeent configutions in Figue 2 to otin {ρ } ccoding to 8. 2 Constuct {ρ s } in 9 fom Clculte R, κ nd ρ 0 fom 4 to 7. It is possile to use othe configutions thn the one in Figue 2 in Algoithm 1 s long s M T M hs full nk 1. 1 The mti M T M hs lys full nk if none of the to sets {ρ } nd {ρ s} e copln.

5 1 2 3 g Fig. 2. Si diffeent configutions of the oot tool used in Algoithm 1. The oienttion of the desied coodinte system O is shon fo ech configution. The se coodinte system O nd the gvity vecto e lso shon. IV. ESTIMATION OF THE POSITION OF THE ACCELEROMETER Using mthemticl model of the oot motion it is possile to compute the cceletion, pmeteised in some unknon pmetes. In the second step of the poposed oienttion nd position estimtion pocess method fo the position estimtion is eplined fo the cceleomete s coodinte system O, epessed in coodinte system Of f f fied to the oot. Fom Section III the oienttion nd senso pmetes e knon, hence the cceletion mesued y the cceleomete hs knon oienttion. To simplify the mthemticl model fo the cceletion nd to mke it possile to pmeteise the unknon pmetes, conside the cse hen the oot is in the configution shon in Figue 3. The figue shos the vecto s, the to coodinte systems Of f f nd O, old fied coodinte system O ttched to the se of the oot, coodinte system O y fied to the end of the oot m, vecto = d 2 dt 2 s desciing the cceletion of O, hich e nt to find n epession fo. The figue lso shos pmete desciing the ottion eteen Of f f nd O, to knon pmetes nd L 2 desciing the m lengths nd thee unknon pmetes l i, i = 1, 2, 3 desciing the vecto s/ in O y. All the clcultions e done in the old fied coodinte d system in ode to otin n epession fo 2 dt 2 s. In d ody fied coodinte system Of f 2 f dt 2 s = 0. The nottion [ s ] i is used to emphsise tht s is epessed in coodinte system i. In Figue 3 e see tht s cn e itten s um of to vectos, hee [ s ] f = [ ] f + [ s/ ] f, 11 [ s/ ] f = l 2 l 1 T, 12 [ ] f = 0 L 2 T. 13 The tnsfomtion of s fom Of f f to O cn e epessed s [ s ] = [Q f/ ] [ ] f + [ s/ ] f, 14 f f f f L 1 s s s s y s / l 1 L 2 Fom the side. l s 2 y l 2 L 2 Fom ove. Fig. 3. f The fist oot configution fo estimtion of the mounting position. The lck cue on the yello o indictes the senso, i.e., the oigin of Of s. The yello o is ttched to the oot in the point L1 0 L T 2 epessed in Of f f. hee f f [Q f/ ] = s cos sin 0 sin s / cos 0 l y f 15 is the ottion mti tht eltes the coodinte system Of f f to O. = t is the ngle elting O nd Of f f ccoding to Figue 3. Tking the deivtive of [ s ] ith espect to time gives d dt [ s] = d [Qf/ ] [ ] f + [ s/ ] f. 16 dt Fom [6] e hve tht d [Qf/ ] = Sω[Qf/ ], 17 dt hee ω = 0 0 T nd Sω = is ke symmetic mti. Hence, the time deivtive of [ s ] cn e itten d dt [ s] = Sω[Q f/ ] [ ] f + [ s/ ] f. 19 The second time deivtive of [ s ] ecomes [ ] = d2 dt 2 [ s] = d dt Sω [Q f/] [ ] f + [ s/ ] f + Sω d [Qf/ ] [ ] f + [ s/ ] f dt =S ω[q f/ ] [ ] f + [ s/ ] f + SωSω[Q f/ ] [ ] f + [ s/ ] f =SωSω[Q f/ ] [ ] f + [ s/ ] f, 20 f f f

6 f f f L 3 s l 1 l 3 L 4 f f f s s y l s 2 l 2 L 2 y f f f Fom the side. f Fom the side. s f f L 3 s l 1 y Fom ove. l 2 Fig. 4. The second oot configution fo estimtion of the mounting position. The lck cue on the yello o indictes the senso, i.e., the oigin of Oy s. The yello o is ttched to the oot in the point T L3 0 L 4 epessed in Of f f. y f f s l 1 Fom ove. Fig. 5. The thid oot configution fo estimtion of the mounting position. The lck cue on the yello o indictes the senso, i.e., the oigin of Oy s. The yello o is ttched to the oot in the point T L1 0 L 2 epessed in Of f f. hee ω = T follos fom the ssumption of constnt ngul velocity. It no emins to tnsfom the mesued cceletion M s fom O to O. Fom Figue 3 e see diectly tht [ M s ] f = M s, M s,y 0 T, 21 hence Equtions 20 nd 22 give [ M s ] = [Q f/ ] [ M s ] f. 22 [Q f/ ] [ M s ] f = SωSω[Q f/ ] [ ] f + [ s/ ] f [ M s ] f = [Q f/ ] T SωSω[Q f/ ] [ ] f + [ s/ ] f 23 since [Q f/ ] T = [Q f/] 1. Cying out the mti multipliction in the ight hnd side epession of 23 gives 2 + [ M s ] f = 2 l 2 0, 24 hee 12, 13, 15 nd 18 hve een used. Equtions 21 nd 24 cn no e itten s ystem of equtions hee l 2 nd e unknon, 0 2 l2 M 2 = s, M s,y It is thus possile to find l 2 nd fom 25 ut unfotuntely not l 1. To find l 1, otte the senso ccoding to Figue 4 nd do the sme kind of movement. The sme clcultions s efoe ith see Figue 4, give [ s/ ] f = l 1 l 2 T, 26 [ ] f = L 3 0 L 4 T, 27 [ M s ] f = M s, M s,y 0 T, l1 M = s, + 2 L l 2 M s,y Equtions 25 nd 29 cn no e used to estimte the unknon pmetes. The estimtion of l i, i = 1, 2, 3 ill e moe ccute if moe dt e used ith diffeent configutions. Theefoe, one moe oot configution is used ccoding to Figue 5, hich gives Fom 23 e no get [ s/ ] f = l 1 l 2 T, 30 [ ] f = 0 L 2 T, 31 [ M s ] f = M s, M s, 0 T l1 2 = 0 M s, + 2 M s,. 33 Equtions 25, 29 nd 33 cn no e itten s one

7 system of equtions ccoding to c1 M s,,c1 + 2 c1 0 2 c1 0 c l M s,y,c1 1 l 2 = M s,,c2 + c2l c c3 2 M, 34 s,y,c2 }{{} M s,,c3 + 2 c3 c l M s,,c3 }{{}}{{} A hee inde ci, i = 1, 2, 3 indictes fom hich oot configution the mesuements come fom. Eqution 34 hs moe os thn unknons, hence the solution to 34 is given y the solution to the optimistion polem g min l hich hs the nlyticl solution Al 2 2, 35 ˆl = A T A 1 A T. 36 Thee eist ette numeicl solutions to 34 thn 36, e.g. l=a\ in MATLAB. The pocedue to estimte the position of the cceleomete is summised in Algoithm 2. Algoithm 2 Estimtion of the mounting position 1 Mesue the cceletion of the tool [ M s ] s nd the ngul velocity fo the thee diffeent configutions in Figues 3, 4 nd 5 hen vies fom min to m ith constnt ngul velocity. 2 Constuct A nd in Solve 34 ith espect to l, fo emple ccoding to 36. V. EXPERIMENTAL RESULTS In this section the poposed oienttion nd position estimtion method descied in the to lgoithms in Sections III nd IV is evluted using epeimentl dt. Fo Algoithm 1, the dt, i.e., the cceletion vlues, e collected duing 4 s fo ech one of the si configutions in Figue 2 using mple te of 2 kh. Fo Algoithm 2, the m ngul velocity fo joint 1 nd the cceletion mesuements e collected hen the oot is in the thee diffeent configutions ccoding to Figues 3, 4 nd 5. The m ngul velocity fo joint 1 is computed fom the moto ngul velocity m using, m = τ, 37 hee τ is the ge tio. In the position estimtion epeiments dt e collected duing 4 s in ech one of the thee configutions, ut it is only the constnt ngul velocity pt of the dt tht is used. The sme smple te s efoe is used, i.e., 2 kh. The cceleomete used in the epeiments is tiil cceleomete fom Cosso Technology, ith nge of ±2 g, nd ensitivity of ppoimtely 1 V/g [7]. The cceleomete is connected to the mesuement system of the oot contolle, nd hence the cceletion nd moto ngul velocity cn e synchonised nd mesued ith the sme smpling te. 1 y 2 3 y 4 5 y Fig. 6. Oienttion fo the five mounting positions tht ee used to evlute the to lgoithms. The oienttion of the se coodinte system nd the desied coodinte system e lso shon. Five diffeent mounting positions nd diffeent oienttions of the cceleomete hve een used fo evlution of Algoithms 1 nd 2. The ctul physicl oienttion of the senso s mesued using potcto, see Figue 6, hee the oienttion of the desied senso coodinte system lso is shon. Algoithm 1 s pplied to the five test cses pesented ove nd the esult ˆR, ˆκ nd ˆρ0 cn e seen in Tle I. Fom Figue 6 e hve tht the ottion mti R in 1 should esemle R 1 = 0 0 1, R 2 = R 3 = , R 4 = c 3 d R 5 = , c 5 d 5 0 hee,, c nd d e positive numes tht should e close to cos The supescipt indictes the test nume. A ottionl diffeence eteen the mesued ottion mti R i nd the estimted mti ˆR i cn e computed using the coesponding unit qutenions q i nd ˆq i. The ottion ngle ϑ i of q i fom qi = q i 1 ˆq i, hich should e smll, is good mesue of the diffeence eteen R i nd ˆR i. See e.g. [5] fo hot intoduction to qutenions. The esulting ottion ngle ϑ i fo the five test cses cn e seen i Tle II. The diffeence is smll in ll cses ut fo test 3 nd 5 lge devition cn e seen. One eplntion fo this is tht it is moe difficult to mount the cceleomete in configution not ligned ith the oot tool, s seen in Figue 1. It is moe difficult to otin tue vlues fo the pmetes κ nd ρ 0. To veify them, the mesued cceletion fo ll five test cses in configution 1, in Figue 2, is tnsfomed fom O y to O, hich esults in thee constnt signls M s,, M s,y nd M s, fo the thee es of the cceleomete. Figue 2 shos tht the mesued cceletion in fme O should esemle, = 0,,y = 0 nd, = g. Sutcting,j fom the men of M s,j, j =, y,, gives n eo fo the tnsfomed cceletion. A digm of the eos fo ech coodinte is in O is shon in Figue 7. The digm shos the medin s the centl mk, the edges of the o e the 25th nd 75th pecentiles nd y s y

8 TABLE I ESTIMATED PARAMETERS IN 1 USING ALGORITHM 1 FOR FIVE DIFFERENT TEST CASES. Test ˆκ ˆρ 0 ˆR TABLE II THE ROTATION ANGLE ϑ INDICATES HOW CLOSE THE ESTIMATED AND MEASURED ROTATION MATRICES ARE TO EACH OTHER. THE MATRICES ARE IDENTICAL IF ϑ = 0 Test ϑ the dshed lines etend to the most eteme eo. The eos e smll nd, s epected, the eos e lge in nd y due to the highe sensitivity to oienttion eos in these is hen mesuing gvity long the -is. The is in cn e eplined y ystemtic eo in oienttion due to the oot elsticity nd gvittionl foce cting on the oot in the evlution position, see Figue 1. Algoithm 2 s lso pplied fo the five test cses. Figue 8 shos ho the mesued dt, i.e., the cceletion in O nd the m ngul velocity, cn look like hen the oot is in the configution ccoding to Figue 3. Note tht it is only the sequence hee the ngul velocity is constnt, in this cse ound 3 d/s, tht is used. Fom Figue 3 e see tht the cceletion in the -diection only oiginte fom the gvity hich is veified y Figue 8. We lso see tht the cceletion due to the cicul motion should e in the negtive -diection nd in the positive y Acceletion [m/s 2 ] Fig. 7. Digm of the tnsfomtion eos in the -, y- nd -diection fo 1 in configution 1 Figue 2 fo ll five test cses. The centl mk is the medin, the edges of the o e the 25th nd 75th pecentiles nd the dshed lines etend to the most eteme eo. TABLE III ESTIMATED POSITIONS ˆl OF THE ACCELEROMETER IN THE COORDINATE SYSTEM O y FOR FIVE DIFFERENT MOUNTING POSITIONS. IS Acceletion [m/s 2 ] THE ERROR RELATIVE THE MEASURED POSITION l M. Test Estimted position l [cm] = ˆl l M [cm] T T T T T T T T T T y Time [s] Mesued cceletion in O. Angul velocity [d/s] Time [s] Mesued m ngul velocity. Fig. 8. Mesued dt, to e used to estimte the position l, fo test 1 hen the oot is in the configution ccoding to Figue 3. y-diection hich is the cse in Figue 8. Hence, the tnsfomtion fom O y to O, given y the identified pmetes in 1, is coect. The estimted position ˆl fo the five test cses cn e seen in Tle III. Note tht ˆl2 fo test five is negtive hich comes fom the fct tht the senso is plced on the othe side of the eight thn s used in the deivtion in Section IV. The tle lso shos the eo eteen ˆl nd the mesued position l M. The position s lys mesued using tpe mesue to the cente of the cceleomete, since the position of the oigin of the cceleomete s coodinte system inside the senso is unspecified. Consideing the ccucy of the mesuements nd the uncetinty of the oigin of the cceleomete coodinte system the esult in Tle III is consideed s cceptle. The ctul equiement of the esult, in tems of position nd oienttion ccucy, ill depend on the ppliction hee the cceleomete is used. A moe detiled investigtion of the equiement fo the ccucy in the dynmic position nd oienttion estimtion of the tool position, such s descied in [8], is left s futue ok. VI. CONCLUSIONS A method to find the position nd oienttion of tiil cceleomete mounted on i DOF oot is pesented. The method is divided into to min steps, hee in the fist step, the oienttion is estimted y finding the tnsfomtion fom the ctul coodinte system of the cceleomete, ith unknon oienttion, to ne coodinte system ith knon oienttion. It is lso possile to find the sensitivity nd the is pmetes. The estimtion of the oienttion is sed on sttic mesuements of the gvity vecto hen

9 the cceleomete is plced in diffeent oienttions using the si DOF oot m. In the second step of the method, the mounting position of the cceleomete in oot fied coodinte system is computed using sevel epeiments hee the oot is moving ith constnt speed. Finlly, the method is evluted on epeimentl dt. The esulting position nd oienttion ccucy e evluted using mesuements on the physicl system. The oienttion eo is in the nge 1 to 6 degees nd the position eo up to 2 cm. The ccucy is sufficient in epeiments ith dynmic position nd oienttion estimtion of the tool position using senso fusion methods, such s etended Klmn filte nd pticle filte. ACKNOWLEDGEMENTS This ok s suppoted y the Vinnov Ecellence Cente LINK-SIC t Linköping Univesity. REFERENCES [1] E. L. Renk, W. Collins, M. Rio, F. Lee, nd D. S. Benstein, Cliting tiil cceleomete-mgnetomete using ootic ctution fo senso eoienttion duing dt collection, Contol Systems Mgine, vol. 25, no. 6, pp , Deceme [2] S.-h. P. Won nd F. Golnghi, A tiil cceleomete clition method using mthemticl model, IEEE Tnsctions on Instumenttion nd Mesuement, vol. 59, no. 8, pp , Aug [3] B. K. P. Hon, H. M. Hilden, nd S. Neghdipou, Closed-fom solution of solute oienttion using othonoml mtices, Jounl of the Opticl Society of Ameic, vol. 5, no. 7, pp , July [4] B. K. P. Hon, Closed-fom solution of solute oienttion using unit qutenions, Jounl of the Opticl Society of Ameic, vol. 4, no. 4, pp , Apil [5] L. Scivicco nd B. Sicilino, Modelling nd Contol of Root Mnipultos, 2nd ed. London, UK: Spinge, [6] M. W. Spong, S. Hutchinson, nd M. Vidysg, Root Modeling nd Contol. John Wiley & Sons, [7] Cosso Technology, Acceleometes, High Sensitivity, LF Seies, CXL02LF3, Jn. 2004, [8] R. Heniksson, M. Nolöf, S. Moeg, E. Wenholt, nd T. B. Schön, Epeimentl compison of oseves fo tool position estimtion of industil oots, in Poceedings of 48th IEEE Confeence on Decision nd Contol, Shnghi, Chin, Deceme 2009, pp

10 Avdelning, Institution Division, Deptment Dtum Dte Division of Automtic Contol Deptment of Electicl Engineeing Spåk Lnguge Rppottyp Repot ctegoy ISBN Svensk/Sedish Licentitvhndling ISRN Engelsk/English Emensete C-uppsts D-uppsts Övig ppot Seietitel och seienumme Title of seies, numeing ISSN URL fö elektonisk vesion LiTH-ISY-R-3025 Titel Title Method to Estimte the Position nd Oienttion of Tiil Acceleomete Mounted to n Industil Mnipulto Föftte Autho Ptik Aelsson, Mikel Nolöf Smmnfttning Astct A novel method to nd the oienttion nd position of tiil cceleomete mounted on i degees-of-feedom industil oot is poposed nd evluted on epeimentl dt. The method consists of to consecutive steps, hee the st is to estimte the oienttion of the senso dt fom sttic epeiments. In the second step the senso position eltive to the oot se is identied using senso edings hen the senso moves in cicul pth nd hee the senso oienttion is kept constnt in pth ed coodinte system. Once the cceleomete position nd oienttion e identied it is possile to use the senso in oot model pmete identiction nd in dvnced contol solutions. Comped to pevious methods, the senso position estimtion is completely ne, hees the oienttion is found using n nlyticl solution to the optimistion polem. Pevious methods use pmeteistion hee the optimistion uses n itetive solve. Nyckelod Keyods Rootics, Acceleomete, Estimtion