Analysis of simultaneous 3D positioning and attitude estimation of a planar coil using inductive coupling

Analysis of simultaneous 3D positioning and attitude estimation of a planar coil using inductive coupling Antonio Mosch itta, Alessio D e Angelis, Marco Dionigi, Pao lo Carbone Dept . of Engineering, Univers ity of Per ugia Perugia , Italy {antonio.moschitta,a lessio.deangelis,marco .dionigi,paolo.carb one}@unipg.it Abs tr ac t — In th is pap er , sim ult ane ous est ima ti on of 3D po sit ion an d at titu d e o f a s ing le coi l u sin g a set of anch or s, w ith kn own po sit io n and ma gne tic dip o le, is ana l yzed . Effe c t of nois e an d geom et ric prope r tie s of the anch ors ’ con st el lat ion is con s ide red . Severa l par am ete rs are analy zed and disc usse d , in clu din g pla cem ent of anc ho rs in a sin gle or in mu ltip le or tho gon al plan e s. It is sho wn tha t addin g spac e and orien ta tio n div e rsi ty a nch or s m ay lead to a m ore rob us t pe r form an ce whe n t he m obi le n od e a tt itu de chan ge s in tim e . Ke yw ord s — 3D pos iti oni ng; attit ude ; m agne ti c ; induc ti ve cou pli ng I. I NTRODUCTI ON Magn et ic fiel ds are a wel l-know n solu ti on for shor t rang e pos it ion ing sy st ems, an d sub ject of rec ent rese arch activ itie s sin ce they consent accurat e range and posi ti on measu rem ent s. Unl ike thos e devel oped usin g RF or ult ra soun d techn olog ies , sys tem s base d on magn eti c fie lds are robus t to mult ip ath and Lin e of Sig ht obs tru cti ons , and ca n be rea liz ed usi ng low- cost ele ct roni c circu it ry. Som e soluti ons in the liter atur e use senso rs to buil d magn eti c m aps of th e env i ronm ent for pos it ion ing pur pos es, as in [1] . Bett er ac cu racy may be o btain ed by des ign ing a m agn eti c po siti on ing sy stem comp ris ing an inf ras tru ctu re with a rtif ic ial m agn etic fiel d sour ces, thus elim in atin g the need for extens ive dataset coll ec ti on [2]. Fu rthe rm ore , oth er pr opos ed solut ion s are bas ed on DC magn eti c fi el ds or perm anen t magn ets, as in [3]. Comp are d to th ese syst ems , arc hit ectures explo iting A C magne tic fie lds , wh ich are typic ally based on coil s, allow for larg er operat ional rang e an d may reduc e pow er con sumpt ion if reson ance is em ployed [4]. Fur therm ore, for specif ic applic ati ons, comm ercia l syst ems using magn etic fie lds are avail able, prov iding high accura cy ove r a s hort range [5][6]. In this cont ext, 3D posit ioning can be achiev ed using tri- ax ial coils , i.e. th ree coil s, orth ogonal t o each oth er, in th e mob ile node and/ or in the anch ors [7][8]. How eve r, this can lim it the usa bility of the system , since the mobi le no de may no t be eas ily manuf acture d or ha ndle d, unle ss its s ize is kep t very sma ll . Moreov er, coil mis alignm ents due to constru ction hav e a sign ific ant im pact on posi tion ing accu racy [ 9] . Note that , to mit igate su ch problem s, a solu tion bas ed on plan ar coils may be im plement ed, als o usin g Prin ted Cir cuit Boa rd (PC B) tec hnolog y. This , in turn, may limit the operati onal range of the sys tem and its accu racy, sin ce, accor ding to the Fara d ay Neu mann Len z law, the coi l size aff ects the magn etic coupl ing. Con sequen tly, sys tems aim ed at locatin g single coil s mov ing in spa ce hav e been analy zed an d publ ished [ 10]-[ 13] . In th is pa per, design cri teria for s hort rang e six-d egrees- of - fr eedom (6DoF ) posi tionin g of a mobil e nod e equi pped with a sim ple plana r coil are inves tigate d. The consi dered system is as sum ed to rely on Recei ved Sign al Streng th me asurem ents, i.e. root me an squ are v oltag e measu rement taken by the mobil e nod e, and aims at accu rately loca ting a node opera ting with in les s th an 2 m off the beacon s, as for in stanc e in bi o me tric appl icat ions. Var ious param eters are cons idere d, tha t inclu de pla cement of fixe d anch ors, Sign al to Noise Rati o (SNR ) at the rec eiver’ s outpu t, and the sen sitiv ity of the posit ion est imation fit ting to ina ccu rate kn o wl edge of anc hors’ positi on and magn etic dipol e moment s . Not e that techn ologi cal param eters suc h as bea cons’ an d m obil e n ode’s implem entati on criter ia hav e been dee ply analyz ed in the lit eratur e [2]-[ 10 ]. This an alysis , base d on Mont e Carl o simu lations , aims at assess ing th e influe nce o f param eters rela ted to anch or plac ement, sen sitiv ity to inac curat e kn owledg e of anch ors param ete rs, and nois e s ensitiv ity . To th e au thors ’ know ledge , this issu e h as not bee n previo usly analy zed. In a sho rt rang e system the po siti on Fig. 1 – The considered system archit ecture . The mobile coil is represe n ted b y th e red circle, while the coils acting as anchors are represe n ted by the blue c ircles. For each coil, a n arro w is show n, describing its orie n tation. © 2017 IEEE. Per sona l use of t his material is permitted. Permiss ion from IEEE m u st be o b tained for all other us es, in any current or future media, incl u ding reprinting/re p ublishing t h is mater i al for adver t ising or promotional pur p oses, cre ating new c ollective works, for re sale or r edistribution to ser vers or lists, or reuse of any copyrighted component of this wo rk i n other w orks Preprint versio n. Presented at:I EEE Inter national Instrumentat i on and Meas urement Technology C onfere n ce (I 2 MTC), Turin, 20 1 7. Link to the fi n al publis h ed versio n: https://doi.org/10. 1109/I2MT C.2017.7969848 of the anch ors can be cons ide r ed as a design deg ree of free dom, bec aus e the envi ronmen t is cont rolled. Cons equen tly the prop ose d analy sis aim s at pro viding usefu l design cri teria w h en rea lizing a shor t ran ge positi oning system based on induct ive cou pling of AC m agne tic fiel ds. It is sh own tha t alloc ating an chors in a tridim ension al patte rn may resu lt in a mor e robust perf orman ce w ith resp ect to plac ing all anc hors in a sing le pla ne, which is the typi cal solu tion men tione d in the lite rature [10] . The rest of this pape r is organ ized as foll ows. Sect ion II des cribes the consi dered sy stem arch itectu re, whi le Secti on III prov ides sim ulation res ults. Fir stly th e eff ect on posi tionin g acc uracy of anch or placem ent, is stud ied, compar ing a pla nar arr ay of anchors to a 3D tri-pl anar array . Then the effe ct of in accura te kn owledge of anch ors’ par ameters is analy zed. Fin ally, the effect ive op era tional area of th e posi tioning arc hitec ture is inves tigat ed. II. S YSTEM A RCHITECT URE The arc hit ectu re of the cons idere d sy stem is descri bed in Fig. 1, an d consis ts in a se t of ancho rs, realiz ed by plan ar coils , des cribe d by th eir mag netic dipol e m oment . The m obile coil is repr esent ed as a red cir cle, wh i le the beac ons are rep res ented as blu e circ les. Withou t loss of gene rality , th roughout this paper th e an chors are assum ed to act as active beac o ns , w hile th e mob ile coil is assum ed to act as a re ceive r. Root me an square vol tage ( V rms ) measu rement s are assum ed to be coll ecte d at the mob ile coil’s output , usin g a hig h input imped ance measu ring dev ice. Each beacon operat es at a known frequ ency, and it is as sum ed that the mobil e node can discrim inate transm ission s fr o m dif ferent bea cons. Throug hout thi s paper, we assu med oper ation s under steady s tate con ditions , and tha t all bea con coi ls were stimul ated by the same sinu soida l current with pe ak va lue 2 0 I I  , w here I is th e rms value , an d f reque ncy f 0 , giv en by ) 2 sin( ) ( 0 0 t f I t I   . (1) Eac h beacon was modele d as a coil of radius r with N w win dings , with center coor dinat es ( x b , y b , z b ). The coil orienta tion wa s descr ibed by a vers or (i .e. a unit length vector ) b n  , mod eling the dire ction of the coil axis. Henc e, usin g phas or not ation , the ma gneti c dipol e momen t of the i -th coil is given by. 1 ,..., 0 , , 2 , ,     B i b w i b N i r S n SI N m    , (2) wh ere S is the coi l’s area , N B is the numbe r of beac ons, an d I is th e curr ent stim ulati ng the coil . By assum ing know n the magne tic dipo le mom ent o f each bea con , the magne tic fie ld produ ced by each beacon in a give n po si tion is                3 , 5 , 0 3 4 i i b i i i b i i d m d d m d B          , (3) wh ere i d  is the dist ance vect or conne cting the cent er of the i -th bea con to the cent er of the mobi le c oil. Eq. (3) can be sim plifie d as fol lows:     i b c bi c bi i b i i m n n m d B , , , , 3 0 3 4           , (4) wh ere c bi n ,  is the unit vecto r ass o ci ate d to i d  . Usin g (4), and as sum ing the mob ile coil o utp ut to be c o nn ected to a high im pedanc e measu ring device, the rms voltag e at the mo bile coil out put, induce d by th e i -th beaco n, is giv en by 1 ,... , 0 , 2 , 0 ,     B c i c c w i rms N i n B S N f V    , (4) wh ere c n  is a uni t vec tor descri bing the m obile coi l orient ation . The simula tion model ass umes that V rms ,i is est imated by sam pling the vol tage sinew a ve at the mobil e coil’s outpu t, corr upte d by an Add itive White Gaus sian Nois e (AWGN ), wit h st andard d eviati on  . Note th at, in a pra ctica l sc enari o, the nois e lev el can be measu red when the activ e beac o ns are turne d off. Thu s, a s et of N B nois y V rms measu remen ts is coll ecte d, one for eac h beacon . Then , these mea surem ents are us ed to evalu ate the fo ll owing cost functi on   ] , [ , ) ( ˆ ) ( 1 0 2 , , e N i i r m s i rm s n P V V F F            , (5) wh ere the argum ent ] , [ e n P     is compos ed b y ] , , [ z y x P   and e n  , th at des cri be the mobi le node posi tion and atti tude, res pectiv ely, i r m s V , ˆ is the me asured rms volt age in duced by the i -t h beac on, an d ) ( ,  i r m s V is th e voltag e th at sh ould be m easu red 0 0.5 1 1.5 2 0 0.5 1 1.5 0 0.5 1 1.5 2 z x y 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 z x y Fig. 2 – Estimat i on o f position and attitude of a mobile coil (taking positions repre sented by black points, with random attitude), using a single planar array of 28 coils. Fig. 3 – Es ti mation of position and a t titude of a mobile coil (taking positions represe n ted by black points, with random attitude), using 3 orthogonal pla n ar array s, for a total of 27 coils. in absenc e of no ise if the mobile node pos ition and att itude w ere ex actly P  and e n  . Note that , when the rece ived sign al power is ve r y low, th e res ult ing V rms mea suremen t can be domin ated by nois e, and using it in (5) can reduc e the accura cy of the fo ll owing fit ting . Con sequ ently, a Signal to Noise Ratio ( SNR ) th reshold SNR th , expres sed in dB, was defin ed, such tha t th e sim ulator can disca rd noisy measu rements . As a consequen ce, (5) ev aluat es a subset of N F measu rement s, out of th e N B av ailabl e ones . This proc edure can also incre ase pr ocessin g spe ed, becau se the comput ati onal com plexity of the num erical fit ting g rows with the nu mber o f V rms me asu rements . Thus, the mobile co il position and attitude can be estimated by minimizing (5 ) with respect to ] , , [ z y x P   and e n  , b y means of numerical techniques. In this paper , (5) was minimiz ed using the Nelder-Mead algorithm. Note that, being the optimization iterative, an initial esti mation of t he position is needed. T he initialization is described in section III.B. The simulation environment, written i n Mat lab, was used to test several scenarios. The results are describ ed in the following section. III. A NALYSIS AND R ESULTS A. Test co nditions an d performance metrics The sim ulati o ns w ere ini tially run by assum ing norm alized magn etic dipol e mom ents (i. e. w ith unita ry magn itude), aiming at com paring th e eff ectiv eness of diffe rent config urati ons of an chors, and at assessin g sensi tivi ty to no ise of the consi dered sys tem. In a pra ctic al scen ario (2) shows that su ch a magne tic dipo le magn itud e may be achi eved on ly using a larg e numb er of win dings , a lar ge coi l area , or a larg e stim ulating curre nt, and th at a low magni tude of (2) resu lts in low coupl ing with the mob ile coil . Moreov er, reducin g the coil size may help spaci ng th e anchor s away, reduc ing unc ertain ty sourc es due to mutu al cou pling betw een the anc hors. How ever, it is wel l know n that th e usag e of high Q reson ant coils can mitig ate this issu e when AC magn etic fi elds are us ed [4 ]. The an alysis w as org anized as fo ll ows: first , the effect of arrang ing an chors in differe nt pat terns was an alyzed . Then , sensi tiv ity to th e selecti on of the SNR thresh old SNR th wa s inv estiga ted. F ollowing that , the effect of inac curacie s in the know ledge of the bea cons posi tions and magn etic dipo le mom ent on the fittin g accura cy was analyz ed (se e subsec tion II I- D) . Throug hout Secti on III, a fr equency f 0 =200 kHz was assum ed, and an AWG N with  =10  V was con sidere d aff ecting th e acqu isi tion of the mo b il e nod e’s si gna l. As a firs t approx imati o n, the selec ted nois e level is compat ible with the output ref erred nois e of an Inst rument ation Amp lifier, boos ting the mo bile coil’s outpu t . Fin ally, a more real istic scena rio w as consi dered (se e su bsecti on III- E) , postu lating spec ific va lues for the coil s radi uses , num ber of windin gs, and currents . Couple d with the se lect ed no ise, these parame ters w ere chosen so as to des cribe a rea listi c scen ario, wher e a mobi le node oper ating w ithin 2 m off th e anch ors is to be locate d. Sim ulatio n resu lts we re com pared against a se t of me aningfu l metri cs. In parti cular, for eac h cons ide red scen ario th e empiri cal Cumu lativ e Distri bution Funct ion (CDF) o f both pos ition ing err or and attitu de err or we re deriv ed. Th e pos ition ing error was defin ed as the Eucl idean dist ance betw een th e true mob ile coil positi o n and the estim ated one , whil e th e angu lar error was defin ed as th e angle between the est imated att itude vers or and the true one. Note that angula r erro rs can tak e value s in the [-90°, 90°] int erval, becaus e an angle of 180° does not corre spond to variati ons in the rec eive d V rms . As an addi tiona l perf ormanc e m etric, t he fra ction of me asurem ents lea ding to a posi tionin g error not exc eed ing 1 cm and to an angu lar error n ot exc eeding 1° w as eval uated. B. Pla cement of an chors In t his test , tw o conf igurat ions of anch ors were cons idered. Wit h respec t to a 3 axis Cartes ian refe rence syst em, the first con sidere d configu ration is an array of 28 coi ls, lyin g on the xy pla ne, in a 7x 4 m atrix cov ering a squ are of abou t 1.5mx 1.5m. The secon d consi dered conf igurati on is a set of 3 pla na r array s, res pectiv ely locat ed on the xy , xz , and yz plan es. Each plana r arr ay is a 3x3 squar e matri x, cov ering an are a of abou t 1. 2m x1.2m. In both conf igurati ons, for each array of coplana r coi ls, the mag netic dipole mom ents are orthog o na l to the plane hos ting th e coil s. Both beac o n confi gurati ons were teste d again st the same set of posit ions assum ed by the mobil e coil , coverin g a squ are area on a verti cal plane, para llel to the xz pl ane, for a total of 400 pos itions . The two scen arios are sh own in Fig . 2 and 3 res pectiv ely, where the beacons are shown as diamon ds and the mob ile coil pos ition s are sh o w n as bla ck poin ts . Note that , for eac h posi tion a ssumed by the mobil e coil , a ran dom bea ring w as con sidere d. Table I-a Configuration P d P  P d,  Tri-planar 0.984 1 0.969 Mono-planar 0.7735 0.9982 0.7398 Table I-b Configuration P d P  P d,  Tri-planar 0.785 0.9995 0.749 Mono-planar 0.681 0.9955 0.609 Fig. 4 – Euclidean error CDF and angular erro r CDF, for both mono-plan a r and t ri-planar beacon arrays, assuming t hat the initial estimation of  i s affected by u ncertainty. 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0. 05 0 0.5 1 x [m ] F x (x) euclidea n err or CD F mono- plana r tri-pla nar 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 x [deg] F x (x) ang ular err or CD F mono- plana r tri-pla nar Un der the sta ted condit ions, two sets of simula tions we re run . The f irst on e (case ‘a’) , aimed a t asses sing th e robus tness t o AWG N of th e cons idered conf igur ations , was perf orm ed by usin g as initi al cond ition the true po sition /attit ude, wh ile th e se cond set of sim ulati ons (case ‘b’ ) wa s run by assum ing unc ertain ty in th e ini tial guess for  . T o this aim, a rand om err or, unif ormly dist ribut ed in [- 10 cm, 10 cm] , was adde d to eac h Car tesian coordi nate of the init ial gues s. Simil arly, b ot h th e eleva tion the and azim uth ang les asso ciated to the attitu de ve r so r wh ere aff ected by a random error, u niform ly distr ibu ted in [ - 18° , 18° ]. In parti cula r, Tabl e I-a and I- b show th e es tim ated prob abi lity P d of positi on estima tions meeting the 1 cm target f or the Eu clide an error e d , th e estima ted pro babil ity P  th at a po sition measu rements meets the t arget angu lar er ror e  of 1°, and th e join t prob abi lity P d,  that a m easurem ents me ets both tar gets simul taneous ly. Tabl e I- a was obtai ned unde r cas e ‘a’ , Tabl e I- b w as ob taine d under case ‘ b’. Un der both scen arios, the tri -plan ar config urat ion result ed in a bette r accu racy with res pect to positi on estimat ion, whi le the two solu tions are com parabl e wh en estim ating the coil at titude . Fig. 4 shows the CDF of both the Euc lid ean and angula r errors obta ined under cas e ‘b’. Using the tri- planar configu rati o n, th e Eu clide an error was less than 25 cm in 99% of th e simu lat ed me asur ements, with a sing le out lier (1 case out of 4000 tr ials ) sh o win g e d =70cm , due to fail ed conv ergen ce o f th e fitt ing algo rith m . When usin g the mon o - plan ar conf igurat ion fai led con vergenc e occ ur red m ore f requ ently (4 case s out of 4000 tri als) w ith larg er pos ition ing err ors. The reas on behin d the bet ter perf ormance of the tri-plan ar conf igura tions is bot h in the redu ced averag e distan ce be t wee n th e mobil e coil and th e coil a ctin g as anch ors, and in the in crease d atti tude dive rsity. In f act, it w as obs erved , tha t, for a giv en posi tion and orienta tion, the mo bil e coil may be scarcely cou pled with most of the coil s lyin g on a plan e. In this cas e, the coi l on the remain ing plan es are st ill likely to be fa irly coup led to the mobile coil, becaus e of their diffe rent ori entati on. Fig . 5 , obta ined when SN R th =10 dB, sh ows the frac tion of V rms me asuremen ts c ollect ed by us ing th e tw o be acon conf igura tions . Note that in most of the pos itions assum ed wit h ran dom atti tude by the mobil e coi l, the tri-pl anar conf iguratio n lea ds to a larg er numbe r of beacon s provi ding mean ingful inf ormati on tha n the mon o-plan ar conf igura tion. C. SNR threshold Fol lowing the com pared analy sis in Su bsect ion III-B , es tim ation of  was fu rther tested assumin g to use the tri-plan ar arr ay of anc hor coils . This tim e, variou s t ests were run by ch anging SN R th , fo r the tr i-plan ar beacon conf igurati on. For th e con sidere d tes t setu p, a SNR thres hold of 15 dB was obs erved to prov ide opt imal perf ormance . Figs . 6, shows the proba bility th at Euc lidean and ang ul ar err or are les s th an 1cm and 1° res pectiv ely, and in Table II , that shows mean valu e an d st andard dev iation of both Eucli dean and angula r error . Note th at the existe nce of an optim al valu e SNR th was expect ed. In fa ct, for low valu es of SNR th many V rms measu remen ts dom inate d by nois e mak e their way into the nume rical fitt ing alg orithm , incre asing the noise eff ect on the estim ation of  . Co nvers ely, w hen a high value of SNR th i s select ed, few er and few er V rms mea suremen ts a re deeme d elig ible for fitting , pos sibly disc arding measu rements carry ing usefu l info r mat ion fo r th e est imation of  . D. Sen sitivity to inaccurate kn owledge of anch ors The sen sitivi ty to anchor s’ param eters was invest igate d usin g Monte Carlo simul ation. At each iterat ion, when sim ulating the measu rement of V rms , the sof tware introd uced a ran dom unif ormly distri b ut ed pert ur bati on in the m agnet ic dipo le mom ents asso ciated to each bea con, whil e the num eric al min imiza tion of (5) use d the nom inal magne tic dipol e m o men t va lues w hen ev alua ting ) ( ,  i r m s V . Fig. 5 - Fraction of beacons providing V rms measurements above a S NR threshold of 10 dB, using the mono-planar beacon array (black) and the tr i- planar beacon co n figurat ion. 0 50 100 150 200 250 300 350 400 0 0.2 0.4 0.6 0.8 1 mobile coil positi on # fract ion of V rms measuremen t s above SNR th triplana r mono plan ar Table I I – Euclidean and angular e rror statistics as a function of SNR th . SNR th e d mean e d std e  mean e  std 0 dB 0.01559 0.044663 0.039248 0.10737 5 dB 0.014779 0.040018 0.040072 0.11001 10 dB 0.0148 0.0444 0.0373 0.0897 15 dB 0.013144 0.040419 0.036089 0.10394 20 dB 0.036005 0.16713 0.059954 0.17993 25 dB 0.72881 5.6534 0.20672 0.44474 30 dB 1,0462 5.1339 0.43236 0.63185 Fig. 6 – Probability that the Euclidean error e d is lower th an 1c m (upper plot), and probability that the angular error e  is lower t han 1° (lower plot), as a funct i on of SNR th . 0 5 10 15 20 25 30 0.4 0.6 0.8 1 SNR th [ dB] P(e d <1c m) 0 5 10 15 20 25 30 0.8 0.85 0.9 0.95 1 SNR th [ dB] P(e  <1°) Thi s proc edure wa s used to simul ate the eff ect of inacc urate know ledge of poin t of appli cation , direct ion and magn itud e of bea cons ’ magne tic dipol e momen ts. Errors in the point of appl icat ion can model errors in beacon s’ placemen t, while err ors in magn etic di pole mom ents bearing can m odel coils fa bricat ion toler anc es, an d m agnitu de er rors can des cribe unc ertain ty in current f eeding the coils . T he Monte Carlo an alysis, covering the scen ario in Fig. 3 un der cas e ‘a’ con dition s, l ed to the results summ arize d in Fig s. 7-12, th at sh o w Euc lid ean a nd a ngular error CDF un der v arious con dition s. In particu lar, Fig. 7 shows tha t the positi oning perf orman ce is mos t sensi tive to error s in the magn etic dipol e dir ection , sin ce for a max imum angu lar erro r of 5° P d drops fr o m 0.98 t o 0.27. Fig. 9 show s that the p ositi o nin g perf ormance is fai rly sensi tive to erro rs in the magn etic dipol e magn itude, sin ce a for max imum magn itude er ror of 5% P d drops from 0.98 to 0.58. Final ly, Fig . 11 show s that the posit ioning perf ormance is fai rly tol erant to beac on placem ent errors . In fact , wh en a max imum p la cement e rror of 5 mm on th e beacon s’ coordin ates is consi dered, P d drop s from 0.98 to 0.77. Simi lar conc lusio ns can be draw n f or the angu lar error , com parin g Fig s. 8, 10, and 12. How ever, the system angu lar accu racy is seemin gly more robu st to inaccu rate know ledge of bea cons, sin ce the obse rved angu lar error w as alw ays low er th an 0.5 °. E. An alysis of a realistic scenari o Fin ally, beac on and mob ile nodes wer e mo deled in gre ater det ail, assu ming bea cons to be 20 w inding coils wit h a radius of 3 cm , fed by a 2 A rms curre nt (th is value may be achi eved usin g high- Q reson ant coi ls). The m obile node was mod eled by a 10 w inding coi l with a radi us of 1 cm, f ollowe d by an am plifie r wi th gain G . A ccurate kn owledg e of the an chors ’ 0 0.02 0. 04 0.06 0.08 0.1 0.12 0. 14 0.16 0. 18 0 0.2 0.4 0.6 0.8 1 x [m] F(x) 0 1% 2% 3% 4% 5% Fig. 9 – Euclidean error CDF obtained by assuming a random uniformly distributed error in the magnitude of the beacons’ magnetic dipole moments, upper bounded b y 1 %, 2%, 3 %, 4%, and 5% respectively . Th e error-free c ase is shown for comparis on purposes. Fig. 10 – Angular error CDF obtai ned b y assuming a random uniformly distributed error in the magnitude of t he beacons’ magnetic dipole moments, upper bounded by 1%, 2%, 3%, 4%, and 5% respectivel y. Th e error-free ca se is shown fo r c omparis on . 0 0.05 0.1 0.15 0.2 0 .25 0.3 0.35 0.4 0 0.2 0.4 0.6 0.8 1 x [deg] F(x) 0 1% 2% 3% 4% 5% Fig. 7 – Euclidean error CDF obtained by assuming a random uniformly distributed error in the attitude of the beacons’ magnetic dipole moments, upper b ounde d by 1°, 2°, 3°, 4°, a nd 5° re sp ective l y. The error-free c ase is shown for comparis on purposes. 0 0.05 0.1 0.15 0.2 0.25 0 0.2 0.4 0.6 0.8 1 x [m] F(x) 0° 1° 2° 3° 4° 5° Fig. 8 – Angular error CDF obt ained by assuming a random uniformly distributed e rror in the attitu d e of th e beaco ns’ magnetic d ipole moments, upper bounded by 1°, 2°, 3 °, 4°, and 5° respectively. The error-free ca se is shown fo r c omparis on purposes. 0 0. 05 0.1 0 . 15 0.2 0.25 0.3 0.35 0 . 4 0.45 0.5 0 0.2 0.4 0.6 0.8 1 x [deg] F(x) 0° 1° 2° 3° 4° 5° Fig. 11 – Euclidean error CDF obtained by assuming a random uniformly distributed err or in the position of the beacons, upper bounde d by 1 mm , 2 mm , 3 mm ,4 mm , and 5 mm respective l y. The error- free case is shown for comparison purpose s. Fig. 12 – Angular erro r CDF obtained by assuming a random uniformly distributed error in the position of the beacons, u pper bounded by 1 mm , 2 mm , 3 mm , 4 mm , and 5 mm respectively. The error-free case is shown for compariso n purpose s . 0 0. 01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0. 09 0.1 0 0.2 0.4 0.6 0.8 1 x [m] F(x) 0 mm 2 mm 3 mm 4 mm 5 mm 0 0. 02 0.04 0. 06 0.08 0.1 0. 12 0.14 0. 16 0.18 0.2 0 0.2 0.4 0.6 0.8 1 x [deg] F(x) 0 mm 2 mm 3 mm 4 mm 5 mm param eter s was assum ed. In order to invest igate th e stabi lity of th e num eri cal fitt ing, the Ne lder-M ead algo rithm was fed wi th a fix ed n umber N of me asur ements, obt ained b y tak ing the N lar gest valu es am o ng the 27 V rms me asurem ents colle cted by the mob ile node . Simula tions we re initi ally run, und er the scen ario depi cted in Fig . 3, assuming G =20 (i. e. 26 dB), and N =7 . A perf orman ce compar able to the erro r fr ee cas e in Fig s. 7 -1 2 was obs erved , w ith an Euc lid ean m ean e rror of 1.8 cm and an angu lar me an error of 0.02° . How ever, by repeat ing the sim ulati o n w ith l arger v alues of N , f ailed conver genc e was o f ten obs erved , with larg e positi oning error s. To gain fu rther insigh t, th e positi o ns corr espond ing to Eucli dean er rors la rger th an 1 cm or to angul ar error s larger than 1° we r e sh o wn as red circ lets i n Fig . 13, obtain ed f or N =1 0 . Fig . 13 sh ows that larg e error s mos tly occu r w hen the mo bil e node is fa r f rom the ma jority of th e anch ors, lea ding only a few anchor s of the tri-p lan ar grid to prov ide u seful inform ation for f itting purpos es . Hence sim ulati o ns w ere repe ated by assum ing a small er operati onal are a, sh o w n in Fig. 14, fo r v arious valu es of G . Th is tim e, th e fit ting was fed wi th up to 12 V rms measu remen ts, chos en among th o se exce eding SNR th . As shown in Tab. III, in all consi dered cas es a m ean e rror of less than 1 cm w as obse rved. IV. C ONCLU S IONS Thi s paper is focu sed on crit eria to desig n and optim ize a pos ition ing sys tem base d on in ductiv e coupl ing, ca pable of es tim ating bo th the positi on and th e attitu de of a sin gle c oil . The pla cement of anchors was discus sed, show ing that placin g all an chors o n diff erent ortho gonal pla nes may lea d to incr eased acc uracy and robu stness . The s elec tion of co llected me asuremen t on a SNR basis w as inv estigat ed as wel l. Fin ally, sen sitiv ity t o inaccu racies in re alizin g the ancho r coi ls was inv estig ated, ke eping into account placemen t, ori enta tion, and cu rrent am plitud e errors . The ope rati on of a re alisti c syst em was als o sim ulated. A CKNOWL E DGEMENT Thi s res earch activi ty was fun ded through grant PRIN 2015C 37B25 by the Italian Min istry of Inst ructio n, Univ ersity an d Resea rch (MIUR ), whose suppo rt the autho rs grat efully ack now ledge . R EFERENCES [1] B. Gozick, K. P. Subbu, R. Dantu and T. Maeshiro, "Magnet i c Maps for Indoor Navigation," in IEEE Trans. Instrum. Meas. , vol. 60, no. 12, pp. 3883-3891, De c. 2011. [2] G. D e Angel i s, V. Pasku, A. De Angelis, M. Dionigi, M. Mongiardo, A. Moschitta and P. Carbone, "An Indoor AC Magnetic Positio ning Syste m," i n IEEE Trans . Instrum. Meas. , vol. 64, no. 5, pp. 1267-1275, May 2015. [3] V. S chlagete r, P. A. Besse, R. S. P opovic and P. K ucera, “ Tracking system with five degr ees of freedom using a 2D-array of Hall sensors and a p erma n ent magnet.” Sensors and Actuators A: Physical , 92 (1), 37- 42, 2001. [4] M. Dionigi, G. De Angelis, A. Moschitta , M. Mongiardo and P. Carbone, "A Simple Ranging S ystem Based on Mutually Coupled Resonating Circuits," in IEEE Trans . Instrum. Meas. , vol. 63, no. 5, pp. 1215-1223, May 2 014. [5] Polhemus Fastrak motion tracking sy s tem, www.pol h emus.com. [6] Ascension Technology Corporation, 3D Guidance t racking system, www . ascension-tech.com [7] T. E. A b rudan, Z. Xiao, A . Markham and N. Trigoni, "Distortion Rejecting Magneto-Inductive Three-Dimensional Localization (MagL oc)," in IEEE J. S el. Areas Commun. , vol . 33, no. 11, pp. 2404- 2417, Nov. 201 5. [8] C. Hu, S. Song, X. Wang, M. Q. H. M eng and B. Li, "A Novel Positioning and Orientatio n System Based on Three-Axis Magnetic Coils," in IEEE Tr a ns. Mag n. , vol. 48, no. 7, pp. 2211-2219, July 2 012. [9] A. Sheinker, B. Ginzburg, N. Salomo n ski, L. Frumkis a nd B. Z. Kaplan, "Localiz at ion in 3 -D Using Beacons of Low Frequency Magnetic F ield," in IEEE Trans. Instr um . Meas. , vol. 62, n o. 12, pp. 3194-3201, Dec. 2013. [10] A. Plotkin and E. Pape rn o, "3-D magnetic t racking of a single subminiature coil with a large 2 -D array of un iaxial transmitters," in IEEE Tra n s. Magn. , vo l. 39, no. 5, pp. 3295-3297, Se p t. 2003. [11] S. So n g, W. Qiao, B. Li, C. H u, H. Ren a nd M. Q. H. Me n g, "An Efficient M agnetic Tracking M etho d Using Uniaxial Sensing Coil," in IEEE Tra n s. Magn. , vo l. 50, no. 1, pp. 1-7, Jan. 20 14. [12] D. D. Arumugam, J. D. Gr iffin, D. D. Stancil and D. S. Ricketts , "Thre e - dimensional position and orientation measurements using magneto - quasistatic fields and comple x image theory ," in IEEE Antennas Propag. Mag. , vol. 56, no. 1 , pp. 16 0 -173, Fe b. 2014. [13] V. Pasku, A. De Angelis, G. De Angelis, A. Moschitta a nd P. Carbone, "Magnetic field analysis for dist ance measureme nt in 3D positioning applications," in IEEE I2MTC 2016 , T a ipei, 2016. 0 0.5 1 1.5 2 0 0.5 1 1.5 0 0.5 1 1.5 2 z y x Fig. 13 – Mobile node positions le ading to positioning outlie rs (red circle t s), obtained by ass u ming re alistic coils when s i mulating t he scenario of F ig. 3. 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 z y x Fig. 14 – Positio n ing outliers (red circlets), obtained by assuming realistic coils w hen simulating the scenario of Fig. 3, when the mobile node takes positions close r to th e area cov ered by the beacons. Table III – Euclidean and ang ular error statistics as a f u nction of G, under the conditions of F ig. 14 G e d mean e d std e  mean e  std 1 0. 00 59 0.03 0.09 0.93 5 0.015 0.0 024 0.0 02 4 0.03 10 0.012 0. 022 0.0 018 0.0 35 15 0.012 0.0 22 0.0 013 0. 027 20 0.012 0.024 0.0014 0.029