Recursive Total Least Squares: An Alternative to Using the Discrete Kalman Filter in Robot Navigation

Transcription

1 Recursive Toal Leas Squares: An Alernaive o Using he Discree Kalman Filer in Robo Navigaion Daniel L. Boley an Erik S. Seinmez Karen T. Suherlan Deparmen of Compuer Science Deparmen of Compuer Science Universiy of Minnesoa Universiy of Wisconsin La Crosse Minneapolis, MN La Crosse, WI Absrac In he robo navigaion problem, noisy sensor aa mus be filere o obain he bes esimae of he robo posiion. The iscree Kalman filer, commonly use for preicion an eecion of signals in communicaion an conrol problems, has become a popular meho o reuce he effec of uncerainy from he sensor aa. However, in he omain of robo navigaion, sensor reaings are no only uncerain, bu can also be relaively infrequen, compare o raiional signal processing applicaions. Hence, here is a nee for a filer ha is capable of converging wih many fewer reaings han he Kalman filer. To his en, we propose he use of a Recursive Toal Leas Squares Filer. This filer is easily upae o incorporae new sensor aa, an in our experimens converge faser an o greaer accuracy han he Kalman filer. 1 Inroucion The iscree Kalman filer, commonly use for preicion an eecion of signals in communicaion an conrol problems, has become a popular meho of reuc- This work was suppore joinly by Minnesoa Deparmen of Transporaion gran an Naional Science Founaion gran CCR

2 ing uncerainy in robo navigaion. A brief summary of he Kalman filer can be foun in [2] an a complee escripion in [9]. One of he main avanages of using he Kalman filer is ha i is recursive, eliminaing he necessiy for soring large amouns of aa. I requires a goo iniial esimae of he soluion. I also assumes ha he noise obeys a weighe whie gaussian isribuion. The Kalman filer is guaranee o be opimal only in ha i is guaranee o fin he bes soluion in he leas squares sense. Alhough originally esigne as an esimaor for ynamical sysems, he filer is use in many applicaions as a saic sae esimaor [13]. Also, ue o he fac ha funcions are frequenly non-linear, he exene Kalman filer (EKF) raher han he Kalman filer iself is ofen use [1, 11]. In his case, he funcion is linearize by aking a firs orer Taylor expansion. This linear approximaion is hen use as he Kalman filer equaion. There are wo basic problems which can occur when using eiher he Kalman or exene Kalman filer in robo navigaion applicaions: Due o he fac ha he filer was evelope for applicaions such as hose in signal processing, i is assume ha many measuremens are aken. Sensing in robo navigaion, ofen one using camera images, is a ime consuming process. To be useful, a meho mus succee wih relaively few reaings. An unerlying assumpion in any leas squares esimaion is ha he enries in he aa marix are error-free [7], e.g., he ime inervals a which measuremens are aken are exac. In many acual applicaions, he errors in he aa marix can be a leas as grea as he measuremen errors. In such cases, he Kalman filer can give poor resuls. Two aiional problems occur when using he EKF: The linearizaion process iself has he poenial o inrouce significan error ino he problem. The EKF is no guaranee o be opimal or o even converge [14]. I can easily fall ino a local minimum when an iniial esimae of he soluion is poor, ofen he ype of siuaion face by robo navigaors. Alhough limie moificaions can be mae o he Kalman approach o improve robusness o noise [12], our work in ouoor navigaion [17], where measuremens are expensive o obain an have significan error inheren o he sysem, moivae 2

3 Figure 1: In an LS soluion, as shown on he lef, he sum of he square verical isances o he line of bes fi is minimize. In a TLS soluion, as shown on he righ, he sum of he square perpenicular isances o he line of bes fi is minimize. us o look for anoher filering meho, preferably one which woul no require numerous measuremens o converge an i no assume an error-free aa marix. As emonsrae by Minz e al [8], he crierion of opimaliy epens criically on he specific moel being use. When error exiss in boh he measuremen an he aa marix, he bes soluion in he leas squares sense is ofen no as goo as he bes soluion in he eigenvecor sense, where he sum of he squares of he perpenicular isances from he poins o he lines are minimize (Fig. 1). This secon meho is known in he saisical lieraure as orhogonal regression an in numerical analysis as oal leas squares (TLS) [18]. In he nex secion, we iscuss he Recursive TLS algorihm, in secion 3 we presen our experimenal resuls, an in secion 4 we offer some concluing remarks. 2 Recursive Toal Leas Squares Algorihm Given an overeermine sysem of equaions Ax = b, he TLS problem, in is simples form, is o fin he smalles perurbaion o A an b o make he sysem of equaions compaible. Specifically, we seek a marix E an vecor f ha minimizes k(e; f )k 2 such ha (A+E)x = b+f for some vecor x. The vecor x corresponing o he opimal (E; f ) is calle he TLS soluion. Recenly, some recursive TLS filers have been evelope for applicaions in signal processing [4, 5, 20]. Davila 3

4 [4] use a Kalman filer o obain a fas upae for he eigenvecor corresponing o he smalles eigenvalue of he covariance marix. This eigenvecor was hen use o solve a symmeric TLS problem for he filer. I was no explaine how he algorihm migh be moifie for he case where he smalles eigenvalue is muliple (i.e., corresponing o a noise subspace of imension higher han one), or variable (i.e., of unknown mulipliciy). In [20], Yu escribe a meho for he fas upae of an approximae eigenecomposiion of a covariance marix. He replace all he eigenvalues in he noise subspace wih heir average, an i he same for he eigenvalues in he signal subspace, obaining an approximaion which woul be accurae if he exac eigenvalues coul be groupe ino wo clusers of known imensions. In [5], DeGroa use his approach combine wih he averaging echnique use in [20], again assuming ha he singular values coul be groupe ino wo clusers. Recenly, Bose e al.[3] applie Davila s algorihm o reconsruc images from noisy, unersample frames afer convering complex-value image aa ino equivalen real aa. All of hese mehos mae some assumpions ha he singular values or eigenvalues coul be well approximae by wo igh clusers, one big an one small. In his paper, we presen a recursive algorihm ha makes very few assumpions abou he isribuion of he singular values. The mos common algorihms o compue he TLS soluion are base on he Singular Value Decomposiion (SVD), a non-recursive marix ecomposiion which is compuaionally expensive o upae. The TLS problem can be solve by he SVD using Algorihm 3.1 of [18]. The main compuaion cos of ha algorihm occurs in he compuaion of he SVD. Tha cos is O(p 3 ) for each upae. The basic soluion meho is skeche as follows. If v = (v 1 ; : : : ; v p ) T is a righ singular vecor corresponing o he smalles singular value of (A; b), hen i is well known ha he TLS soluion can be obaine by seing x =?(v 1 ; : : : ; v p?1 ) T =v p. If he smalles singular value is muliple, hen here are muliple TLS soluions, in which case one usually seeks he soluion of smalles norm. If v p is oo small or zero, hen he TLS soluion may be oo big or nonexisen, in which case an approximae soluion of reasonable size can be obaine by using he nex smalles singular values(s) [18]. In cases such as he applicaions consiere in his paper where he exac TLS soluion is sill corrupe by exernal effecs such as noise, i suffices o obain an approximae TLS soluion a much less cos. We seek a meho ha can obain a goo approximaion o he TLS soluion, bu which amis rapi upaing as new aa samples arrive. One such meho is he so-calle ULV Decomposiion, firs inrouce by Sewar [15] as a means o obain an approximae SVD which can be 4

5 easily upae as new aa arrives, wihou making any a priori assumpions abou he overall isribuion of he singular values. The ULV Decomposiion of a real n p marix A (where n p) is a riple of 3 marices U, L, V plus a rank inex r, where A = ULV T, V is p p an orhogonal, L is p p an lower riangular, U has he same shape as A wih orhonormal columns, an where L has he form C 0 L = E F where C (r r) encapsulaes he large singular values of A, (E; F ) ((p? r) p) approximaely encapsulae he p? r smalles singular values of A, an he las p? r columns of V encapsulae he corresponing railing righ singular vecors. To solve he TLS problem, he U marix is no require, hence we nee o carry only L, V, an he effecive rank r. Therefore, a given ULV Decomposiion can be represene jus by he riple [L; V; r]. The feaure ha makes his ecomposiion of ineres is he fac ha, when a new row of coefficiens is appene o he A marix, he L, V an r can be upae in only O(p 2 ) operaions, wih L resore o he sanar form above, as oppose o he O(p 3 ) cos for an SVD. In his way, i is possible o rack he leaing r- imensional signal subspace or he remaining noise subspace relaively cheaply. Deails on he upaing process can be foun in [15, 10]. We can aap he ULV Decomposiion o solve he Toal Leas Squares (TLS) problem Ax b, where new measuremens b are coninually being ae, as originally propose in [2]. The aapaion of he ULV o he TLS problem has also been analyze inepenenly in grea eail in [19], hough he recursive upaing process was no iscusse a lengh. For our specific purposes, le A be an n(p?1) marix an b be an n-vecor, where p is fixe an n is growing as new measuremens arrive. Then given a ULV Decomposiion of he marix (A; b) an an approximae TLS soluion o Ax b, our ask is o fin a TLS soluion bx o he augmene sysem b Abx b b, where ba = A a T an b b = b an is an opional exponenial forgeing facor [9]. The RTLS Algorihm: Sar wih [L; V; r], he ULV Decomposiion of (A; b), an he coefficiens a T ; for he new incoming equaion a T x =. 5 ;

6 Compue he upae ULV Decomposiion for he sysem augmene wih he new incoming equaion. Denoe he new ecomposiion by [ b L; b V ; br]. Pariion V b bv b 11 V 12 = bv b ; 21 V 22 where b V 22 is 1 (p? br). If k b V 22 k is oo close o zero (accoring o a user supplie olerance), hen we can ajus he rank bounary br own o obain a more robus, bu approximae soluion [2]. Fin an orhogonal marix Q such ha b V 22 Q = (0; : : : ; 0; ), an le v be he las column of b V 12 Q. Then compue he new approximae TLS soluion accoring o he formula bx =?v=. This RTLS Algorihm makes very few assumpions abou he unerlying sysem, hough he user mus supply a zero olerance an a gap olerance for k b V 22 k. For he applicaion here, i suffice o se he zero olerance o zero an epen on jus he gap olerance of Experimenal Resuls To compare he performance of he Kalman filer an RTLS in pracice, we ran hree ses of experimens, incluing one wih a physical mobile robo an camera, an wo in simulaion. In he firs se of experimens, we simulae a simple robo navigaion problem ypical of ha face by an acual mobile robo [1, 6, 11]. The robo has ienifie a single lanmark in a wo-imensional environmen an knows he lanmark locaion on a map. I oes no know is own posiion. I moves in a sraigh line an wih a known uniform velociy. Is goal is o esimae is own sar posiion relaive o he lanmark by measuring he visual angle beween is irecion of heaing an he lanmark. Measuremens are aken perioically as i moves. Figure 2 shows a iagram of he problem. For simplificaion, i is assume ha he lanmark is locae a (0,0), ha he y coorinae of he robo s sar posiion oes no change as he robo moves (i.e. he robo heaing efines he x axis), an ha he robo knows wha sie of he lanmark i is on. To map his robobase coorinae sysem o he groun coorinae sysem, i suffices o know only he robo s compass heaing from, say, an inernal compass. Below we iscuss a 6

7 simple way o ynamically incorporae reaings from wo lanmarks, avoiing he nee o know he compass heaing inepenenly. Lanmark (x,y) α 1 α Robo Moves Figure 2: Diagram of a simulae robo navigaion problem. The robo moves along he horizonal line. Lanmark locaion an velociy are known. Angle i is he angle from robo heaing o he lanmark a ime i. The goal is o esimae he iniial robo locaion (x,y). In our experimens, i was assume ha he y coorinae of he robo pah was negaive (i.e., he pah, as shown in Figure 2, was on he sie below he lanmark), ha robo velociy was 20 per uni of ime an ha measuremens of were aken a uni ime inervals. A any ime i : co( i ) = x + i velociy y where (x; y) is he robo sar posiion an i is he angle from he robo heaing o he lanmark. Ranom error wih a uniform isribuion was ae o he angle measures an a normally isribue ranom error was ae o he ime measuremen. We formulae he problem so ha he aa marix, as well as he measuremen vecor conaine error: h A i = 1?co( i ) " i x ; x i = y # ; b i =? i velociy where, a ime i, A i is he aa marix, b i is he measuremen vecor, an x i is he esimae sae vecor consising of he coorinaes (x; y) of he robo sar posiion. The Kalman filer was given an esimae sar of (0,0). The RTLS algorihm 7

8 ha no esimae sar posiion provie. The leaing column of he aa marix was scale by = 100 o reuce he allowe errors. Resuls are summarize in Figure Figure 3: Comparison of mean eviaions from esimae o acual sar posiion. Measuremens were aken a uni ime inervals (horizonal axis). The verical axis gives he mean eviaion. The op hree graphs have uniformly isribue error in of 2 an normally isribue error in wih s = 0,.05 an.1. The boom hree graphs have uniformly isribue error in of 4 an normally isribue error in wih s = 0,.05 an.1. Resuls using he RTLS algorihm are shown in black. Resuls using he Kalman filer are shown in grey. Error in Error in Kalman RTLS Kalman RTLS Table 1: Mean eviaion of esimae from acual locaion afer 15 measuremens. The mean eviaions (of 10 rials) of he esimaes from he acual sar locaion of (-460, -455) are compare for six ifferen error amouns. The op hree graphs have uniformly isribue error in of 2 an normally isribue error in wih sanar eviaion s = 0,.05, an.1. The boom hree graphs have 8

9 uniformly isribue error in of 4 an normally isribue error in wih s = 0,.05 an.1. The jump in he RTLS isance a he secon measure is ue o he fac ha he RTLS filer oes no require, an is no given, an iniial esimae of locaion. The velociy/ime inerval use, combine wih he error isribuion use, prouce error on some runs ha gave reaings of 2 < 1 (see Figure 2). Since here were only wo measuremens aken a his poin, he sysem was no ye overeermine, an he erroneous measures were given significan weigh. This emonsraes how quickly he RTLS filer can recover from such errors. Table 1 gives he mean eviaion from he acual locaion afer 15 measuremens. For all six groups of experimens, he RTLS filer converge more quickly han i he Kalman filer. Afer 15 measuremens, he RTLS esimae was closer o he acual locaion han was he Kalman in five of he six groups. The secon se of experimens consise of a sequence of inoor robo runs. As in he firs se of experimens, he robo i no know is own posiion on he map, bu i know he locaion of a single lanmark. Is ask was o ake an image, fin he lanmark in he image, an use he resul o eermine is sar posiion relaive o he lanmark. Movemen Labmae Camera β β 1 2 Lanmark Figure 4: TRC Labmae wih camera moune a for he given fiel of view. Angle measure is boun by A Panasonic WV-BL202 camera was moune on a TRC Labmae a an angle of 90 o robo bearing. Horizonal fiel of view was Lanmarks were mini Maglie high inensiy flashligh canles. The angular posiion of he lanmark was measure in a sequence of images aken while he robo move across he room a a consan velociy. In aiion o he error in angle measure, error also occurre 9

10 in velociy, robo bearing an in he imes a which he images were aken. I is no possible o preic an moel hese errors. For example, velociy was se a 20mm/secon, bu average rue velociy across runs range from 21.4mm/secon o 22.5mm/secon. In aiion, he suppose consan velociy was no consan hroughou a single run, varying in an unpreicable manner. I woul be unrealisic o assume any of hese errors or heir combine resul o have a gaussian isribuion. Figure 4 shows a iagram of how he angles are measure. When he lanmark is in he lef of he camera image, he angle ( 1 in he iagram) is negaive. When he lanmark is in he righ of he camera image, he angle ( 2 in he iagram) is posiive. Angle measure is hus boun by for he given fiel of view Figure 5: Comparison of filers wih acual robo runs: Images were grabbe a ime inervals (horizonal axis) 12 secons apar. The verical axis gives he eviaion of he esimae sar posiion from he acual sar posiion in millimeers. The lanmark was place a a ifferen locaion for each run. Resuls using he RTLS algorihm are shown in black. Resuls using he Kalman filer are shown in grey. I is again assume ha he lanmark is locae a (0,0), ha he y coorinae of he robo s posiion oes no change as he robo moves, an ha he robo knows which sie of he lanmark i is on. A any sep i: an( i ) = x + ( 0 + i inerval) velociy y 10

11 where (x; y) is he robo sar posiion, i is he measure angle, 0 is robo sar ime, inerval is he inerval a which images are grabbe an velociy is he robo velociy. The problem was expresse as a linear funcion so ha no accuracy was los by linearizing. However, he aa marix as well as he measuremen vecor conaine error: " # h i x A i = 1?an( i ) ; x i = ; b i =?( 0 + i inerval) velociy y where a any sep i, A i is he aa marix, b i is he measuremen vecor an x i is he esimae sae vecor consising of he coorinaes (x; y) of he robo sar posiion. As in he previous se of experimens, he Kalman filer was given an esimae sar posiion of (0,0) an he leaing column of he aa marix was weighe by = 100. Figure 5 shows a comparison of four of he robo runs. The robo velociy was se o 20mm/sec. Five images were grabbe 12 secons apar. Robo sar posiion relaive o he lanmark use for localizaion was ifferen in each run. The eviaions of he esimae of sar locaion from acual sar locaion a each 12 secon ime inerval are compare. As in he simulae runs, he RTLS filer converge faser an o more accuracy han i he Kalman. The hir se of experimens was again run in simulaion, bu use wo lanmarks wihou assuming any prior knowlege of he robo s heaing. We assume ha he robo has no insrumen such as a compass which coul be use o regiser is compass heaing. Such insrumens can give varying, incorrec reaings in ouoor, unsrucure environmens [17], so ha i is useful o esign an evaluae mehos o obain heaing informaion from exernal sources. Such heaing informaion coul be use inepenenly or as correcions o esimaes from inernal sources. The robo knows he locaion of he wo lanmarks on a map (groun coorinae sysem). A coorinae sysem is arbirarily cenere a one lanmark. The goal is o eermine he robo sar posiion plus he locaion of he secon lanmark. Knowing which lanmark is which in he view will allow he robo o uniquely eermine is posiion, excep for cerain egenerae configuraions, bu even if he robo oes no know he orer of he wo lanmarks in is view, i can limi is sar posiion o only wo possible locaions in he groun coorinae sysem, symmerically locae on eiher sie of he line joining he lanmarks, wihou any a priori knowlege of irecion. The coorinae sysem is efine by placing lanmark 1 a (0; 0) an lanmark 2 a coorinaes (l; m) o be eermine by he filer. The x-axis is efine by he 11

12 irecion of he robo heaing. The compue coorinaes (l; m) permi mapping his coorinae sysem o he groun coorinae sysem. Generalizing figure 2, we le 1i, 2i be he angles from he robo heaing o each of he lanmarks a ime i. We have he following relaionships:?sin( 1i ) x + cos( 1i ) y = i velociy sin( 1i )?sin( 2i ) x + cos( 2i ) y + sin( 2i ) l? cos( 2i ) m = i velociy sin( 2i ) where (x; y) is he robo sar posiion. Ranom error wih a uniform isribuion was ae o he angle measures an a normally isribue ranom error was ae o he ime measuremen. As in he previous experimens, he problem was expresse as a linear funcion wih he aa marix as well as he measuremen vecor conaining error: A i = " x i =?sin( 1i ) cos( 1i ) 0 0?sin( 2i ) cos( 2i ) sin( 2i )?cos( 2i ) x y l m 3 " ; 5 b i = i velociy sin( 1i ) i velociy sin( 2i ) # # ; where a any sep i, A i is he aa marix, b i is he measuremen vecor an x i is he esimae sae vecor consising of he coorinaes (x; y) of he robo sar posiion an he coorinaes (l; m) of he secon lanmark. Resuls are summarize in Figure 6. The mean eviaions (of 19 rials) of he esimaes from he acual sar locaion of (-460, -455) are compare for six ifferen error amouns. As in he firs se of simulaions, he RTLS algorihm quickly recovers from he jump ue o is lack of an iniial esimae. Furhermore, in he regions where he RTLS error excees he Kalman filer error, neiher filer yiels any accuracy a all, since boh errors are larger han he values being esimae. 4 Conclusion In his paper, we have propose a Recursive Toal Leas Squares (RTLS) filer. This filer is easily upae as new aa arrives, ye makes very few assumpions abou 12

13 RTLS Kalman RTLS Kalman RTLS Kalman RTLS Kalman RTLS Kalman RTLS Kalman Figure 6: Comparison of mean eviaions from esimae o acual sar posiion. Measuremens were aken a uni ime inervals (horizonal axis). The verical axis gives he mean eviaion. The op hree graphs have uniformly isribue error in boh 1 an 2 of 2 an normally isribue error in wih s = 0,.05 an.1. The boom hree graphs have uniformly isribue error in boh 1 an 2 of 4 an normally isribue error in wih s = 0,.05 an.1. he aa or he problem being solve. The meho was base on he ULV Decomposiion. We sugges is use as an alernaive o he Kalman filer in reucing uncerainy in robo navigaion. In his conex RTLS oes no require an iniial sae esimae, avois moeling errors inrouce by he exene Kalman filer, oes no suffer he raps of local minima, an converges quickly. We have illusrae he meho wih simulae as well as acual robo runs. I is emonsrae ha in he omain of robo navigaion he RTLS can ofen provie more accurae esimaes in fewer ime seps han he Kalman filer, especially when errors are presen in boh he measuremen vecor an he aa marix. Fuure work inclues uilizing he filer in navigaion problems wih acual ouoor errain aa an combining is use wih he higher level reasoning escribe in [16]. 13

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