Cooperative localization method for multi-robot based on PF-EKF

Transcription

1 Scence n Chna Seres F: Informaton Scences 008 SCIENCE IN CHINA PRESS Sprnger nfo.scchna.com Cooperatve localzaton method for mult-robot based on PF-EKF WANG Lng, WAN JanWe, LIU YunHu & SHAO JnXn School of Electronc Scence and Engneerng, Natonal Unversty of Defense Technology, Changsha , Chna A method of cooperatve localzaton for mult-robot n an unnown envronment s descrbed. They share nformaton and perform localzaton by usng relatve observatons and necessary communcaton. At ntal tme, robots do not now ther postons. Once the robot that can obtan the absolute poston nformaton has ts poston, other robots use partcle flter to fuse relatve observatons and mantan a set of samples respectvely representng ther postons. When the partcles are close to a Gaussan dstrbuton after a number of steps, we swtch to an EKF to trac the pose of the robots. Smulaton results and real experment show that PF-EKF method combnes the robustness of PF and the effcency of EKF. Robots can share the absolute poston nformaton and effectvely localze themselves n an unnown envronment. cooperatve localzaton, relatve observaton, partcle flter, EKF 1 Introducton Nowadays, the requrement to complete complex tass s ncreasngly becomng desrable. Wth the development of robot technologes, the capabltes of robots have been mproved and the applcaton areas have been enlarged. For some complex tass, such as cooperatng n war feld, RoboCup, mult-robots worng together, cooperaton among robots are becomng more and more mportant. Mult-robot system can ncrease ts autonomy and robustness by fusng nformaton from every sngle robot. The poston of each robot n the system s basc and crucal nformaton for autonomy capablty. Thus, the research of cooperatve localzaton for mult-robot has become an actve area n recent years [1 5]. In ref. [4], a centralzed Kalman flter s used to fuse relatve observatons and localze every member of the robot group smultaneously. Roumelots et al. [5] decompose the centralzed Kalman flter nto M smaller communcatng flters. Every fl- Receved July 18, 006; accepted August 6, 007 do: /s Correspondng author (emal: wl_anne@sohu.com) Supported partally by the Mnstry Research Fund Project (Grant No KG0180) and the Natonal Natural Scence Foundaton of Chna (Grant Nos and ) Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no

2 ter processes nformaton from ts own host. Only when two robots detect each other and measure ther relatve postons (relatve dstance and bearng) do they exchange necessary nformaton, and then update correlatve elements of the poston and covarance of the whole group. Fox et al. [6] use a probablstc approach to collaboratve localzaton n a nown envronment. Informaton of robot movement, envronment measurements, and relatve poston measurements are fused to mprove localzaton accuracy. Howard [7] presents a dstrbuted MLE method to mult-robot relatve localzaton. When a robot detects another robot and measures ther relatve poston, the observer wll send the measurements to the observed. They share the nformaton between them and update the postons of themselves by fusng movements and measurements, respectvely. Then, they broadcast ther new postons among the group perodcally. In ref. [8], a relatve localzaton approach for mult-robot group based on partcle flter s presented. In ths method, every robot has to mantan M 1 partcle flters (M s the number of the robots), each of whch descrbes the poston of one other robot relatve to tself. Each set of samples are updated n response to dfferent observatons, so the correspondng pose estmate of the robot s updated. As the number of robots ncreases, the number of partcle flters, whch are mantaned by one robot, s also ncreased. The computatonal cost wll be expanded n an exponental mode, and the real-tme and practcablty of the localzaton method s decreased. Montesano [9] presents a method about two robots relatve localzaton by fusng bearng nformaton and motons. The performances of partcle flter, EKF, and a combnaton of them are compared n ref. [9]. In ths paper, what we focus on s the absolute localzaton for mult-robot wthout orgnal postons n an unnown envronment. The robots of the group have dfferent measure abltes. Only one of them has absolute localzaton ablty. Each robot mantans one partcle flter, the samples of whch represent ts global poston, and t uses two nds of measurements to update the flter. Under such condtons, each set of partcles are constantly updated and re-sampled based on relatve observatons and motons. They gradually converge to the real poston of the robot. In ths process, each robot shares the only absolute localzaton method, fusng relatve observatons, and cooperatvely localzes ts poston. It s well nown that a partcle flter has partcular advantages n a non-lnear, non-gaussan system. Partcles can resemble any probablty dstrbuton, thus a partcle flter s more robust and relable n such a system. Because robots do not now ther orgnal postons, a partcle flter s used to estmate the postons of robots n the frst steps. When the samples converge to a certan degree and approxmate to a Gaussan dstrbuton, then an EKF s adopted to trac the robots. The advantages of the two flter algorthms are combned, and the localzaton of the robot group wll be mproved n real-tme and robustness. Localzaton method based on PF-EKF In order to specfy our research problem and set up the localzaton frame of the system, frst we state the followng assumptons: 1) a group of M robots move n D envronment wthout obstacles. Only one of them has absolute localzaton ablty, such as usng GPS; ) each robot carres proproceptve sensors that can propagate and update ts own poston estmate, such as an encoder; 3) each robot carres exteroceptve sensors that can detect and dentfy other robots around t and measure ther relatve postons; 116 WANG Lng et al. Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no

3 4) each robot s equpped wth communcaton devces that can communcate wth other robots n the group..1 Relatve observatons Because the robots do not now ther orgnal postons, we use a unform dstrbuton to represent the orgnal postons of the robots. Fgure 1(a) depcts the orgnal dstrbuton of partcles. Once the robot R 0, whch has absolute poston ablty, acqures ts global poston, ts localzaton can be determned n certan accuracy. Here, we separate the relatve observatons among robots nto two nds: 1) relatve observatons ncludng R 0, that s, R 0 measures other robots relatve postons, or other robots measure the relatve poston of R 0 ; ) relatve observatons wthout R 0, t means the relatve measurements among other robots except R 0. At the frst step other robots do not now ther postons exactly, and ther localzaton errors are very large. If the relatve measurements among them are used to update the flters at ths tme, t often maes the partcles converge to wrong postons. Therefore, at the frst step, we do not utlze the second nd of measurements. Only the frst nd measurements are used to estmate the postons of the robots. When R 0 measures the bearng to R 1 at the frst tme, the partcle dstrbuton of R 1 changes to the form as shown n Fgure 1(b). After a certan varable number of steps the partcles converge to an approxmate Gaussan dstrbuton (Fgure 1(c)), and then the second nd of measurements are used to update the postons. Now, the poston errors of the robots are determned n a certan range. The partcles wll converge to the real postons more qucly by usng the second nd of measurements at ths tme, and t s not easy to result n a wrong way. The convergence velocty Fgure 1 Partcle dstrbuton. (a) Intal dstrbuton; (b) partcle dstrbuton after the frst bearng measurement; (c) partcles converge to an approxmate Gaussan dstrbuton after 10 update steps. + represents the real poston of a robot. WANG Lng et al. Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no

4 of the partcles depends on the relatve motons of the robots. The motons whch produce large changes on bearngs provde more nformaton for localzaton, thus the partcles converge more qucly to a Gaussan dstrbuton. On the other hand, those motons whch produce less changes on bearngs provde less nformaton, and they are not effectve for partcles convergence. Only when two robots detect each other and measure the relatve postons do they exchange necessary nformaton, such as ther new postons and relatve measurements, so that they can update ther postons respectvely. Therefore, communcaton between robots s the base for collaboratve localzaton. The communcaton requrements are lmted, about a few hundred bytes per second. The total needed communcaton bandwdth wll ncrease lnearly wth the number of the robots.. Partcle flter (PF) A partcle flter represents the requred posteror probablty dstrbuton by a set of random samples wth weghts and the estmates based on the samples. The samples are descrbed as follows: T s = { X, w}, X = ( x, y, φ), w are weghts. When the number of the samples s large enough, these estmates wll be equal to posteror probablty dstrbuton. Its advantage s that the samples can approxmate any posteror probablty dstrbuton. The frame of a PF s shown n Fgure. Fgure The frame of a partcle flter...1 Drawng samples from the proposal dstrbuton. As we do not now the posteror dstrbuton of the robot s current poston, t s often mpossble to sample drectly from the true posteror densty. However, we can rather sample from a nown, easy to sample, proposal dstrbuton, qx ( X 1, Z). Ths proposal dstrbuton s the mportance densty functon [10]. Then, at the current tme predcton partcles are ˆ { X, = 1,, N}~ q( X X 1, Z), and N s the number of partcles. The weght of each partcle s as follows: ˆ ˆ pz ( X) PX ( X 1) wˆ = wˆ 1. (1) ( ˆ qx X 1, Z) Dfferent mportance densty functon s correspondng to dfferent partcle flter. In refs. [10, 11] the unscented partcle flter (UPF) and the Gauss-Hermt partcle flter (GHPF) are presented. They are sutable for accurate measure models, that s, the stuaton of the lelhood s a narrow pea. Here, we use the standard partcle flter; let ˆ ˆ qx ( X 1, Z) = px ( X 1). The new partcles are generated accordng to the last set of samples and the moton model. It s easy to mplement. Then, eq. (1) becomes the followng ˆ wˆ = wˆ 1 p( Z X). () When a robot detects another one and measures the relatve dstance and the bearng of them, 118 WANG Lng et al. Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no

5 the mportance weghts of partcles can be evaluated by eq. (). We assume the measure nose of relatve dstance and bearng s an ndependent error wth Gaussan dstrbuton, zero mean value, and standard devaton σ, σ, respectvely. Thus, ρ θ ( ρ ρ ) ( θ θ ) σ 1 1 ρ σθ wˆ = wˆ 1 e e, (3) πσ πσ ρ where ρ s the measurement of the relatve dstance at tme. ρ s the predcton of relatve dstance between partcle and the observed robot. It s X ( xˆt x) ( yˆt y ρ = + ), (4) and T(Target) presents the observed robot, and ( xˆ, y ˆ ) s ts latest estmate poston. θ s the bearng at tme. θ s the predcton of bearng whch partcle s relatve to the observed robot. It s yˆ T y θ = a tan φ. xˆ T x (5) If at tme a robot measures M other robots, and these measurements are ndependent, and they have the same standard devaton, then eq. (3) becomes M 1 j= 1 T ( ρ ) j ρ ( θj θ) σ ρ σθ T 1 1 wˆ = wˆ e e ( = 1, N), (6) πσ πσ ρ where ρj, θ j( j = 1, M ) are the measurements of the relatve dstances and bearngs. The normalzed mportance weghts are θ 1 θ X N l w = wˆ wˆ, (7) l= 1.. Resample the partcles. The partcles are resampled accordng to ther normalzed mportance weghts. Keep and copy those partcles wth larger weghts and abandon those ones wth smaller weghts. The new set of partcles are mapped nto equally weghted samples, { ˆ X, w }, 1 = 1,,, N { X, N }, = 1,,, N. The estmate of the posteror dstrbuton s N 1 = δ N = 1 px ˆ( Z) ( X X). (8)..3 Mean estmaton of the posteror. When we obtan the new set of samples wth equal weghts, the mean estmaton of the posteror for the robot poston s N ˆ 1 X = X. (9) N = 1 After a number of update steps, the partcles gradually converge to the real poston of the robot. We perodcally compute the mean and covarance of the partcles and ft a Gaussan dstrbuton to the partcles based on the covarance of R 0, whch has the absolute localzaton ablty. When the partcles are close to a Gaussan dstrbuton, we swtch to an EKF to trac the poston WANG Lng et al. Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no

6 of the robot. The computaton complexty of the EKF s ο( n ), where n= dm( X), the dmenson of the robot state. The complexty of the partcle flter s o(n), where N s the number of partcles. The number of partcles N s much larger than the state dmenson; therefore, the EKF method s much more effcent than the partcle flter n computaton cost and has strong real tme capablty. We combne the flexblty, robustness of the PF and the effcency, real tme of the EKF, and mae full use of ther advantages to obtan better effect for the mult-robot localzaton..3 Extended Kalman flter method (EKF).3.1 The state predcton and propagaton. In the robot group, each robot R,( = 1, M), M s the number of robots, has the same moton model x D cos( φ + α ) + 1/ X = f( X, u ) = y + D sn( φ + α ), (10) φ α u s the moton measurements (evaluated by an encoder), the dsplacement and rotaton angle from tme to +1 ( D, α ), and the error model s the Gaussan dstrbuton wth zero mean value, covarance matrx Q σ 0 D =. 0 σα For a lnear system, Kalman flter s an optmum state estmate n statstcal mean. For the nonlnear system presented by eq. (10), we have to do lnearzaton to the state equaton and the measure equaton, that s, the extended Kalman flter (EKF). The state estmate of robot based on the moton measurements s as follows: Covarance s as follows: where x R ˆ + 1/ ( ˆ X = f X, u ). + 1/ T T x x u u (11) P = F P F + F Q F, (1) F and F are the Jacoban of the functon f wth respect to the state X and u. u.3. Update the estmated state. In the robot group, at one tme robot R detects robot R j and measures the relatve poston to tself by the exteroceptve sensors. Relatve poston ncludes the relatve dstance between two robots and the bearng of them. The measure methods are dfferent accordng to the dfferent sensors. In ref. [6], the relatve dstance and bearng are obtaned by fusng nformaton from a laser range-fnder wth vsual nformaton from an on-board camera. In RoboCup, the omndrectonal vsual system s often used to obtan the relatve locaton of other robots [3,1]. The relatve dstance between robots R and R j s The bearng s j ( x j x ) ( yj y ). ρ = + (13) y 1 j y θ = tan φ, xj x θ 1 j j π y y = + tan j, xj x φ (14) 1130 WANG Lng et al. Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no

7 θ s the bearng of robot R j relatve to R, θ j s that of R to R j. We can wrte the measure equaton as the common form z = hx (, X) + v, (15) H, H j are respectvely the Jacoban of the functon h wth respect to the state X and X j. Then, the Kalman flter equatons are as follows: ˆ + 1/ + 1 ˆ + 1/ + 1/ T + 1/ T 1 X = X + P H [ H P H + R] [ z h( X, X j)], (16) + 1/ / + 1/ T + 1/ T 1 + 1/ j P = P P H [ HP H + R] HP, (17) ˆ + 1/ + 1 ˆ + 1/ + 1/ T + 1/ T 1 X = X + P H [ H P H + R] [ z h( X, X )], (18) j j j j j j j + 1/ / + 1/ T + 1/ T 1 + 1/ j j j j j j j j j P = P P H [ H P H + R] H P, (19) j where R s the covarance of v,, ρ σ and σ, wth respect to ρ and θ. θ.4 Comparson of the approaches In ref. [8], the Ego-Centrc approach based on PF for mult-robots relatve localzaton s presented. In ths method, each robot determnes the pose of every other robot n the group, relatve to tself. Each robot mantans M 1 partcle flters, (M s the number of robots). Each set of samples represent the pose of every other robot relatve to tself. Each set of samples are updated accordng to dfferent measurements, thus the pose of the correspondng robot s updated. Supposng at tme robot R obtans M 1 relatve dstances to other robots. We compare the approach n ref. [8] wth the approach n ths paper (Table 1). The largest computaton dscrepancy s shown n Table 1. We assume that L (L M 1) dfferent measurements relatve to dfferent robots are obtaned. Then, the computaton cost of the Ego-Centrc approach s L tmes of that of our approach. In ref. [8], accordng to dfferent measurements relatve to dfferent robots, a dfferent set of samples s updated to estmate the postons of dfferent robots. In the approach of ths paper, no matter how many measurements are obtaned, all of them are used to updated the one set of samples whch represent the absolute pose of tself. Furthermore, when t swtches to the EKF method, the computaton effcency wll be mproved largely. In the computer of Pentum IV.8 GHz, 51 M, by Matlab, the mean tme for one update step of the PF s 86 ms, and 0. ms for the EKF. Table 1 Comparson of Ego-Centrc and PF-EKF Approach Locaton type Computaton of one update step PF-EKF Absolute locaton One partcle flter (wth N samples) wth M 1 measurements. We wrte the computaton as q Ego-Centrc Relatve locaton M 1 partcle flters (wth N samples), every one has to update by correspondng measurements. The computaton s (M 1) q 3 Smulaton and experment 3.1 Smulaton results We focus on 4 robots movng randomly at the same velocty 0.5 m/s n an area. The rotatonal speed s ω = (0. rad / s) n0,1, n0,1 s a random value from the Gaussan dstrbuton wth zero mean value, and standard devaton σ = 1. The velocty and rotatonal speed are nfluenced by a WANG Lng et al. Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no

8 V Gaussan nose wth zero mean value, and the standard devatons are σ = (0.015 m / s) ω σ = ( rad / s), respectvely. The measurements, relatve dstance, and bearng are also affected by a Gaussan nose wth zero mean value, and the standard devatons are σ = (0.053 rad) and σ = (0.08 m). The measure frequency s 1 Hz. At the ntal steps, θ ρ each robot does not now ts ntal poston. When robot R 0 obtans ts absolute poston for the frst tme, ts locaton can be determned n a certan error range. We assume the errors are σ = σ = 0.5 m, σ = 3. Robot R 0 cannot obtan ts absolute poston contnuously, that s, x y φ le GPS cannot obtan contnuous vald nformaton under some envronment condtons. We suppose the nterval of the absolute poston nformaton s 0 s. Before obtanng ts new absolute poston, robot R 0 propagates ts poston by dead reconng. As the encoder error accumulates, ts poston error s growng gradually untl the next absolute poston nformaton corrects t. Therefore, the poston error of R 0 wll be lmted n a certan degree (Fgure 3). We use the set of samples that dstrbute unformly n the area to represent the ntal postons of other robots (R 1, R, R 3 ). The sample number s N =500. We dscuss n two stuatons. (1) Usng bearng. At the ntal steps, when R 0 observes R 1, R, R 3, or R 1, R, R 3 observe R 0, R 1, R, R 3 update ther postons by the measurements. Along wth the moton of R 0, ts locaton error s growng. After a number of update steps, the samples of R 1, R, R 3 are close to a Gaussan dstrbuton (Fgure 1(c)). When the poston errors of R 1, R, R 3 are near the error of R 0, we begn to use eq. () nd of observatons, and then swtch to the EKF method. As the error of R 0 ncreases, t needs to use ts measurements to update ts poston. Under the error condtons for smulaton, trajectores of R 0 and R 1 are shown n Fgure 4. Because R 0 can obtan the absolute poston nformaton perodcally, the whole system wll acheve a stable state gradually. Once the postons converge, the accuraces based on PF and PF-EKF are very near. Ths s depcted n the poston error curves (Fgure 5), and the results are 100 Monte-Carlo. and Fgure 3 Poston error of R 0. Fgure 4 The trajectores of R 0 and R 1. () Usng relatve dstance. If R 1, R, R 3 can obtan ther orentatons (by a smulated compass wth σ φ =0.054 rad), the errors of ther moton drectons are lmted. In ths stuaton the poston error of the whole system s confned n a certan range (Fgure 6(a)). If R 1, R, R 3 cannot obtan ther orentatons, as the relatve dstances do not nclude the nformaton about the orentatons, 113 WANG Lng et al. Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no

9 the error of moton drecton s accumulated gradually. Therefore, the error of the whole system s contnually ncreasng. Ths can be seen from the trend of the mean error (Fgure 6(b)). Fgure 5 The mean poston error usng bearngs. (a) The mean poston error of all robots based on PF-EKF; (b) the mean poston error of all robots based on PF. Fgure 6 The mean poston error usng relatve dstances based on PF-EKF. (a) Wth orentatons; (b) wthout orentatons. On the scene we focus, as the robots R 1, R, R 3 do not now ther ntal postons, f we use the EKF method to update ther postons at the frst steps, the localzatons are often not converged to the real postons, only about 60% success n our smulaton. Thus, at the ntal tme, t s necessary to use the PF method. Once the samples converge to a certan degree, under the smulaton condtons, the poston accuracy of the PF and the EKF are almost on the same level. However the EKF method has hgh effcency on computaton. 3. Experment results We do real experments on the platform of RoboCup, Nubot. Nubot (Fgure 7) adopts four-wheel omndrectonal drve system, thus t s hghly deft. Its moton model has some characterstcs [13]. It s X = X + C d (0) T 1, where X = ( x, y, φ), the posture of the robot at tme. d = ( D1, D, D3, D4) s the four encoders measurements n a samplng perod. C s a 3 4 matrx, the measure couplng matrx. It s T WANG Lng et al. Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no

10 cos( φ + ϕ) cos( φ ϕ) cos( φ + ϕ) cos( φ ϕ) sn( φ + ϕ) sn( φ ϕ) sn( φ + ϕ) sn( φ ϕ) C =, (1) R 4R 4R 4R where ϕ = π / 4, and R=0. m s the dstance between the drve wheel and the machne centre. The measure data d s nfluenced by nose, manly caused by the wheel slp and loc. We measure the devaton σ1 = σ = σ3 = σ4 = (0.007 m) of d through many tmes experments. The exteroceptve sensors of Nubot are an omndrectonal reflector and a 1394 dgtal camera, wthout magng dstorton n a certan range. A Nubot can dentfy other Nubots n the feld by color mars and measure the bearngs of the robots [13,14] wth σ θ = (0.053 rad) by experments. The pcture taen by the omndrectonal camera s shown n Fgure 8. We valdate the PF-EKF method through the real data from the Nubots. Fgure 7 Three Nubots wth dfferent color mars. Fgure 8 The pcture taen by the omndrectonal camera. Three Nubots eep movng randomly n the feld at the velocty V=0.5 m/s for mn. In the process the relatve dstances among Nubots are not too large so that they can dentfy other Nubots and obtan the bearngs. The measurements are avalable at 1 Hz. On the other hand, a Nubot can locate tself n the feld by ts postonng system, whch fuses nformaton of vson, encoder, compass, and acceleraton sensor, and the error s less than 50 mm. We tae the recorded trajectores by ts own postonng system as the ground truth. By usng the bearng measurements relatve to the other two Nubots, we perform localzaton based on the PF-EKF approach, and then compare the poston trajectores of the ground truth and the PF-EKF approach. We suppose the Nubot wth blue mar s R1. R1 has the ablty to obtan the absolute poston nformaton perodcally to correct ts poston. The other two Nubots, one wth red mar, the other wth pn mar, are R and R3, respectvely. They do not now ther ntal postons and cannot obtan absolute poston nformaton. They need to fuse relatve measurements and ther motons to locate themselves. We use the partcles dstrbuted unformly at the area of 6 m 4 m to represent the ntal postons of R and R3. The number of partcles s N =500. At the begnnng steps, we only ntegrate the bearng measurements between R1 and R, and R1 and R3 to update 1134 WANG Lng et al. Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no

11 the postons of R and R3 usng the partcle flter. We calculate the mean square error of the partcles ε = ( x x). When ε < 3σ (σ s the devaton of R1), the devatons of about 95% N 1 N = 1 partcles are less than 3σ. Then, we swtch to the EKF method to trac R and R3. Fgure 9 depcts the poston error of R1. Because R1 can obtan the absolute poston nformaton perodcally, ts error s lmted n a certan range. The trajectores and poston errors of R and R3 are shown n Fgure 10. We can see that the errors based on the PF are very large at the begnnng steps, as they do not now ther ntal postons. Along wth fusng the bearng measurements relatve to R1, the partcles gradually converge to the real poston. In our experment, when the partcle flter processes about 0 s, the poston errors of R and R3 are already near to that of R1. The mean square error of the partclesε s less than the threshold. Then, we swtch to the EKF to trac the robots and fuse the bearng measurements between R and R3. The convergence velocty depends on the relatve motons of robots, that s, the relatve postons of robots when measures occur. We compare the PF and the PF-EKF method, fndng that when the postonng converges, the poston accuraces of the two methods are at the same level. In the computer of Pentum IV.8 GHz, 51 M, by Matlab, the mean tme for one update step of the PF s 86 ms, and 0. ms for the EKF. If we use the EKF method to update ther postons at the frst steps, only part of the trals converges to the real postons. Through the experment, we mae sure that n the group of mult-robots wthout ntal poston nformaton, once one of them obtans ts absolute poston, other members can use relatve measurements and exchange nformaton to locate themselves by sharng the absolute poston means. We mae use of the advantages of the PF to process the ntal steps wth large errors, and then combne the effcency and real tme of the EKF to trac the robots. Thus, the poston errors of the whole group are mantaned n certan accuracy. 4 Conclusons Fgure 9 The poston error of R1. We have presented a method to localze a group of robots wth dfferent measure abltes n an unnown envronment. The robots use relatve observatons and necessary nformaton exchange to locate ther postons under the condton that only one robot has the absolute poston means. We combne the advantages of the PF and the EKF to obtan a robust and effcent method. The WANG Lng et al. Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no

12 robots can share the unque absolute poston nformaton and mae the localzaton accuracy of the whole group reach a certan level. Ths s sutable for the scene that a group of robots wor cooperatvely n an area. It s not necessary that each of them has the absolute poston ablty. Fgure 10 The poston trajectores and errors of R, R3. (a) The trajectory of R; (b) the trajectory of R3; (c) the poston error of R; (d) the poston error of R3. We have performed the smulatons and the experments to valdate the soluton to the mult-robots localzaton. Bearngs are obtaned by the omndrectonal camera n the experment. We compare the Ego-Centrc approach wth the PF-EKF method and evaluate the dfferent performances of them. The results of the smulatons and the experments show that the mult-robots localzaton based on relatve observatons can acheve a better effect by combnng the benefts of the PF and the EKF. 1 Cao Y U, Fuunaga A S, Kahng A B. Cooperatve moble robotcs: antecedents and drectons. Autonomous Robots, 1997, 4(1): 7 7 [DOI] Stroupe A W, Martn M C, Balch T. Dstrbuted sensor fuson for object poston estmaton by mult-robot systems. In: Proceedngs of the IEEE Internatonal Conference on Robotcs and Automaton (ICRA 01), Seoul, Korea, May Relets I M, Dude G, Mlos E E. Mult-robot cooperatve localzaton: a study of trade-off between effcency and accuracy. In: Proceedngs of the IEEE Internatonal Conference on Intellgent Robot and Systems (IROS0), Lausanne, Swtzerland, Sep. 30 Oct. 5, Martnell A, Pont F, Segwart R. Mult-robot localzaton usng relatve observatons. In: Proceedngs of the IEEE Internatonal Conference on Robotcs and Automaton, Barcelona, Span, Aprl Roumelots S I, Beey G A. Dstrbuted multrobot localzaton. IEEE Trans Robot Autom, 00, 18(5): [DOI] 6 Fox D, Burgard W, Kruppa H, et al. A probablstc approach to collaboratve mult-robot localzaton. Specal Issue of Auton Robot Heterogeneous Mult-Robot Syst, 000, 8(3): WANG Lng et al. Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no

13 7 Howard A, Matarc M J, Suhatme G S. Localzaton for moble robot teams: a dstrbuted MLE approach. Experm Robot, 003, 5: Howard A, Matarc M J, Suhatme G S. Cooperatve relatve localzaton for moble robot teams: an EGO-Centrc approach. In: Proceedngs of Naval Research Laboratory Worshop on Mult-Robot Systems. Washngton DC, March 17-19, Montesano L, Gaspar J, Santos-Vctor J, et al. Cooperatve localzaton by fusng vson-based bearng measurements and moton. In: Proceedngs of the IEEE/RSJ Internatonal Conference on Intellgent Robots and Systems (IROS'05), August Merwe R V D, Doucet A. The unscented partcle flter. Adv Neural Infor Proc Syst. MIT, Yuan Z J, Zheng N N, Ja X C. The Guass-Hermte partcle flter. Acta Elec Sn, 003, (7): L S, Xu X M, Ye B, et al. Internatonal RoboCup and the technologes. Robot, 000, (5): Lu Y P. The desgn of mult-sensor system and the mplementaton n robot localzaton. Master thess. Natonal Unversty of Defense Technology, Changsha, Hunan, Hao Y M, Dong Z L, Meng K. An omndrectonal locaton system for autonomous moble robot. 30th ISR, Japan, 1999 WANG Lng et al. Sc Chna Ser F-Inf Sc Aug. 008 vol. 51 no