Accurate Odometry and Error Modelling for a Mobile Robot

Transcription

1 Aurate Odometry and Error Modelling for a Mobile Robot Ko Seng CHONG Lindsay KLEEMAN o.seng.hong@eng.monash.edu.au lindsay.leeman@eng.monash.edu.au Intelligent Robotis Researh Centre (IRRC Department of Eletrial and Computer Systems Engineering Monash University, Clayton Vitoria 368, Australia Abstrat his paper presents the ey steps involved in the design, alibration and error modelling of a low ost odometry system apable of ahieving high auray dead-reoning. A onsistent error model for estimating position and orientation errors has been developed. Previous wor on propagating odometry error ovariane relies on inrementally updating the ovariane matrix in small time steps. he approah taen here sums the noise theoretially over the entire path length to produe simple losed form expressions, allowing effiient ovariane matrix updating after the ompletion of path segments. Systemati errors due to wheel radius and wheel base measurement were first alibrated with UMmar test [4]. Experimental results show that, despite its low ost, our system s performane, with regard to dead-reoning auray, is omparable to some of the best reported odometry vehile. Introdution One of the major tass of autonomous robot navigation is loalisation. In a typial indoor environment with a flat floorplan, loalisation beomes a matter of determining the Cartesian oordinates (x,y and the orientation of the robot on a two dimensional floorplan. For a wheeled robot, odometry (also nown as dead-reoning is one of the most important means of ahieving this tas. In pratie, optial enoders that are mounted on both drive wheels feed disretised wheel inrement information to a proessor, whih ontinually updates the robot s state using geometri equations. However, with time, odometri loalisation aumulates errors in an unbounded fashion due to wheel slippage, floor roughness and disretised sampling of wheel inrements. At the hardware level, gyrosopes and/or aelerometers [] are used to determine the position of the robot based on inertial navigation, but this method is suseptible to drift. lanhe [6] uses a single steerable drive wheel with a pair of nife-edge unloaded bearing wheels solely for odometry. Dead-reoning implementations at the University of Mihigan inlude [3]: Cybermotion KA utilises synhro-drive, maing it insensitive to non-systemati errors; CLAPPER, essentially two RC LabMates onneted by a ompliant linage, uses two rotary enoders to measure the rotation of the labmates relative to the ompliant linage, and a linear enoder to measure the relative distane between their entrepoints, giving it the unique ability to measure and orret non-systemati dead-reoning errors during motion; Andros, a traed vehile, has a two-wheeled enoder trailer attahed to its ba whih allows free horizontal rotation. he rotations of the trailer wheels and the trailer with respet to Andros are measured with the attahed optial enoders. heoretial researh normally involves error modelling, so that a mathematial treatment is possible. For example, the Extended Kalman Filter requires that the odometry errors be expressed in the form of an error ovariane matrix. Odometry errors an be lassified as being systemati or non-systemati. In the wor by orenstein and Feng [3], a alibration tehnique alled UMmar test has been developed to alibrate for the systemati errors of a two wheel robot. he dominant systemati error soures are identified as the differene in wheel diameter and the unertainty about the effetive wheel base. In our wor, this method has been used to alibrate systemati errors in our robot. For dealing with non-systemati errors, all lassial models [4,8,9,,3] either suffer from a la of physial basis or inonsisteny. o illustrate, suppose that at stage S x y, whih -, the state of the robot is [ ] omprises Cartesian oordinates (x -, y - and orientation - with respet to a global referene frame. A rotation followed by a translation D moves the robot to a new state S.. D os( S f( S u S D (, sin( o propagate the error ovariane matrix of S - to S, the error inurred is assumed to be small so that first order aylor s expansion does not introdue signifiant higher order errors. y assuming that the error in stage - is not orrelated with the error introdued by the input,

2 the ovariane matrix of stage, Cov(S, an be evaluated as follows, Cov( S S fcov( S S f u fcov( u u f ( where, f are the Jaobians of f with respet to S f U S and U, respetively, and D Cov( u (3 In the authors opinion, the major problem with this treatment is that there is no physial basis in assuming that the translation error is unorrelated with the rotation error [8,4,,3]. Model parameters do not reflet the physial harateristis of the system, hene they are diffiult to obtain experimentally. he model is also inonsistent. For the same path, if propagation of error is done in multiple parts, the model yields different solutions. o illustrate, suppose that S [ ] and Cov(S-. Compare the following two senarios : D (i u [ D ] with Cov( u. D From equation (, Cov( S 4D D D (ii [ D ] followed by u [ D ] Cov( u u D twie, Cov( S D, with. y applying equation ( 5D 3D 3D Even by setting Cov(u Cov(u, the two ases yield different final state error ovariane, even though they lead to the same final state by following exatly the same path. his problem has been resolved [4] by performing error propagation for every time inrement on the wheel enoders. his approah is oneptually similar to numerial integration but suffers a high omputational ost. Also no physial justifiation for the error model has been offered. he theoretial wor by [4] has introdued a more realisti, physially-based error model for an arbitrary irular ar motion. he result is a model whih is very aurate for large wheel turn variane, but limited in its appliability to a range of rotation angles. For large rotation angles, the robot path has to be divided into small segments in whih the total turning angle is within the limitation of the model. Other methods of representing position error inludes the irular-error probable (CPE by [] whih is questionable beause it is well nown that the position error is usually not equal in all diretions. [] has proposed the use of equal-error probability isoline and has outlined some ways of growing the isoline as the vehile moves. It remains unertain whether new and sound methods ould be developed to mae use of them. he new non-systemati error model developed by the authors has a strong physial basis whih is losely related to the odometry design of our robot, Werrimbi. he model also generates error representation in the form of a error ovariane matrix, whih is the standard operating blo for a multitude of noise filtering tools. Unlie the model illustrated earlier, the new model is onsistent in a multiple path segments senario. he omputational load in inrementally updating the ovariane matrix in small time steps, as done in [4] has been removed beause simple lose form formulae have been derived for three simple path types: (I irular ar motion (II straight line (III rotation about the entre of the axle. Complex paths an be divided into a small number of setions whih an be approximated by the aforementioned ases, hene the model an be applied on a setion by setion basis. Unlie the model in [4], it is valid for arbitrary distane and rotation angle. Even though the model by [4] is more aurate when the errors are large, ompared to the first order auray of the new model, in the authors opinion, should the robot be operating in onditions liely to inur large errors, an aurate representation of these large errors is inonsequential. Instead, preventive measures should be adopted to avoid suh errors, suh as employing external referening. Our odometry model an then be used to optimally ombine other sensors with odometry. he remainder of this paper is organised into six setions: Setion presents the odometry system, and states, with justifiation, the ey assumptions being inorporated into the model. Setion 3 desribes the UMmar test used for reduing systemati errors and lists the ey equations used. his is followed by setion 4 on the derivation of the proposed non-systemati error model and its properties. he details and results of alibration of systemati errors and validation of the proposed error model onstitute setion 5. Lastly, setion 6 is the onlusion whih suggests possible future extensions of the model. SEP ONE: Robot Design and Assumptions Werrimbi has two pairs of wheels: drive wheels and enoder wheels. he enoder wheels are as sharp-edged as pratially possible to redue the wheel base ( unertainty, and are unloaded beause they are independently mounted on linear bearings whih allow vertial motion, hene the problem of wheel distortion and slippage is minimised. he idea of unloaded wheels an

3 to assume that for a short unit of travel, the error inurred on both wheels are unorrelated. beause the two drive separate optial shaft enoders are used to gather odometry information. his assumption is also adopted in y enoder wheel drive wheel x motor astor astor optial shaft enoder Figure : (Left Werrimbi, the sonar sensing robot (Right Design of the aurate odometry system Our wor taes the assumption one step further. For a short unit of travel, the error is assumed to be zero mean, and white, that is, unorrelated with the previous or next unit of travel. he variane of the umulative error is then the sum of the variane of eah statistially independent unit. his leads to a reasonable assumption that the variane of eah unit of travel is proportional to the distane travelled d d (4 L L L R R R where dl and dr are the distanes travelled by eah wheel, and L and R are onstants with unit m /. 3 SEP WO: Calibration of Systemati Error Using UMmar est UMmar test [3] is used to alibrate wheel base error and unequal wheel diameter error. riefly, the robot is programmed to travel a square path of side D in the lowise sense (CW n times, and the offsets of the final Cartesian oordinates from the initial Cartesian oordinates, ex i,cw, ey i,cw are reorded. he experiment is repeated for the ounterlowise sense ( and ex i,, ey i, are reorded. he tuning fators required to be inorporated into the software to ounterat the effet of the systemati errors are alulated from the weighted Cartesian offsets in both senses. In summary, the entres of gravity of the offsets an be omputed from their averages n. g., CW/ n exi, CW/ i n. g., CW/ n eyi, CW/ i x y With the two pairs of entres of gravity, the tuning fators for the wheel base, the radius of left wheel and the radius of right wheel, b, l and r, an be found by following this sequene of omputations: average x x y y. g., CW. g.,. g., CW. g.,, (6 4D 4 D β average x x y y. g., CW. g.,. g., CW. g.,, (7 4D 4D D E sin( β / (8 d D sin( β / b π / ( π (9 l / ( E ( d E ( r d l and finally, the measure of dead-reoning auray for systemati errors is defined in [3] as ( E max x y, x y ( max, syst. g., CW. g., CW. g.,. g., 4 SEP HREE: he New Non-systemati Error Model With the new non-systemati error model, the entire path travelled by the robot is treated as onsisting of small segments. Propagation of error ovariane is required to be done times to obtain the error ovariane of the final state. his setion shows that it is possible to obtain a losed form solution for this model, as approahes infinity. he solution for a general irular ar motion is first developed. he solutions for two speial ases, straight line motion and on-spot turn are then obtained by suitably taing limits. Figure : Initial and final state of the robot after following a irular ar trajetory Suppose that at segment -, the state of the robot is S -. It then propels its left wheel by L metre and its right wheel by R metre, to bring the robot to a new state (5

4 S. Over an infinitesimal time inrement, the speed of the wheels an be assumed onstant, hene the path taes on a irular ar trajetory with onstant radius of urvature r. Refer to Figure. ased on the onstant radius of urvature assumption, the following transition equation an be derived: R L r [ sin ( ] sin (3 R L S f( S, u S r [ os( os ] R L where r is the instantaneous radius of urvature, ( L R r L R (4 where [ L R ] u. he next step is to propagate the error ovariane matrix assoiated with (- th stage to the th stage using equation (. Now suppose that the ar segment is infinitesimally small, and the full path atually omprises suh segments being onatenated from end to end. he initial state of the robot is S whih is at the starting end of the first segment and the last segment is S whih is the destination of the last segment. he expression for ovariane propagation an be reursively expanded (lie a Marov proess, S S u u Cov( S fcov( S f fcov( u f f Cov( S i S i i f Cov( u i S j S j j i j i S f i f (5 Let Li, Ri denote the small inrements in wheel turn at for the i th segment, and for irular ar motion, L Li, R Ri. he Cov(S an be further evaluated to Cov( S r(os os r(os os r(sin sin Cov( S r(sin sin L Li f i S j S j j i R Ri j i f (6 he sum of produts part of equation (6 gives rise to an error ovariane matrix heneforth nown as Cov(U. Let Cov(U i,j be the i th row, j th olumn omponent of Cov(U, after taing (and onsiderable effort!, then r ( [ ( sin( sin( ] r ( r Cov( U (sin sin os os, ( L R 3 4( L R 3 r Cov( U (os os sin sin Cov( U,, ( L R [ ( sin( sin( ] ( L R r r Cov( U (os( os( sin(, ( L R [ os( os( ] 3 3 4( L R r Cov( U, 3 (sin sin L R os ( r Cov( U 3 (os os sin (7 where R L R L R R L R L R R L 3 R, ( L R L L L ( (8 hese equations remove the need to inrementally update the ovariane matrix in small time steps. As losed form expressions, they are appliable to any irular ar motion with onstant radius of urvature. For simple paths, the matrix terms an be further simplified by suitably taing limits. 4. Speial Cases : Straight Line ranslation and Rotation about the Centre of Wheel Axle For a straight line path of length D, both wheels rotate by the same amount and the initial angle is approximately the same as the finally angle, the limits to be taen are L,R D and. Contrary to popular assumptions [7,], our model predits that variane in the diretion perpendiular to the diretion of motion is proportional to the ube of the distane travelled, whereas the variane in the diretion of motion is only proportional to the distane travelled. For rotation about the entre of wheel axle, both wheels still rotate by the same amount, but in opposite diretion. he limits to be taen are L ( and R (. All simplified expressions an also be found in [5]. his model is onsistent unlie the models presented in the introdution of this paper [4,8,9,,3]. hat is, if propagation is done in multiple parts, the model generates exatly the same result. his is beause the model itself is founded upon the onept of inrementally propagating error ovariane from one infinitesimal setion to the next. 5 Implementation and Results 5. Calibration for Systemati Error he experiments were onduted in a lab with a parquetry floor. he wheel enoder measurements were used to

5 alulate the pereived final state of the robot, whereas a sonar array [5] mounted on top of the robot was to used to estimate its atual state by sensing some referene walls plaed lose to the initial position. wo referene walls were used to establish the robot s oordinates and orientation..4 CW Figure 3 : Result of UMmar test, before and after alibration. (metre units he distribution of Cartesian offsets after the ompletion of D4m square path for 5 runs in eah sense (lowise and ounterlowise, before and after alibration, are shown in Figure 3. he value of D has been hosen as suh in order to mae benhmar omparison with the results presented in [3]. he alibration results are presented in able. Comparison with other robot vehiles are made in able. All exept Cybermotion KA are alibrated. It an be seen that the measure of dead-reoning auray for systemati errors, E max, syst, of Werrimbi is omparable to those ahievable by many advaned odometry systems. Further alibration has been arried out with the ompensated parameters but no signifiant improvement has been made. -.5 able : Key results before and after alibration efore After x.g., (mm 97-6 y.g., (mm x.g.,cw (mm 3.5 y.g.,cw (mm 3 E max, syst (mm 35 3 able : Comparison of dead-reoning auray and approximate ost of our design with four vehiles. he first four sets of figures are obtained from [3] Vehile E max, syst Cost (US$ RC LabMate average 7 K Cybermotion KA 63 <5K CLAPPER 3K Andros with railer 74? Werrimbi 3 ~4K -.5 CW 5. Computation of Non-systemati Error Parameters After alibration, Werrimbi was programmed suh that it sanned two referene walls, moved forwards metres, moved baward metres, re-sanned the two referene walls and ompared the position estimation from sonar sensing and odometry reading. he proess was repeated 6 times (5 hours onsumed and the differene in the Cartesian oordinates, (ex, ey and the differene in orientation et for all 6 runs are plotted against eah other in Figure 4(a-(. he values of L and R have been obtained by fitting error ellipses to the data, and they are L.4m /, R.58m /. In Figure 4, both the 95% onfidene error ellipses of the atual data and the 95% onfidene error ellipses generated with L and R are overlayed on the plots for omparison. 6 Conlusion and Future Wor his wor is the preliminary stage of a robotis mapping projet [5]. It draws together the ey onsiderations and proedures involved in the alibration and error modelling for an odometry design. An aurate odometry system has been presented whih is omparable to the best reported system but an be fabriated at low ost. A new first order odometry error model has been derived to propagate the state error ovariane following a motion. he error model taes the form of a ovariane matrix whih is prevalent in statistis and filtering theory, hene ould easily fit into many powerful tools suh as the Kalman Filter. his model annot aount for unexpeted errors suh as hitting a bump on the floor, whih violates the flat floorplan assumption. For ertain appliations suh as mapping, external referening should be deployed to detet suh errors. 7 Anowledgment he authors are grateful to Mr. Greg Curmi who too part in the design of the robot ey ex (a ex versus ey

6 .E E-3 et.e- 5.E-3 -.E- -.5E- -.E- -.5E- -3.E- -3.5E- ex (b ex versus et E-3 et ey ( ey versus et.e- 5.E-3.E -.E- -.5E- -.E- -.5E- -3.E- -3.5E- Figure 4 : Errors in the along-tra diretion (ex, the off-tra diretion (ey and orientation (et, for 6 runs, are plotted against eah other. he solid ellipses belong to the real data whereas the dashed ellipses are generated with the R and L obtained by trial and error. 8 Referene [] ar-shalom, Y. and Li, X.R. "Estimation and raing: Priniples, ehniques and software", oston, London: Arteh House In., 993 [] arshan,. and Durrant-Whyte, H. An Inertial Navigation System for a Mobile Robot, IEEE ransations on Robotis and Automation, Vol, No. 3, June 995, pp [3] orenstein, J. and Feng, L. UMmar - A method for Measuring, Comparing, and Correting Deadreoning Errors in Mobile Robots., ehnial Report UM-MEAM-94-, University of Mihigan [4] Crowley, J.L. and Chenavier, F. "Position Estimation for a Mobile Robot Using Vision and Odometry", Proeedings 99 IEEE International Conferene on Robotis and Automation, pp [5] Chong, K.S. and Kleeman, L, Aurate Odometry and Error Modelling for a Mobile Robot, ehnial Report MECSE-996-6, 996, Intelligent Robotis Researh Centre, Department of Eletrial and Computer Systems Engineering, Monash University [6] Chong, K.S. and Kleeman, L. Sonar ased Map uilding for a Mobile Robot, 997 International Conferene on Robotis and Automation (this issue. [7] Cox, I.J. "lanhe - An Experiment in Guidane and Navigation of an Autonomous Robot Vehile", IEEE ransation on Robotis and Automation, 99, Vol 3, pp [8] Durrant-Whyte, H.F. et al "Position Estimation And raing Using Optial Range Data", Proeedings 993 IEEE International Conferene On Robotis and Automation, Vol 3, pp [9] Jenin, M. et al "Global Navigation for ARK", Proeedings 993 IEEE International Conferene On Robotis and Automation, Vol 3, pp [] Krantz, D. Non-Uniform Dead-Reoning Position Estimate Updates, IEEE International Conferenve on Robotis and Automation, April 996, Vol, pp [] Leenhouts, P. On the Computation of i-normal Radial Error, Navigation: Journal of he Institute of Navigation, Vol 3, No, Spring 985, pp [] Lu, F. and Milios, E.E. "Optimal Global Pose Estimation for Consistent Sensor Data Registration", IEEE International Conferene on Robotis and Automation, 995, pp.93-. [3] Nishizawa,. et al "An Implementation of On-board Position Estimation for a Mobile Robot", IEEE International Conferene on Robotis and Automation, 995, pp [4] Wang, C.M., Loation Estimation and Unertainty Analysis for Mobile Robots, in IEEE Conferene on Robotis and Automation, 988, pp [5] Kleeman, L. and Ku, R. "Mobile Robot Sonar for arget Loalization and Classifiation", he International Journal of Robotis Researh, Vol. 4, No 4, August 995, pp