Planning with Uncertainty in Position Using High-Resolution Maps

Transcription

1 Planning wih Unerainy in Posiion Using High-Resoluion Maps Juan Pablo Gonzalez and Anhony (Tony) Senz The Robois Insiue Carnegie Mellon Universiy 5000 Forbes Ave, Pisburgh PA 523, USA {jgonzale, Absra We presen a novel approah o mobile robo navigaion ha enables navigaion in oudoor environmens wihou GPS. The approah uses a pah planner ha alulaes opimal pahs while onsidering unerainy in posiion, and ha uses landmars o loalize he vehile as par of he planning proess. The landmars are simple, possibly aliased, feaures ha have been previously idenified in a highresoluion map. These landmars are ombined wih an esimae of he posiion of he vehile o reae unique and robus feaures. This approah redues or eliminaes he need for GPS and enables he use of prior maps wih imperfe map regisraion. I. INTRODUCTION Navigaing auonomously is probably he mos imporan problem faing oudoor mobile robos. This as an be exremely diffiul if no prior informaion is available, and would be rivial if perfe prior informaion exised. In praie prior maps are usually available, bu heir qualiy and resoluion varies signifianly. When aurae, high-resoluion prior maps are available and he posiion of he robo is preisely nown, many exising approahes an reliably perform he navigaion as for an auonomous robo. However, if he posiion of he robo is no preisely nown, mos exising approahes would fail or would have o disard he prior map and perform he muh harder as of navigaing wihou prior informaion. Mos oudoor roboi plaforms have wo ways of deermining heir posiion: a dead-reoning sysem and a Global Posiion Sysem (GPS). The dead reoning sysem provides a loally aurae and loally onsisen esimae ha drifs slowly, and he GPS provides globally aurae esimae ha does no drif, bu is no neessarily loally onsisen. A Kalman filer usually ombines hese wo This wor was sponsored by he U.S. Army Researh Laboraory under onra Robois Collaboraive Tehnology Alliane (onra number DAAD ). The views and onlusions onained in his doumen do no represen he offiial poliies or endorsemens of he U.S. Governmen. esimaes o provide an esimae ha has he bes of boh posiion esimaes. While for many senarios his ombinaion suffies, here are many ohers in whih GPS is no available, or is reliabiliy is ompromised by differen ypes of inerferene suh as mounains, buildings, foliage or jamming. In hese ases, he only posiion esimae available is ha of he deadreoning sysem whih drifs wih ime and does no provide a posiion esimae aurae enough for mos navigaion approahes. This paper presens a new approah o mobile robo navigaion ha addresses some of he issues menioned above. We propose a pah planner for auonomous ground vehiles ha alulaes resoluion-opimal pahs while onsidering unerainy in posiion, and uses landmars o loalize he vehile as par of he planning proess. The planner uses simple, possibly aliased, feaures ha have been previously idenified in a high-resoluion map, and ombines hem wih an esimae of he posiion of he vehile o reae unique and robus feaures. This approah redues or eliminaes he need for GPS and enables he use of prior maps wih imperfe map regisraion. II. RELATED WORK While many exising approahes suh as Simulaneous Loalizaion and Mapping (SLAM) provide robus loalizaion, few ombine loalizaion wih he pah planning proess. To he bes of our nowledge, he approah presened here is he firs o address he ombined hallenges of planning wih unerainy and landmars o redue unerainy while opimizing a oninuous objeive funion in an oudoor seing. There is however, signifian wor in some of he pars of he problem: In he field of lassial pah planning, Laombe [3]0 has an exensive review on he sae of he ar as of 99. Sine hen, imporan onribuions by Lazanas and Laombe [4], Bouilly [5][6], Haï [7], Fraihard [8] and ohers have no only expanded he heoreial approahes o planning wih unerainy, bu have also addressed some of is praial limiaions. There is, however, lile wor aimed

2 a reaing pahs ha are opimal wih respe o more general objeive funions. Alhough he planner proposed by Bouilly [5] alulaes an opimal pah wih respe o unerainy or pah lengh, he approah is no appliable o finding opimal pahs wih respe o oher imporan rieria suh as mobiliy os, ris, or energy expended. Gonzalez and Senz [3] proposed a planner ha onsiders unerainy in posiion while opimizing a non-binary objeive funion. Their approah, however, does no use landmars, does no deal wih aliasing, and is limied o resuls in simulaion. While some of he approahes menioned above use landmars as par of he planning proess, none address he possibiliy of aliasing in he landmars. In he field of Parially Observable Marov Deision Proesses (POMDPs), he problem of planning wih unerainy has been frequenly addressed. However, mos algorihms beome ompuaionally inraable when dealing wih worlds wih a large number of saes. Only Roy and Thrun [9] have solved he problem of finding opimal pahs for large, oninuous-os worlds in he presene of unerainy. The planner hey propose inludes some of he elemens of he planner proposed here bu is based on an approximae soluion o a POMDP. This approah requires pre-proessing of all he saes in he searh spae, whih laer allows for very fas planning. However, he oal planning ime (inluding he pre-proessing sage) an ae from several minues o a few hours [0]. The planner opimizes unerainy raher han expeed os, and does no deal wih aliasing of feaures. The researh presened here exends he wor of Gonzalez and Senz [3] by adding landmars and providing ools o deal wih ambiguiy and aliasing. We also presen some experimenal resuls ha show he feasibiliy of he approah. III. PROBLEM STATEMENT The problem we are rying o solve is navigaing auonomously beween wo poins wihou GPS and using a high-resoluion prior map. We assume an aurae, high-resoluion map ha allows he idenifiaion of landmars and he approximae esimaion of errain ypes by auomai or manual mehods. The high-resoluion map is ranslaed ino a os grid, in whih he value of eah ell orresponds o he os of raveling from he ener of he ell o is neares edge. Nonraversable areas are assigned infinie os and onsidered obsales. The resuling pah should minimize he expeed value of he objeive funion along he pah, while ensuring ha he unerainy in he posiion of he robo does no ompromise is safey or he reahabiliy of he goal. IV. PROPOSED APPROACH Wihou simplifying assumpions, he soluion of he problem desribed above would require solving a Parially Observable Marov Deision Proess (POMDP), sine we are ombining planning, unerainy, and sensing. However, POMDPs are inraable for mos large problems, and alhough effiien approximaions exis, hey are no as effiien as deerminisi searh, espeially A*. In order o solve he problem in a deerminisi way, we use he following simplifying assumpions: - Low iniial unerainy (smaller han he sensor range of he vehile) - Low unerainy rae (less han 0% of disane raveled) - Availabiliy of basi landmars ha an be reliably deeed. These assumpions are easily me in a mobile robo ha has wheel enoders and a fiber-opi gyro for deadreoning, and when here is a good iniial esimae of he posiion of he vehile. A. Perepion Model The vehile is assumed o have a range sensor wih a maximum deeion range R and 360 o field of view. The vehile is assumed o be able o dee obsales no presen in he prior map, and mos imporanly, o be able o reliably dee he landmars in he map. Vandapel [4] has shown ha many naural and man-made sruures an be reliably deeed in laser daa. The resuls presened in his paper use eleri poles as landmars sine hey widely available in our es loaion and an be reliably deeed a disanes of up o 0 meers. Wih lile modifiaion he approah ould be modified o dee ree runs and oher similar feaures. B. Moion Model and Unerainy Propagaion The firs-order moion model for a poin-sized robo moving in wo dimensions is: x () = v ()os θ() y () = v ()sin θ() θ () = ω() where he sae of he robo is represened by x(), y() and θ() (x-posiion, y-posiion and heading respeively), and he inpus o he model are represened by v() and ω() (longiudinal speed and rae of hange for he heading respeively). Equaion () an also be expressed as: () q () = f ( q(), u()) (2) where q () = ( x (), y (), θ()) and u () = ( v (), ω()). A ypial sensor onfiguraion for a mobile robo is o have an odomery sensor and an onboard gyro. We an model he errors in he odomery and he gyro as errors in he inpus where wv () is he error in v ()(error due o he

3 longiudinal speed onrol), and wω () is he error in ω () (error due o he gyro random wal). Inorporaing hese error erms ino () yields: or, in disree-ime: x () = ( v () + w ())os θ() y () = ( v () + wv ())sin θ() θ () = ω() + w () v ω (3) x = x + ( v + w )osθ + v y = y + ( v + w )sinθ + v θ = θ + ( ω + w ) + ω (4) Using he exended Kalman filer (EKF) analysis for his sysem, whih assumes ha he random errors are zero-mean Gaussian disribuions, we an model he error propagaion as follows: where P + = F P ( F ) T + L Q ( L ) T (5) P T σ vv 0 = E( q ˆ ( q ˆ ) ) Q = 0 σ (6) ωω f( qi(), uj()) Fij = q = 0 v sin( θ ) F = 0 v os( θ ) 0 0 L C. Prior Map ij f( qi(), uj()) = u os( θ ) 0 L = sin( θ ) 0 0 i j = A prior map is neessary o provide esimaes of he os o raverse differen areas and o provide landmars for navigaion. ) Cos Map The os map is he represenaion of he environmen ha he planner uses. I is represened as a grid, in whih he os of eah ell orresponds o he os of raveling from he ener of he ell o is neares edge. Non-raversable areas are assigned infinie os and onsidered obsales. The resuling pah alulaed by he planner minimizes he expeed value of he os along he pah, while ensuring (7) (8) Fig. Cos map: ligher regions represen lower os, and darer regions represen higher os. Green areas are manually labeled buildings. ha he unerainy in he posiion of he robo does no ompromise is safey or he reahabiliy of he goal. The proedure o reae a os map from a prior map depends on he ype of prior map used. If elevaion maps are available, os is usually alulaed from he slope of he errain. When only aerial maps are available, mahine learning ehniques suh as hose in [5] an be used. Fig shows he os map for he es area used in he experimenal resuls presened here. The os map was reaed by raining a Bayes lassifier and adding manual annoaions o he resuling map. The able below shows he os assigned o he differen errain ypes. TABLE I COST VALUES FOR DIFFERENT TERRAIN TYPES. Terrain Type Cos Paved Road* 5 Paved Road 2 0 Dir Road 5 Grass 30 Trees 40 Waer 250 Buildings* 255 * Iems manually labeled. 2) Landmars Landmars are feaures in he prior map ha an be deeed wih he on-board sensors. They an ome from a separae daabase of feaures, or an be exraed direly from he prior map if he resoluion of he map is high enough. In our approah, we use aerial maps wih a resoluion of 0.3 meers per ell. A his resoluion, many feaures are learly visible and an be idenified using

4 manual labeling. Auomai exraion is also possible for some ypes of feaures, bu he sae-of-he-ar for auomai feaure deeion does no ye allow for reliable exraion of mos feaures. Fig 2 shows a small seion of our es area wih some eleri poles labeled as landmars. 3) Map Regisraion A prior map ha is no orrely regisered o he posiion of he vehile is of lile use for mos planning approahes. The error in map regisraion usually omes from wo main soures: error in he esimaion of he posiion of he vehile, and error in he esimaion of he posiion of he map. The approah presened here uses he prior map as he referene for all planning and exeuion. Sine he planner onsiders unerainy in posiion, he error in map regisraion an be modeled as being par of he error in he posiion of he robo, herefore maing use of he informaion of he map in a way ha inludes he oal unerainy in he posiion of he robo. D. Unique Deeion Regions and Aliasing The planning approah proposed requires he presene of reliable landmars. Reliable landmars are feaures in he map ha an be reliably deeed in boh aerial images and he onboard perepion sysem. In order o improve he reliabiliy in he deeion of he landmars we hose very simple, ye easy o dee, verial feaures suh as eleri poles and rees. Deeion of eleri poles and rees an be reliably ahieved wih exising approahes suh as he one presened in [4]. The hallenge wih simple feaures suh as eleri poles and rees is ha anno be uniquely idenified in a reliable manner. I is easy o find an eleri pole in a high-resoluion aerial image and in a range image from he on-board perepion of he vehile, bu is very hard o uniquely idenify whih eleri pole or ree we are looing a. However, if we now ha our posiion is wihin erain error disribuion, he number of feaures ha are be visible wihin a given deeion range are signifianly fewer. And if we hoose our feaures and our posiions well, we an ofen mae sure ha here is only one feaure wihin he deeion range of he robo, in whih ase he feaure deeed beomes a unique feaure. The ey idea is o idenify hose areas in whih a given feaure an be uniquely idenified. We all hese regions unique deeion regions. Assuming fla errain, 360 o field of view, and a deeion range R, he deeion region for a poin feaure i suh as an eleri pole is a irle of radius R. If he robo is loaed wihin his region we an guaranee ha only feaure i an be deeed. If here is no overlap beween deeion regions, eah irle would be a unique deeion region. However, if here are oher feaures wihin a 2R radius of he feaure, he oher feaures would redue he unique deeion region of he Fig 2. Deail of es area showing feaures of ineres original feaure. These overlapping regions would be he unique deeion regions for groups of wo or more feaures. However, muliple feaures are harder o dee han individual ones beause of olusions and visibiliy onsrains. For his reason, he urren approah only uses unique deeion regions generaed by single feaures. Fig 3 shows he same area as in Fig 2, wih he unique deeion regions highlighed for a deeion range R=0 meers. The dar(blue) shading shown in feaure number indiaes unique deeion regions. In his ase, sine here are no oher feaures in a 2R radius he whole irular deeion region is a unique deeion region. The ligh shading(red) shown in pars of he deeion regions of all he oher feaures indiaes a par of he deeion region where more han one feaure an be deeed, herefore exluding ha area from he unique deeion region. Fig 3. Unique deeion regions

5 E. Sae Spae Represenaion Gonzalez and Senz [3] showed ha an isomeri Gaussian is an appropriae upper bound in he unerainy propagaion for planning horizons up o a few ilomeers. We model he probabiliy densiy funion (pdf) of he error as a Gaussian disribuion, enered a he mos liely loaion of he robo a sep : q = ( xy, ) q: N( q, σ ) where q is he mos liely loaion of he robo a sep, and σ = σx = σ y is he sandard deviaion of he disribuion a sep. Le us define: ε (9) = 2 σ (0) We an hen model he boundary of he unerainy region as a dis enered a q wih a radius ε. This model is a onservaive esimae of he rue error propagaion model and, depending on he ype of error ha is dominan in he sysem, an provide an aurae approximaion of he rue model. F. Sae Propagaion In order o use a deerminisi planner o plan we need o define he ransiion os beween adjaen ells. In our 3-D onfiguraion spae, we are ineresed in alulaing he os of moving beween he onfiguraion r (a pah sep ) and an adjaen onfiguraion r + (a pah sep +). This is equivalen o alulaing he expeed os of going from a mos liely worspae loaion q, wih unerainy ε o + an adjaen mos liely worspae loaion q wih unerainy ε + : ( ) C( r, r ) = E C (, ε ),(, ε ) q q () Equivalenly, ( ) E + + C ( q, ε ),( q, ε ) (2) = i j C ( q, q ) p( q, q q, ε, q, ε ) o i j i j i where q is eah of he i possible saes a pah sep, + q j is eah of he j possible saes a pah sep +, and + C o ( q i, q j ) is he deerminisi os of raveling from + q i o q j (see Fig 4). Sine we are assuming a low unerainy rae ( α u < 0.), we an mae addiional simplifiaions ha ransform (2) ino: As menioned previously, we assume a disreized grid of saes orresponding o a nown map. y Fig 4. p( i, ε) ε q x q + q i + ((, ),( + q ε q, ε )) E C = ε + q + j q q and he sae ransiions from (, r = q ε ) o + ( +, ε + r = q ) + E q ε bce q ε a C (, ) + (, ) (3) where a and b are onsans deermined by he relaive + posiion of q and q, and ( ) ( ε ) E o i i i C ( q, ε ) = C q p q q, (4) is he expeed os of raversing ell a his loaion is ε. Therefore, + + (, ) = E(, ε) + E(, ε ) q if he unerainy C r r a C q bc q. (5) The planner used for planning wih unerainy is a modified version of A* in 3-D in whih he suessors of eah sae are alulaed only in a 2-D plane, and sae dominane is used o prune unneessary saes [3]. ) Ouside of Unique Deeion Regions Sine he dominan erm in he error propagaion for our planning horizon is linear wih disane raveled, equaion (5) an be simplified o he following model o propagae unerainy: ud ε( q )= ε( q ) + α ( q, q ) (6) where α u is he unerainy arued per uni of disane raveled, q is he previous posiion along he pah, and d( q, q ) is he disane beween he wo adjaen pah loaions q and q. The unerainy rae α u is ypially beween 0.0 and 0. (% o 0%) of disane raveled. By modeling he error propagaion in his manner, we are assuming ha he dominan erm is he unerainy in he iniial angle. Even hough we are no expliily modeling θ as a sae variable, he effes of unerainy in his variable

6 are aouned for in he unerainy propagaion model for q=(x,y). Sae propagaion ouside unique deeion regions uses (5) for os propagaion, and (6) o updae he unerainy a eah sep. No sensing of landmars aes plae when planning ouside of unique deeion regions, whih allows he planner o use deerminisi searh in he sae expansion. 2) Inside Unique Deeion Regions If all he possible loaions for a onfiguraion r are inside a unique deeion region, we an guaranee ha he feaure ha reaed he region an be deeed, and ha no oher feaures will be visible wihin he field of view of he robo. For praial purposes we mae he simplifying assumpion ha a irle wih radius ε = 2 σ ompleely onains all possible loaions on (x,y) of a given sae r. Therefore, if a irle of radius ε enered a q is ompleely onained wihin a unique deeion region i, we an guaranee ha feaure i will be deeed. This approximaion allows us o model he deeion of landmars in a deerminisi way, herefore allowing he use of deerminisi searh for his par of he sae propagaion as well. This assumpion is only valid if we an reliably dee landmar i. G. Disussion The approah proposed here finds he pah ha has he lowes expeed os and guaranees he reahabiliy of he goal wihin he given error bounds. The soluion is resoluion-opimal as long as he landmars an be reliably deeed. However, in some senarios, he bes approah would be o have a poliy insead of a pah. A poliy would onsider he deeion of feaures as a non-deerminisi even and would produe a se of aions o be performed depending on he ouome of he deeion of feaures. Beause of his added flexibiliy a poliy ould have a lower expeed os han he pah found by our approah. Bu finding an opimal poliy would require solving a POMDP, whih would be inraable o solve or would ae signifianly longer o plan even wih an approximae soluion as in [9] disane raveled. The yellow irles indiae he ε = 2σ onours of he error disribuion a eah sep along he pah. The expeed os for his pah is and he unerainy a he goal is ε f = 2σ f = 3.8 m. The following figures show he pahs found by our approah under differen onsrains for unerainy a he goal. They also illusrae he advanages of minimizing he expeed os of he pah insead of minimizing he pah lengh or he unerainy of he pah. Fig 6 shows he lowes expeed os pah wih an unerainy rae u =0% if he maximum unerainy allowed a he goal is 2 m. The pah found has an expeed os of 485 (82% lower han he shores pah) and he unerainy a he goal is m. Beause he unerainy allowed a he goal is large, he planner has enough freedom o loo for a low os pah, even if a low os pah is longer and has higher unerainy. Only one of he loalizaion regions an provide an improvemen in he oal os, herefore he planner only inludes ha landmar in he final pah. The planner also avoids he aliased region beween he wo landmars on he lef, and loalizes only wih he lefmos landmar. If he maximum unerainy allowed a he goal is small, he planner rades off lower os soluions in order o saisfy Fig 5. Planning wih unerainy rae u =0% and using landmars for loalizaion (shores pah). V. RESULTS A. Simulaion Resuls Fig 5 shows a sample os map wih some landmars. Shades of gray indiae differen oss in he os map: areas wih ligher olor have lower os, and areas wih darer olor have higher os. Solid green areas are obsales. The sar loaion is a small square on he lef, and he goal is a small irle on he righ. As a referene, his figure also shows he shores pah ha guaranees reahabiliy of he goal for his os map. The unerainy rae is 0% of Fig 6. Planning wih unerainy rae u =0% and maximum unerainy a he goal of 2 m.

7 auonomous vehile shown in Fig 8: a pah was planned beween a loaion S and a loaion G, assuming iniial unerainy σ=2.5m, unerainy rae of 5% of disane raveled and maximum unerainy of 0 m, using eleri poles for loalizaion (Fig 9). Noie how he pah follows a road in order o minimize he expeed os along he pah (insead of jus minimizing he lengh of he pah). The pah also visis deeion regions as needed o mainain a low os pah and avoids narrow areas ha ould no be safely avoided if he posiion of he robo is no auraely nown. Also noie ha he final unerainy of he pah is relaively high (ε = 5.4 m). This is beause he maximum Fig 7. Planning wih unerainy rae u =0% and maximum unerainy a he goal of 3.8 m he unerainy onsrain. Fig 7 shows he lowes expeed os pah when he maximum unerainy allowed a he goal is redued o 3.8 m. Even wih a maximum unerainy a he goal equal o ha of he shores pah here an be signifian advanages in minimizing he expeed os insead of he unerainy or he pah lengh. The expeed os is now 470 (sill 47% lower han he shores pah) and he final unerainy is ε =3.8 m. Alhough he las segmen of he pah is he shores pah for ha segmen, he firs segmen is able o loo for a less expensive pah han he shores pah and he resuling pah is signifianly less osly han he shores pah. S G B. Field Tess In order o validae he resuls experimenally he following field es was arried ou on he e-gaor 50 m G S 50 m Fig 8. E-gaor auonomous vehile used for esing and eleri poles used for loalizaion a es sie. The vehile equipped wih wheel enoders and a KVH E- ore 000 fiber-opi gyro for dead reoning, and a iling SICK ladar and onboard ompuing for navigaion and obsale deeion Fig 9. Pah planned assuming iniial unerainy σ=2.5m, unerainy rae of 5% of disane raveled and maximum unerainy of 0 m. The expeed os of he pah is 3232, and he final unerainy is σ=2.7m. Top: aerial image and unique deeion regions. Boom: os map used.

8 inludes using oher ommon feaures for loalizaion suh as rees, buildings and roads. This will require a version of he algorihm ha allows for more omplex represenaions of he error propagaion model. Relaxing he assumpion ha landmars will always be deeed in planning and exeuion is anoher area of fuure wor ha we will explore. Fig 0. Pah planned and exeued wihou GPS. Blue dos show he loaion of landmars. The blue line is he posiion esimae of he Kalman filer on he robo and he green line is he posiion repored by a WAAS differenial GPS wih auray of approximaely 2 meers (for referene only). unerainy allowed was se o 0 meers, and he planner will only ry o redue he unerainy if he reduion in unerainy will redue he expeed os of he pah. In he las segmen he pah is going hrough a large paved area and here is no inrease in os beause of he higher unerainy. For his reason, he planner does no ry o dee any feaures in las 50 meers of he pah. Fig 0 shows he pah exeued by he robo. The blue line is he posiion esimae of he robo aording o he onboard Kalman filer ha ombines he dead reoning sensors and he landmar deeions (no GPS). The green line shows he posiion esimae aording o he GPS (shown as a referene only). Noie how he blue line says very lose o he GPS esimae, and jumps in a few plaes afer deeing a landmar. VI. CONCLUSIONS AND FUTURE WORK We have inrodued a novel approah for mobile robo navigaion ha allows robus and effiien navigaion wihou GPS. The approah uses landmars in he environmen ha have been manually idenified in a highresoluion prior map o redue he unerainy in he robo s posiion as par of he planning proess. The resuling pah minimizes he expeed os along he roue onsidering he unerainy in he posiion of he robo. We have also shown experimenal resuls of he sysem, showing navigaion apabiliies similar o hose of a robo equipped wih GPS. The urren version of he algorihm uses ligh poles as is landmars, and assumes ha he landmars will always be deeed in boh planning and exeuion. Fuure wor REFERENCES [] B. Bone and H. Geffner, "Planning wih inomplee informaion as heurisi searh in belief spae," in Proeedings of he 6h Inernaional Conferene on Arifiial Inelligene in Planning Sysems (AIPS), pp. 52-6, AAAI Press, 2000 [2] J.C. Laombe, A. Lazanas, and S. Shehar, "Robo Moion Planning wih Unerainy in Conrol and Sensing," Arifiial Inelligene J., 52(), 99, pp [3] J.C. Laombe, Robo Moion Planning. Kluwer Aademi Publishers. 990 [4] A. Lazanas, and J.C. Laombe, Landmar-based robo navigaion. In Pro. 0h Naional Conf. on Arifiial Inelligene (AAAI-92), Cambridge, MA: AAAI Press/The MIT Press. hp://ieseer.nj.ne.om/lazanas92landmarbased.hml [5] B. Bouilly. Planifiaion de Sraegies de Deplaemen Robuse pour Robo Mobile. PhD hesis, Insiu Naional Polyehnique, Tolouse, Frane, 997 [6] B. Bouilly, T. Siméon, and R. Alami. A numerial ehnique for planning moion sraegies of a mobile robo in presene of unerainy. In Pro. of he IEEE In. Conf. on Robois and Auomaion, volume 2, pages , Nagoya (JP), May 995. hp://ieseer.nj.ne.om/bouilly95numerial.hml [7] A. Haï, T. Simeon, and M. Taïx, "Robus moion planning for rough errain navigaion".published in IEEE In. Conf. on Inelligen Robos and Sysems Kyongju, Korea, 999. hp://ieseer.is.psu.edu/39787.hml [8] Th. Fraihard and R. Mermond "Inegraing Unerainy And Landmars In Pah Planning For Car-Lie Robos" Pro. IFAC Symp. on Inelligen Auonomous Vehiles Marh 25-27, 998. hp://ieseer.nj.ne.om/fraihard98inegraing.hml [9] N. Roy, and S. Thrun, Coasal navigaion wih mobile robos. In Advanes in Neural Proessing Sysems 2, volume 2, pages , 999. [0] N. Roy, Deparmen of Aeronauis and Asronauis, MIT. Privae Conversaion. Sepember, 2004 [] K. Goldberg, M. Mason, and A. Requiha. Geomeri unerainy in moion planning. Summary repor and bibliography. IRIS TR. USC Los Angeles, 992. [2] A. Kelly, "Linearized Error Propagaion in Odomery", The Inernaional Journal of Robois Researh. February 2004, vol. 23, no. 2, pp.79-28(40). [3] J.P. Gonzalez and A. Senz, "Planning wih Unerainy in Posiion: An Opimal and Effiien Planner," Proeedings of he IEEE Inernaional Conferene on Inelligen Robos and Sysems (IROS '05), Augus, [4] N. Vandapel, D. Huber, A. Kapuria, and M. Heber, "Naural Terrain Classifiaion using 3-D Ladar Daa," IEEE Inernaional Conferene on Robois and Auomaion, April, [5] B. Sofman, E. Lin, J. Bagnell, N. Vandapel, and A. Senz, "Improving Robo Navigaion Through Self-Supervised Online Learning," Proeedings of Robois: Siene and Sysems, Augus, 2006.