GAUSSIAN KERNEL CONTROLLER FOR PATH TRACKING IN MOBILE ROBOTS

Transcription

1 Proceedngs o the ASME 2018 Internatonal Desgn Engneerng Techncal Conerences and Computers and Inormaton n Engneerng Conerence IDETC/CIE 2018 August 26-29, 2018, Quebec Cty, Quebec, Canada DETC GAUSSIAN KERNEL CONTROLLER FOR PATH TRACKING IN MOBILE ROBOTS Bjo Sebastan Robotcs and Mechatroncs Lab Mechancal Engneerng Dept. Vrgna Tech Blacksburg, VA bjo7@vt.edu Adam Wllams Robotcs and Mechatroncs Lab Mechancal Engneerng Dept. Vrgna Tech Blacksburg, VA aw13@vt.edu Pnhas Ben-Tzv Robotcs and Mechatroncs Lab Mechancal Engneerng Dept. Vrgna Tech Blacksburg, VA bentzv@vt.edu ABSTRACT Ths paper descrbes the desgn o a Gaussan kernel based path trackng controller or moble robots. In order to acheve successul navgaton under hybrd navgaton archtectures, t s crtcal or the robot to ollow the path provded by a hghlevel planner even whle movng between wayponts. Ths s partcularly dcult n real le stuatons nvolvng robot moton n challengng terrans. Exstng controllers or ths purpose such as the pure pursut does not ensure smooth moton o the robot or accurate trackng whle movng between path segments. Ths paper descrbes the desgn o a controller that can ensure accurate path trackng even n the presence o dsturbances, by utlzng the gradents o movng Gaussan kernels. In order to characterze the perormance o the proposed controller, two derent sets o smulatons are conducted. Based on the results o the smulatons, the Gaussan kernel controller ensures accurate trackng o the provded reerence path whle addressng the shortcomngs o exstng controllers. The paper concludes wth a dscusson on uture drectons or mprovement. NOMENCLATURE l Look ahead dstance (m) Gr Radus o the goal regon (m) N Number o Path segments Center o the Gaussan Standard devaton or the Gaussan G Gradent o the Gaussan F Normalzed 2D crcular Gaussan V V max K p L D l r Desred lnear velocty o the robot (m/s) Maxmum lnear velocty o the robot (m/s) Desred angular velocty o the robot (rad/s) Proportonal gan Wdth o the Derental drve robot (m) Dameter o the wheels o derental drve robot (m) Angular velocty o the let wheel o the derental (rad/s) Angular velocty o the rght wheel o the derental (rad/s) 1 INTRODUCTION One o the most popular topcs n robotcs and control s the exploraton o varous modes o autonomous navgaton o moble robots. An eectve strategy s the mplementaton o hybrd navgaton archtecture, consstng o a delberate hghlevel planner ollowed by a reactve low level controller [1, 2]. The hgh-level planner takes nto account the map o the envronment and develops a path to gude the robot to the goal. The path s generally output as a set o wayponts leadng to the goal. The man responsblty o the reactve controller s to drve the robot through these wayponts. In addton, the reactve controller s also responsble or handlng sudden changes n the envronment that may requre the robot to devate rom the path, or even execute an mmedate stop. Such behavor can be trggered by movng obstacles or even by dsturbances such as the slope o the terran, slp or actuator lmtatons o the robot that the planner aled to consder. In order to handle these vared responsbltes, the reactve 1 Copyrght 2018 ASME

2 controller s oten mplemented as a state machne or hybrd automaton [3]. These states consst o at mnmum a Go-to- Goal behavor as well as an Avod Obstacle behavor. One major advantage o such hybrd mplementaton archtectures are that they enable the computatonally expensve planner to be run at a very low requency as compared to the reactve controller, whle stll provdng accurate navgatonal capabltes. However, a negatve aspect o ths archtecture s that planners oten can only guarantee a easble or optmal path provded that the robot ollows the path accurately, ncludng whle t s movng between suppled wayponts. I the robot devates rom the proposed path whle executng the moton under the acton o the reactve controller, ths can result n the path becomng suboptmal or n puttng the robot n a poston where t s unable to reach the goal. For nstance, when the robot devates rom the path there s the possblty the robot may encounter already planned- or obstacles. In order to overcome ths challenge, exstng works have tred to ncorporate more accurate models o the envronment and the robot s moton, takng nto account the knematc constrants o the robot as well as the dynamcs o robot terran nteractons, allowng the planner to predct a pror possble devatons that the robot mght take [4 6]. In spte o the best eorts o researchers, the robots stll oten devate rom the proposed path n real le applcatons, especally when traversng challengng terrans. Under such condtons t becomes necessary to have a robust path trackng controller wthn the Go-to- Goal behavor that wll drve the robot back onto the proposed path, rather than just drvng the robot towards the next waypont. Ths guarantees that the optmal path, already havng been computed, wll be ollowed whle stll mantanng the hybrd navgaton archtecture. In addton, or autonomous platorms that are desgned to operate alongsde human bengs, such as autonomous wheel chars, nventory handlng robots, and payload-transportng drones, moton produced by the path-trackng controller must be smooth and contnuous. In other words, the controller should lmt the velocty, acceleraton and even jerk o the system whle producng vsbly sae, comortable and ntutve moton o the robot to be consdered [7, 8]. In ths paper we propose a path trackng controller that produces smooth motons whle meetng the aorementoned requrements. 2 LITERATURE REVIEW Path trackng or autonomous moble robots can be dened by the problem o stablzng the robot on a reerence path such that the trackng error converges to zero, and the progress o the robot along the reerence path tends toward a nomnal, nonzero rate [2]. One o the earlest path trackng strateges that showed promsng results was the pure pursut algorthm [9 10]. Other methods or path trackng nvolve modyng the reerence path [11, 12] such that a eedback controller can converge the system onto the path. In addton to beng more computatonally ntensve, these methods can al when the path s desgned ndependent o the dynamcs o the robot usng parametrc curves such as B-splnes. LQR based path trackng methods, whle beng optmal, requre a lnearzed model o the system that possesses path dependent gans that must be adjusted or each path type [13]. Due to these complextes, pure pursut remans as one o the most smple and eectve algorthms n path trackng or a moble robot. Due to ts ease o mplementaton and robust nature o ths method t s stll very wdely used n path trackng, especally or car-lke robots [14, 15]. In the smplest orm, pure pursut (PP) can be consdered as drvng the robot to a constantly movng temporary goal pont along the reerence trajectory or path [10, 16]. For a gven locaton o the robot, the algorthm starts wth ndng the closest pont on the provded path. Next, the temporary goal pont or the robot s chosen wth a certan look ahead dstance ( l ) n ront o the closest pont along the path, n the drecton o movng towards the nal destnaton. Ths acts to gude the robot orward along the path. Fnally, based on the type o the robot, control nputs (let and rght wheel veloctes or derental drve robot, steerng angle and orward velocty or a car-lke robot) are computed that wll drve the robot to the temporary goal pont. The whole process s repeated untl the robot arrves wthn a gven dstance o the nal goal pont, reerred to as the radus o the goal regon (Gr). The look ahead dstance s generally chosen to be a constant; however there are mplementatons where the look ahead dstance s vared onlne based on the curvature o the path [17, 18]. The two man challenges arsng n the orgnal mplementaton o PP as descrbed n lterature are: 1. The algorthm can request sharp turns whle the robot s movng at very hgh speeds. 2. The control acton produced vares wth the chosen look ahead dstance. Both ssues must be addressed when mplemented n hybrd archtecture, as they may result n oscllatory moton o the robot when combned wth state-o-the-art plannng algorthms. Common mplemented planners or autonomous robots nclude D*Lte [19] and RRT* [20], whch n ther smplest orm approxmate the moton o the robot wth straght lnes. As a result, the reerence path s usually generated as a seres o wayponts connected by lne segments (henceorth reerred to as path segments) leadng to the goal. The pure pursut algorthm wll acheve zero trackng error whle ollowng a straght lne segment, as the curvature o the path s zero. However, the changes n slope encountered whle movng between path segments result n jerky moton o the robot. Such behavor s due to the act that the eedback n pure pursut s a uncton o the nearest pont on the proposed path, as determned by the look ahead dstance. As explaned n [10, 16] changng the value o the look ahead dstance changes the behavor o the controller. A small look ahead wll allow the robot to ollow the desred path more closely, but wll produce oscllatons n the moton o the robot. On the other hand, a large look ahead dstance produces smooth moton but oten 2 Copyrght 2018 ASME

3 Fgure 1. Commanded moton and gradent o Gaussan Kernel controller durng derent stages o path trackng. The robot path s shown n red, goal pont n green and Gaussan gradent s shown n blue results n cases where the robot does not ollow the path very accurately, especally at sharp corners or when transtonng rom one path segment to another. In such scenaros, the pure pursut controller tends to cut the corners early, whch s an especally dangerous behavor when operatng n conned envronments such as warehouses or hosptals. Exstng works n ths doman have tred to solve the problem ether by usng smooth trajectores nsde the planner or ncorporatng an external path smoothng operaton perormed beore executng the path wth a trackng controller. However, such smoothened paths can result n nonzero trackng error due to the curvature o the path, as mentoned above. A better approach would be to mody the pure pursut algorthm tsel to handle the above ssues so that the algorthm mplctly produces smooth trackng or the robot. In the ollowng sectons, we propose modcatons to the pure pursut controller that result n an ecent path trackng algorthm whch handles these shortcomngs. The proposed algorthm s speccally desgned and tested or derental drve robots, but can easly be generalzed or other robot models. Under these assumptons, the workng o the controller can be explaned as ollows: Let N be the number o lne segments on the path proposed by the planner. For any gven poston o the robot, the GK controller wll nd the closest pont on each o the N segments. Ths can be done ether analytcally or through geometrc methods. Temporary goal ponts are then chosen l dstance n ront o the closest pont on each o the path segments. However, the closest pont s the end pont on the goal sde o a path segment, the temporary goal s chosen to be the end pont tsel. For each temporary goal pont, a normalzed 2D Gaussan dstrbuton s created centered at that pont. Each dstrbuton has a standard devaton equal to the squared dstance rom the robot to the respectve temporary goal pont. The controller conssts o two dmensonal crcular Gaussan unctons where the standard devaton along the X axs ( x ) s equal to the standard devaton along Y axs ( y ), allowng or the use o a sngle standard devaton term, y = 3 PROPOSED CONTROLLER The central dea behnd the proposed controller s to generate control nputs usng a Gaussan kernel centered at the temporary goal pont to whch the robot needs to be drven. In addton, the temporary goal pont s computed as a weghted sum o the closest ponts to the robot on each o the ndvdual path segments, wth weghts based on the relatve dstance o the robot rom these ponts. The underlyng assumptons or the Gaussan kernel (GK) controller workng are detaled below: 1. A path leadng to the goal has been computed by a planner, such as A* or RRT*. The path s then provded as a set o wayponts leadng rom a start locaton to the nal goal pont. 2. The knematc parameters o the robot,.e. the robot wdth and the wheel radus n the case o a derental drve robot, are assumed to be known. x =. The normalzed crcular Gaussan s gven by F ( x, y ) = Ae a (( x x )2 + ( y y )2 ) (1) where, A= a= 2 2 where, ( x, y ) s the center o the dstrbuton. The N Gaussan dstrbutons can then be combned usng the product rule, whch or the specal case o 2D crcular Gaussan unctons s gven by 3 Copyrght 2018 ASME

4 Fgure 2. Plot showng the varaton o V wth or the proposed velocty controller where ( μ, N μ = μ (2) 1 N (3) ) denote the resultng nal Gaussan and ( μ, ) are the ndvdual Gaussan dstrbutons 2 correspondng to the N path segments. Note that μ, μ R. Combnng the N temporary goal ponts or the robot, n ths manner, results n the smooth moton o the center o the resultng Gaussan, gradually transtonng rom one path segment to another as the robot progresses along the path. The drecton o the gradent o the dstrbuton determnes the desred orentaton o the robot n order to approach or ollow the path. The gradent ( G ( x, y) ) o the resultng Gaussan can be calculated usng the ollowng equaton G( x, y) = 2 a( Xμ ) F( x, y) (4) G X R. 2 Note that ( xy, ), For a uncycle robot model, the desred angular velocty ( ) o the robot can be obtaned rom a smple proportonal controller on the error n robot orentaton as per the ollowng equaton; 1 Gy ( x, y) Gx ( x, y) = tan ( ) (5) = K ( ); K 0 (6) p where s the current orentaton o the robot, K s the proportonal gan and desred orentaton, p p s the denotes derence takng nto account the wraparound o angles. Fgure 3. Path produced by the proposed controller or derent start postons. Robot path s shown n red, reerence path n black and goal pont n green The desred lnear velocty (V ) o the robot s vared based on the desred angular velocty o the robot; 1 2tan ( ) V = Vmax (1 ) (7) The above ormulaton ensures a maxmum lnear robot velocty ov max, whle down scalng nonlnearly or any nonzero, as shown n Fg. 2. Once the lnear and angular veloctes or a uncycle model are determned, they can be transormed nto let and rght wheel angular veloctes ( l, r) or a derental drve robot usng the transormaton equatons; 2V L l = D (8) 2V L r = D where L s the wdth o the robot and D s the dameter o the wheels. The workng o the controller at varous stages o trackng a path s shown n Fg. 1. The motons produced by the controller trackng a sngle path rom random start ponts are shown n Fg. 3. In both the gures, the path to be tracked s shown n black, the robot s shown n red and the nal goal s shown n green. Fg. 1 also llustrates the varatons n gradent o the resultng Gaussan n blue or each stage. As shown by the gradent n Fg. 1(A) the nal combned Gaussan dstrbuton has a very hgh standard devaton when the robot s ar away rom the path. The dstrbuton becomes more 4 Copyrght 2018 ASME

5 localzed as the robot gets closer to the path, as shown n Fg. 1(B) and (C). Ths changng Gaussan dstrbuton produced by the GK controller drves the robot smoothly to the proposed path as demonstrated n Fg. 1 and Fg. 3. For generatng both numercal smulatons, a smple pont robot model capable o reely movng n x and y drecton was assumed. At each teraton, the robot was moved along the gradent vector as gven by Eqn. (4). Ths was done solely or llustratng the path dctated by the controller. Proper smulatons nvolvng a derental drve robot model wth control nputs or angular and lnear velocty as gven n Eqn. (5,7) are shown n the later sectons o the paper. Lke the tradtonal pure pursut algorthm, the proposed GK algorthm has only one tunng parameter; namely the look ahead dstance ( l ). The smlartes n mplementaton mean that ths controller s easly substtuted nto systems where pure pursut was typcally utlzed. Furthermore, the advantages o the controller are as ollows: 1. The use o a separate velocty controller as gven n Eqn. (7) scales the orward velocty o the robot so that the algorthm wll request sharp turns only at low speed. 2. The consderaton o the closest ponts n all o the path segments to produce the nal Gaussan allows the algorthm to work wth a small look ahead dstance wthout producng oscllatons n the moton o the robot. Ths n turn enables the algorthm to track the path more accurately, whle stll mantanng smooth moton. These advantages partcularly address the shortcomngs o pure pursut as descrbed prevously. The ollowng secton llustrates these actors by comparng the perormance o the proposed controller wth pure pursut n smulaton. 4 SIMULATION For the purpose o valdatng the proposed control scheme and to compare ts perormance wth pure pursut, the two controllers were utlzed to drve a smulated Poneer P3DX robot and a tracked robot n the V-REP robotc smulator [21]. For the smulatons, the controllers (both pure pursut and Gaussan kernel) were run n MATLAB and co-smulated wth V-REP through the remote API Bndngs. Two sets o smulatons were conducted, the rst to compare the perormance o the controllers whle trackng the reerence path when the robots are ntalzed at derent start ponts and the second to evaluate ther perormance whle trackng the reerence path n the presence o dsturbances. 4.1 Path trackng wth derent start ponts In the ntal smulaton, the envronment conssted o a Poneer P3DX on a lat terran 20m X 20m n sze. A seres o connected path segments, analogous to those provded by a hgh level planner such as RRT*, were passed to the controller as the desred path. The waypont coordnates n meters were [(2, 2), (5, 8), (10, 8), (10, 12)], n that order, wth (10, 12) beng the nal goal poston. For ths smulaton, the robot tracks the path whle startng rom nne derent ntal start ponts, as shown Table 1. For each o the trals, the robot was rst made to ollow the reerence path under the acton o the pure pursut algorthm provded n the MATLAB Robotcs Toolbox [16]. TABLE 1. Results o path trackng smulaton wth derent start ponts Pure Pursut Gaussan Kernel Tral Startng MCTE Tme MCTE Tme No: Pont (m) (s) (m) (s) 1 (0,0) (4,0) (0,5) (10,4) (4,10) (7,5) (8,10) (12,5) (10,10) Once that acton was completed, the robot then utlzed the Gaussan kernel controller n order to track the path rom the same seres o startng ponts. In both cases, the controllers updated at a requency o 50 Hz. For the purpose o ths smulaton, precse postonng normaton obtaned rom the smulators s used wthout any nose. The look ahead parameter was manually tuned or the pure pursut mplementaton as descrbed n [10, 16] n order to produce optmal trackng behavor. The complete set o parameters or the pure pursut mplementaton s as ollows: Desred lnear velocty = 0.05m/s Maxmum angular velocty = 1.0rad/s Look ahead dstance = 0.8m Radus o Goal regon = 0.1m The look ahead dstance or the pure pursut controller was ound to be optmal at 0.8m, as any lower value produced oscllatons n the moton o the smulated robot. As stated prevously, t s desrable to have the smallest possble look ahead dstance n order to more closely track the reerence path. Smlarly the parameters were tuned or the Gaussan kernel controller and the values are: 5 Copyrght 2018 ASME

6 Fgure 4. Comparson o the controller perormance: (A) Pure pursut, (B) Proposed controller Maxmum lnear velocty = 0.05m/s Proportonal gan on angular velocty = 0.6 Look ahead dstance = 0.1m Radus o Goal regon = 0.1m controller as gven by Eqn. (7) reduces the velocty n order to enable the robot to make the turn n place. Ths behavor then contrbutes to the act that the proposed controller takes more tme to reach the goal when compared to pure pursut controller, as shown n Table 1. Both pure pursut and the proposed controller produce V and w control commands or the smulated robot. These commands are rst converted nto let and rght wheel veloctes ( l, r ) or the smulated robot usng the 4.2 Path trackng under dsturbance Autonomous moton under challengng terran condtons, such as the moton o a tracked robot n rough terran, can result n hgh levels o dsturbance beng ntroduced to the system. Some examples o such dsturbances nclude slp experenced by the robot, actuator lmtatons restrctng robot moton n hgh slope regons, and sensor nose. In all these crcumstances, t s extremely valuable to have a robust path trackng controller to ensure that the robot ollows the path accurately. In order to study path trackng under dsturbance, the proposed controller was smulated on rough terran, and ts perormance was compared to that o pure pursut. The smulaton envronment conssted o a derental drve tracked robot model operatng on a rough terran o sze 9m X 7m, as shown n Fg. 5. The waypont coordnates or the reerence path, n meters, were [(2, 2), (0, 8), (6, 8)], wth (6, 8) beng the nal goal. For ths smulaton, the robot was made to track the path startng rom (0, 0), rst under the acton o a pure pursut controller and then under the proposed GK controller. As beore, the controllers were updated at a requency o 50 Hz and precse postonng data s used. The desred lnear velocty or pure pursut and the maxmum lnear velocty or the proposed controller were ncreased to 0.06m/s n order to allow the robot handle the rough terran. All other parameters were unchanged rom the pror smulaton. The heght o the terran vares rom 0m to 3m and the maxmum torque on each o the motors on the tracked robot was lmted to 10 Nm nsde the V-REP smulator. transormaton gven n Eqn. (8) and then are executed by the nbult jont velocty controller n VREP. The trajectores ollowed by the robot are shown n Fg. 4. The varous start ponts o the robot are shown n blue and labeled by ther respectve tral numbers, as n Table 1. The reerence path s shown n black, whle the actual path ollowed by the robot s n red and the nal goal pont s n green. In order to quanty the perormance o the controllers or path trackng, we ntroduce the cross-track error, whch s dened as the Eucldean dstance between the robot and the closest pont on the path computed or every teraton o the controller. The mean cross-track error (MCTE), the cumulatve per tral cross track error averaged over the total number o controller teratons, s then calculated. The values o the MCTE, along wth the tme taken by the controller to drve the robot to the goal, or each tral are gven n Table 1. As shown by Fg. 4 the proposed controller results n closer path trackng as compared to pure pursut method. Ths clam s also supported numercally by the mean cross track error values presented n Table 1. For all start ponts except that o tral 4, the GK controller results n smaller mean cross track error than pure pursut. In the case o tral 4, both trajectores are smlar n orm, thus resultng n nearly equal mean cross track error. Whle Fg. 4 shows that the robot makes sharp turns under the acton o the proposed controller, the moton produced s stll smooth due to the act the lnear velocty 6 Copyrght 2018 ASME

7 Fgure 5. Smulated envronment utlzed or studyng path trackng perormance under dsturbance n V-REP The trajectores ollowed by the robot under the acton o both the controllers are shown overlad wth the terran topography map n Fg. 6. The start pont s shown n cyan, reerence path n black and the goal pont n green. The path ollowed by the robot under the acton o the pure pursut controller s shown n red and the proposed controller s shown n blue. As shown by Fg. 6, even n the presence o dsturbance, the GK controller enables the robot to ollow the path more closely as compared to pure pursut. For ths smulaton case, the startng pont o the last path segment s n a regon o steep slope due to the large gradent n terran heght. As shown n Fg. 6, the steep terran causes the robot to devate rom the desred path at ths pont n both cases. But the GK controller recovers much aster as compared to the pure Fgure 7. Torque appled on the let and rght tracks o the robot or the path trackng under dsturbance smulaton pursut method. Fgure 7 shows the torque actng on the let and rght tracks or the above smulaton case. The results o both smulatons demonstrate that the proposed GK controller has superor path trackng characterstcs as compared to the pure pursut algorthm. Ths behavor s demonstrated both when navgatng rom a varety o ntalzaton ponts as well as n a non-unorm terran. 5 CONCLUSION AND FUTURE WORK Ths paper presented a novel Gaussan kernel controller or path trackng n moble robots. The controller was desgned to address shortcomngs n exstng pure pursut mplementatons n order to produce smooth motons whle ensurng accurate trackng o the provded reerence path utlzng the gradents o movng Gaussan kernels. Based on the smulaton results, the proposed controller demonstrated superor path trackng perormance as compared to the tradtonal pure pursut algorthm. The smulatons compared the perormance o controllers or two derent cases, one where the robots tracked a sngle reerence path when ntalzed rom a varety o start ponts and the other where the robots tracked a reerence path through a rough terran n the presence o envronmental dsturbance. Future plans or contnued research nto the applcatons and eatures o the proposed controller wll ocus on real-le mplementaton. Hardware tests wll nclude the usage o the GK controller by a varety o unque moble robots, ncludng the Hybrd Mechansm Moble Robot (HMMR) [22] and STORM [23]. Tests wll be conducted to very the perormance o the controller n the presence o both sensory and envronmental nose, such as n hgh slp condtons or wth an ntentonally nosy eedback devce. In addton to derental drve robots, the controller wll also be explored or Fgure 6. Comparson o path trackng n the presence o dsturbance. Path ollowed under the acton o the proposed controller s shown n blue and pure pursut controller s shown n red 7 Copyrght 2018 ASME

8 path trackng n other robot models, such as car-lke robots and quadcopters. Addtonally, the use o an ellpsod Gaussan n the controller, and the attendant addtonal tunng adjustments requred to redesgn the controller n such a orm wll be explored. The author s hypothesze that the use o an ellpsod Gaussan can lend tsel to larger gradents adjacent to the path, should the two axes be chosen such that the larger standard devaton (and thus the major axs o the ellpsod dstrbuton) s parallel to the path and the smaller standard devaton (and mnor ellpsod axs) s perpendcular to the path. Ths may provde a stronger propensty o the controller to keep the robot on the path at all tmes. ACKNOWLEDGMENT Ths work s supported by the US Army Medcal Research & Materel Command s Telemedcne & Advanced Technology Research Center (TATRC), under Contract No. W81XWH-16- C The vews, opnons, and/or ndngs contaned n ths report are those o the authors and should not be construed as an ocal Department o the Army poston, polcy, or decson unless so desgnated by other documentaton. REFERENCES [1] Kruse, T., Pandey, A. K., Alam, R., and Krsch, A., Human-aware robot navgaton: A survey. Robotcs and Autonomous Systems, 61(12), dec, pp [2] Paden, B., Cap, M., Yong, S. Z., Yershov, D., and Frazzol, E., A Survey o Moton Plannng and Control Technques or Sel-Drvng Urban Vehcles. IEEE Transactons on Intellgent Vehcles, 1(1), mar, pp [3] Egerstedt, M., Behavor based robotcs usng regularzed hybrd automata. In Internatonal Workshop on Hybrd Systems: Computaton and Control. IEEE, pp [4] Howard, T. M., and Kelly, A., Optmal rough terran trajectory generaton or wheeled moble robots. The Internatonal Journal o Robotcs Research, 26(2), pp [5] Rena, G., Bellone, M., Spedcato, L., and Gannoccaro, N. I., D traversablty awareness or rough terran moble robots. Sensor Revew, 34(2), pp [6] Currer, P. N., andwcks, A. L., A novel method or predcton o moble robot maneuverng spaces. Journal o Terramechancs, 50(2), apr, pp [7] Park, J. J., and Kupers, B., A smooth control law or graceul moton o derental wheeled moble robots n 2D envronment. Proceedngs - IEEE Internatonal Conerence on Robotcs and Automaton, pp [8] Gulat, S., and Kupers, B., Hgh perormance control or graceul moton o an ntellgent wheelchar. Proceedngs - IEEE Internatonal Conerence on Robotcs and Automaton, pp [9] Wallace, R., Stentz, A., Thorpe, C. E., Maravec, H., Whttaker, W., and Kanade, T., Frst Results n Robot Road-Followng.. Ijca, pp [10] Coulter, R. C., Implementaton o the Pure Pursut Path Trackng Algorthm. Communcaton(January). [12] Amd, O., and Thorpe, C. E., Integrated moble robot control. Tech. Rep. May, Carnege Mellon Unversty, Pttsburgh, PA, may. [11] Mcaell, A., Samson, C., Mcaell, A., and Samson, C., Trajectory trackng or uncycle-type and two steerng- wheels moble robots. Tech. rep., INRIA. [12] Pazz, A., Lo Banco, C., and Romano, M., Splnes or the Smooth Path Generaton o Wheeled Moble Robots. IEEE Transactons on Robotcs, 23(5), oct, pp [13] Cordero, R. A., Aznhera, J. R., De Pava, E. C., and Bueno, S. S., Dynamc modelng and bo-nspred LQR approach or o-road robotc vehcle path trackng th Internatonal Conerence on Advanced Robotcs, ICAR [14] Buehler, M., Iagnemma, K., and Sngh, S., eds., The DARPA Urban Challenge, Vol. 56 o Sprnger Tracts n Advanced Robotcs. Sprnger Berln Hedelberg, Berln, Hedelberg. [15] Ohta, H., Aka, N., Takeuch, E., Kato, S., and Edahro, M., Pure Pursut Revsted: Feld Testng o Autonomous Vehcles n Urban Areas. In 2016 IEEE 4 th Internatonal Conerence on Cyber-Physcal Systems, Networks, and Applcatons (CPSNA), IEEE, pp [16] Corke, P., Robotcs, Vson and Control, Vol. 118 o Sprnger Tracts n Advanced Robotcs. Sprnger Internatonal Publshng, Cham. [17] Wt, J., Crane, C. D., and Armstrong, D., Autonomous ground vehcle path trackng. Journal o Robotc Systems, 21(8), pp [18] Gesbrecht, J. L., Mackay, D., Coller, J., and Verret, S., Path trackng or unmanned ground vehcle navgaton. DRDC Sueld TM(December). [19] Koeng, S., and Lkhachev, M., D* Lte. Proceedngs o the Eghteenth Natonal Conerence on Artcal Intellgence, pp [20] Karaman, S., and Frazzol, E., Incremental Samplng-based Algorthms or Optmal Moton Plannng. CoRR, abs/1005.0, may. [21] Coppela Robotcs, V-REP. [22] Ben-Tzv, P., Goldenberg, A. A., and Zu, J. W., Artculated hybrd moble robot mechansm wth compounded moblty and manpulaton and on-board wreless sensor/actuator control nteraces. Mechatroncs, 20(6), sep, pp [23] Kumar, P., Saab, W., Ben-Tzv, P., A Hybrd Tracked-Wheeled Mult-Drectonal Moble Robot. ASME Journal o Mechansms and Robotcs, Under revew. 8 Copyrght 2018 ASME