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Adaptive control for car like vehicles guidance relying on
RTK GPS: rejection of sliding effects in agricultural
applications
R. Lenain, B. Thuilot, C. Cariou, Philippe Martinet
To cite this version:
R. Lenain, B. Thuilot, C. Cariou, Philippe Martinet. Adaptive control for car like vehicles guidance
relying on RTK GPS: rejection of sliding effects in agricultural applications. IEEE International
Conference on Robotics and Automation. IEEE ICRA 2003, Sep 2003, Taipei, Taiwan. pp.115-120,
�10.1109/ROBOT.2003.1241582�. �hal-02466451�
rejection of sliding e ects in agricultural applications R.Lenain ? ,B. Thuilot ,C. Cariou ? ,P.Martinet ? Cemagref LASMEA BP50085 -24, av. desLandais 24, av. desLandais
63172AubiereCedex France 63177 AubiereCedexFrance roland.lenain@cemagref.fr Benoit.Thuilot@lasmea.univ-bpclermont.fr
Abstract- Numerous agricultural applications re-quireveryaccurateguidanceoffarmvehicles. Current workshaveestablishedthatRTKGPSwasavery suit-able sensor in order to meet the expected precision: severalcontrol lawshavebeen designed for vehicles equipped with such asensor, andsatisfactory results havebeen achieved as long as vehiclesdo not slide. Nevertheless, in actual workingconditions (sloping elds,entering intocurvesonawetland,:::), sliding inevitably occurs. Inthis paper, wedesigna nonlin-earadaptivecontrollawinordertopreserveguidance precision in presence of sliding: realtime sliding es-timation is used to correct vehicleevolution. Field experiments, demonstrating the capabilities of that control schemeare reported anddiscussed.
Keywords: mobile robots, nonlinear control systems, adaptivecontrollaws,kinematic GPS,agriculture.
I INTRODUCTION
Many researchgroups and manufacturers are currently workingonvehiclesautomaticguidance,devotedto agri-cultural tasks. The main objective is to increase the driving accuracy. Potential applicationsare forinstance achieving perfectly parallel runs when seeding, harv est-ing,rowcropping,:::oreliminatingskipsand/oroverlaps whensprayingfertilizersorpesticides,:::Inaddition, au-tomatic guidance delivers the humanoperatorfrom the tiring driving task. He can then fully devotehis time to the toolsmonitoring. Quality of the agronomicwork carriedoutandproductivity canclearlybeimproved. SinceRTKGPS(alsonamedkinematicGPS)canprovide realtimeabsolutepositionwithacentimeteraccuracy,it is often used as the keystoneof the perception sensing systemintheseautomaticguidedvehicles. Thissensoris here fully reliable, since interruptionsin GPS signal re-ception,whichisonemajorconcern,donotoccurin agri-culturaltaskswherevehiclesmoveonopen elds.Various systemsrelyinguponaGPSantennaandadditional iner-tialsensors(e.g. [7]),onmultipleGPSantennas(e.g. [8]), oruponauniqueGPSantenna(e.g. [11])havebeen
inves-tigated. Someguidancedevices have alreadybeen mar-keted: e.g. the BEELINE Navigator (relyingon 1GPS antenna and an Inertial NavigationSystem (INS)) and the AutoFarm System (making use of 3 GPS antennas) havebeenintroducedfromthe latenineties,respectively bytheAustraliancompanyAgSystemsandtheAmerican companyIntegriNautics. GPSsystemssuppliersaswellas agriculturalmanufacturersarealsoinvestinginthis mar-ket:Trimbleisselling the AgGPSAutopilotwhen man-ufacturer John Deereproposes a completely automated tractor,usingseveralsensors(GPS,vision,NIR)[9]. Thesedevices are mainlydevotedto applications where thevehiclemustexecuteperfectlystraightlines,andmake useofseveralsensors. Current developmentsaimat ex -tendingthe performancesof the automaticguided vehi-clesandsimultaneouslyreducingtheir cost. Inprevious works(see [11]and[2]),wehaveaddressedthis twofold challenge. On one hand, curved path following c ap abil-ity has been addressed. A nonlinearcontrollaw,taking into account for the reference path curvature, has been designed. Automatichalf-turnsand eldboundaries fol-lowinghavebeensuccessfullyexperimented,see[11]. On the other hand, our sensingdevice consists in a unique kinematicGPS. MultiplesensorsareeÆcient, sincethey providecontroldesignerwith thewholevehicleattitude, butarequiteexpensiv e. In [11]and[2],thewhole state vector is derivedfrom the information providedbythe uniqueGPSantennaviaaKalmanstatereconstructor. Satisfactoryexperimentsreportedin[11]havebeen con-ducted ona dryandevenground. However,it has also been observedthat the guidanceaccuracyis slightly re-ducedwhenthevehicleentersintoacurveonawetland, orwhenthevehiclemovesonsloping elds.Inbothcases, thetracking errororiginatesfrom theoccurrenceof slid-ing. The aimof this paperis to present further dev el-opments of the guidance algorithm described in[11], in order to reject sliding e ects.The paperis organizedas follows:farm tractor modelling andcontrol under non-sliding assumption is rst recalled in section II. Next,
oftheadaptivecontrollaw,whichallowstorejectsliding e ects,isthendetailedinsectionIV. Finally, experimen-tal resultsaredisplayedinsectionV.
II VEHICLEGUIDANCEUNDERNON-SLIPPING ASSUMPTION
A Vehicle modelling
Vehicle modelling relies upon celebrated Ackermann's model,alsonamedbicyclemodel. Moreprecisely,thetwo frontwheelsandthetworearwheelsareconsidered equiv-alentasuniquevirtualwheelslocatedatmid-distance be-tween respectively actual front wheels and rear wheels, see Figure 1. These assumptions are quite common in Automaticliterature,seeforinstance[13].
Figure 1:Vehiclemodellingparameters Ournotationsaredetailedhereafter(seealsoFigure1):
- Cisthepathtobefollowed,
- Oisthecenterofvehiclevirtualrearwheel, - MisthepointonC whichisthe closesttoO.
Misassumedtobeunique,whichisrealisticwhen thevehicleremainsquiteclosefromC.
- sis thecurvilinearcoordinateofpoint M alongC, andc(s)denotesthe curvatureofC atthatpoint. - yand
~
arerespectivelylateralandangular devia-tionofthevehiclewithrespectto referencepathC (seeFigure1).
- Æisthevirtualfrontwheelsteeringangle.
- visthevehiclelinearvelocity,consideredhereasa parameter,whosevaluemaybetime-varyingduring thevehicleevolution.
- Listhevehiclewheelbase.
bythestatevector(s;y; ~
): thetwo rstvariablesprovide point Olocation, andthe lastone describes the vehicle heading. Sincev is considered asaparameter, the only vehicle control variable is Æ. It can thenbeestablished, seeforinstance[11], thatundernon-slidingassumption, vehiclekinematicstatespacemodelis:
8 > < > : _ s = vcos ~ 1 c(s)y _ y = vsin ~ _ ~ = v( tanÆ L c(s)cos ~ 1 c(s)y ) (1)
Model (1) is clearly singularwhen y = 1 c(s)
, i.e. when pointO is superposed with the pathC curvature center at abscissa s. However, this con guration is never en-counteredinpracticalsituations.
B Curvedpathfollowing
The objective of curved path following is to bring and keep y and
~
equal to 0, independently from variable s evolution(whichmainlydependsonthevalueof parame-terv). Thecontrolapproachproposedin[11]consistsin pointingoutthatnonlinear model(1)canbe converted, without any approximation, into linear equations. More precisely,reportingintomodel(1)theinvertiblenonlinear statetransformation: ((s;y; ~ ))=(a 1 ;a 2 ;a 3 ) =(s;y;(1 c(s)y)tan ~ ) (2) anddescribingthevehicleevolutionwithrespecttos (in-steadofwithrespectto time)leadstothelinearmodel:
( da2 ds = a 3 da 3 ds = m 3 (3) Computations show thatthe newcontrol variable m
3 is related inan invertible way with the actual one,i.e. Æ. Such a transformationcan be achieved, sincemodel (1) entersinto the class of nonlinear systems which can be convertedintochainedform,seeforinstance[10]. Celebratedlinearcontroltheorycanthen beusedto de-signcontrollawm
3 inordertobring(a 2 ;a 3 )to0. Inview of(2)yand ~
arealsobroughtto0,asdesired. The inver-sionofthenonlinearrelationbetweenm
3
andÆ provides uswiththeactualnonlinearcontrollaw:
Æ(y; ~ ) = arctan L h cos 3 ~ (1 c(s)y) 2 dc(s) ds ytan ~ K d (1 c(s)y)tan ~ K p y +c(s)(1 c(s)y)tan 2 ~ )+ c(s)cos ~ 1 c(s)y ]) (4)
Closed-loopperformancescanbe adjustedbytuning pa-rameters (K
p ;K
d
). The attractive feature of control law(4)isthatnonlinearsystem(1)iscontrolledjustasa linearone,thustakingadvantageofLinearSystems
The-A rstpossibility,inordertotakeintoaccountforsliding phenomenon, is to considerdynamic models, see for in-stance[4],[12],:::Howeversincesuchmodelsdescribeall tractorfeatures(inertia,slipping,springing,:::),theyare very large,and therefore not verytractable. Moreover, they encompass numerous parameters (masses, wheel-groundcontactconditions,springsti ness,:::)whose val-uesarebadlyknown,andvery diÆculttoreachthrough experimental identi cation. Lastly, if we turn towards dynamic models, we couldno longerrely onthe curved pathfollowingcontrollaw(4),sinceithasbeendesigned speci callyfromakinematicmodel.
Asecondpossibility,moreconvenientwithrespecttoour objective, is to re nekinematicmodel (1)in orderthat itcanaccountforsliding.Inaerodynamicalandnautical applications,ithasbeenproposedtotakeintoaccountfor wind perturbation by adding velocity terms into vessels kinematicmodel,see[6],[5]. Sincewinde ectsonvessels evolutionarequitesimilartoslidinge ectsonland vehi-cles evolution, we proposeherebelowto follow the same approach.
A Modelderivation
Let usagainconsiderAckermann'smodel. Whensliding occurs,thegroundreactionisnolongerequaltothewheel actionontheground. Figure2displays
~ F front and ~ F rear , the resultantforcesperpendiculartopathC.
Figure2:Forcesappliedonvehiclewheels Thesetwononequalslidingforcesclearlygenerate:
- a resultant force which leads the vehicle to move sideways.
- aresultanttorquewhichleadsthevehicletoturnon itself.
Therefore,weproposehereaftertoaccountforthese phe-nomena by introducing into the vehicle model a linear lateral velocity
_ Y p
andanangular one _ p . Model(1) is ( _ y = vsin ~ + _ Y p _ ~ = v tanÆ L c(s)cos ~ 1 c(s)y + _ p (5) B Modelvalidation
Let us rstconsideranacademiccasewheresliding per-turbations _ Y p and _ p
areconstant(forinstanceduringan evolution ona eldwith a perfectly constant slope). In thatcase,itcanbeeasilyderivedthatthecontrollaw(4) designed under non-sliding assumption leads to asymp-toticguidanceerrors. Moreprecisely,from(5),we imme-diatelyobtainthat:
~ t!1 ! arcsin _ Y p v ! (6) tanÆ L t!1 ! _ p v + c(s)cos ~ 1 c(s)y (7) Reporting(7)into(4)showsthat:
cos 3 ~ (1 c(s)y) 2 ( y+ ) t!1 ! _ p v (8) where: = dc(s) ds tan ~ +c(s)tan ~ (K d c(s)tan ~ ) K p = tan ~ (c(s)tan ~ K d ) And nally,byneglectingtermy
2 in(8),weprovethat: y t!1 ! + _ p vcos 3~ 2c(s) _ p vcos 3~ =y c (9)
Provided that c(s) is constant or slowly varying, rela-tions (9) and (6) and nally relation (7) establish that statevariablesyand
~
aswellascontrolvariableÆ asymp-toticallyconvergeto constant orslowly varyingnonnull values. Thisindicatesclearlythat,inresponsetosliding occurrence,the vehicle movescrabways,whichis consis-tentwithexperimentalobservations.
Consistency of model (5) has also been investigated throughexperimentationcarriedoutwiththefarm trac-tordepicted onforthcomingFigure9. The solid lineon Figure3displaystheevolutionofthe lateraldeviationy whenautomaticguidancelaw(4)wastryingtokeepthe vehicle along astraight line recorded ona sloping eld. Lateraldeviationyclearlyexhibitsano set,whichisnot constantsincetheslope,inactualsituation,isnever per-fectly constant. Thedashedline onthesame gure dis-plays the evolution of y simulated from control law (4) andextendedmodel(5). Constantperturbation parame-ters _ Y p and _ p
havebeenchosenempirically,inorderthat experimentalandsimulatedasymptoticdeviationroughly
ttogether( _ Y p = 0:11m.s and _ p =0:022rad.s ). It has been observed, onone hand thatthe asymptotic valuesofstatevariable
~
andcontrolvariableÆsimulated from model (5) also roughly t with the experimental asymptotic values, andonthe otherhand thatthe time evolutionofallvariablesprovidedbythesimulation coin-cideswiththeexperimentaltimeevolution,seeFigure3. Extendedmodel(5)appearsthereforeconsistent.
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Figure 3:Comparisonoflateraldeviationundersliding be-tweenmodelandexperiment
IV GENERALVEHICLEGUIDANCELAW Sinceslidinge ectshavebeenincorporatedintomodel(5) as additional terms, it might be seen quite natural to reject these perturbations by introducing into control law (4)integralcorrection terms. However,during eld works,slidingis de nitely notastaticperturbation. On the contrary,itsdynamicmight be sometimesvery fast. Therefore, integral correctionterms do not appear very suitable for this application: they cannot prevent from transient largeguidanceerrorwhenslidingoccurs. This hasalsobeencorroboratedbyfullscaleexperiments. Adaptivecontrolcanbeseenasanalternativeapproach: for instance, a guidance systemdedicated to harvesting tasks, and relying on a video camera sensor, has been proposedin[3]. Akinematicvariable,namelytheheading deviation, isconsidered asrepresentative ofvehicle slid-ing. Then,acorrectionterm,computedfromthisvariable accordingtoanempiricalrelation,isincorporatedintothe guidance lawdesigned under non-sliding assumption, in order to achieveslidingrejection. In thispaper,a more generaladaptivecontrolframework,namelymodel-based techniques (see [1]), is used to deal with sliding e ects. Moreover,theproposedcontrollawstillreliesupon non-linearcontrollaw(4),thuspreservingallitsadvantages. A Adaptivecontrollawdesign
As it has been above-established, in the academic case whereslidingperturbationterms
_ Y p and _ p areconstant,
control law (4) leads to an asymptotic guidance error, whosevalue y
c
is givenby (9). If c(s) is also constant, thisguidanceerrorcanbeeasilycancelledbytakinginto accountforitinsidecontrollaw(4): ifthevehicleis con-trolled via Æ(y+y
c ; ~ ) instead of Æ(y; ~ ), computations similar to (8)-(9) establishthaty convergesto 0, as de-sired. Guidance accuracyis then preserved,despite the vehicleisstillmovingcrabwaysduetoslidingoccurrence (sincerelations(6)-(7) areunchanged).
In the actual case, _ Y p , _ p
and c(s) are not constant. However, in view of the above discussion, control law Æ(y+y
c ;
~
) canclearlyimproveguidanceaccuracy,once y
c
can be estimated in realtime. Such a control law is nowdetailed. Slidingdetectionandslidingcorrectionrely uponmodel-basedadaptivecontroltechniques.
Slidingdetection: the controllaw sent to the vehicle ac-tualsteeringsystemisalsousedtodrivetheonline simula-tionofmodel(1). Sincethismodeldescribesthevehicle evolution in slidingabsence, any di erence between the simulated and the actual vehicle evolution reveals slid-ingoccurrence. Thetwoevolutionsarecomparedateach time sample,see Figure4, anddisagreementiny and
~ evolution provideswiththe current valuesof the sliding parameters _ Y p and _ p .
Figure4: Slidingdetection Sliding correction:thecorrectivevaluey
c
tobeintroduced intothevehicleguidancelawÆ(y+y
c ;
~
)canbederived ac-cordingtoaModel Reference AdaptiveControl (MRAC) structureoraInternalModelControl (IMC)structure. MRAC structure is displayed on Figure 5. Vehicle ex-tended model (5), driven by control law Æ(y;
~ ) (rela-tion(4)),andfedbythecurrentvaluesofslidingvariables
_ Y p and _ p
,issimulatedinrealtime. When _ Y p and _ p are constant,thesimulatedstatevariabley convergestothe desired value y
c
(see above-discussion). In the general
case, where _ Y p and _ p
are varying,the simulated state variable y is chasing, as close aspossible, the optimum value of y
c
to be introduced into the adaptive guidance lawÆ(y+y
c ;
~ ).
IMCstructureisdepictedonFigure6. Inthisapproach, y
c
isdirectlycomputedfromrelation(9).
Figure6: IMCstructure
TheoveralladaptivecontrolschemeisshownonFigure7: the adaptationmodule is eitherthe MRAC orthe IMC structuredepictedonFigure5or6. Theybothprovidey
c valuetobeusedintheadaptivecontrollawÆ(y+y
c ;
~ )
Figure7: Overalladaptivecontrolscheme
B Simulationresults
Firstly, simulation results in the academic case, where sliding parameters and reference path curvature are all constant,aredisplayedonFigure8. Moreprecisely, sim-ulation parameters are
_ Y p = 0:1 m.s 1 , _ p = 0:03 rad.s 1 ,K p =0:09,K d
=0:6andc(s)=0(i.e. thepath tobefollowedisastraightline). Thesolidlineshowsthe evolution of the guidance errorwhen model (5) simula-tionisdrivenviacontrollawÆ(y;
~
),whichdoesnottake into account for slidingoccurrence. The guidanceerror asymptotically convergesto anon nullvalue, more pre-cisely to48cm. Onthe contrary,whenadaptivecontrol lawÆ(y+y
c ;
~
)isused, relyinguponaMRACstructure (dash-dottedline)oruponaIMCstructure(dashedline), the guidanceerrorconvergesto0,asexpected.
We can observe thatthe guidance errorsettling time is shorterwhentheadaptivecontrollawreliesupontheIMC structure. Itisquitenaturalinacademiccases:withIMC structure, the required valueforthe correctiveactiony
c is immediately used in control law (see Figure 6). On
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Figure 8: Simulation results whensliding parameters are constant
the contrary,with MRACstructure, thisrequired value for y
c
is only obtained when the model (5) simulation (seeFigure5)hasconverged. Therefore,thesettlingtime cannotbeshorterthanthesystemsettlingtime(asitcan be observedonFigure8,whencomparing solidlineand dash-dottedlineevolutions).
However,thisdi erenceintheeÆciencyofthetwo adap-tationstructuresvanisheswhenconsideringactual exper-imentations. Since actual measurements are inevitably corruptedbynoise,relation(9)onwhichreliesIMC struc-ture, cannot be used straightforwardly: all actual data have to be carefully ltered out, in order to observe a soundvehiclebehaviour.Theguidanceerrorsettlingtime is then equivalent to those obtained with MRAC struc-ture,wheremodel(5)simulationactsasanatural lter.
V EXPERIMENTALRESULTS
Experimentshavebeencarriedoutinourlaboratoryfarm in Montoldre, France. The farm tractor is depicted on Figure9. The kinematicGPSreceiveris aThales Nav-igation dual frequency "Aquarius5002" unit, providing position and velocity measurements with a 2 cm accu-racy, ata 10Hz samplingfrequency. TheGPS antenna has been located on the tractor cabin, straight up the point O (see Figure 1), inorder that the GPS receiver providesdirectlythelocationofthatpoint. Theadaptive control laws Æ(y +y
c ;
~
) has been implemented in high levellanguage(C++)onaPentiumbasedcomputer. Land vehicles undergo sliding e ects, either when they enterinto acurveonawetland, orwhentheymove on asloping eld. Theexperimentalresultsreportedbelow addressthe rst situation. Experimentationson sloping eldshavehoweverbeencarriedout,anddisplaythesame positivebehaviour.
con-Figure9: Experimentalvehicle
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Figure10: Experimentonacurve
sists mainlyin a constant curve on a gravelledground. Control law (4), which encloses no sliding corrective terms, leadsto analmostconstant lateraldeviation (re- ecting constant sliding conditions). On the contrary, bothadaptivestructuresensurethatthelateraldeviation returnsto0. Asexpectedfromprevioussimulations(see Figure8), IMCstructure exhibitsonFigure10 ahigher convergenceratethantheMRACone.However,itismore sensitivetonoiseandperturbations:whenthetractorhas covered17m(seethearrowonFigure10),ithasruninto a hole. Sliding correction via IMC structure is altered, whereasMRAConestillshowsasatisfactorybehaviour.
VI CONCLUSION
Inthispaper,model-basedadaptivetechniqueshavebeen usedinordertotakeintoaccountforslidinge ectsin
ve-MRACandIMCstructureshavebeeninvestigatedto re- nea nonlinear control law, designed from chained sys-temstheory,andthuspropagatingalllinearsystems fea-turestovehiclenonlinearmodel. Theoreticalresultshave been validated by experimentations carried out with a farm tractor,using akinematicGPS asunique localiza-tionsensor.
Experimentalresultsarequitesatisfactoryonalmost reg-ular ground. However, on uneven grounds, the GPS antenna located on the top of tractor cabin undergoes shocks,which arewrongly understoodas slidingby the adaptive controllaws. Speci c lteringoperationshave then to be considered. We are alsoworkingon the in-tegrationofalow-costGPSattitudesensor,whichcould provideuswithcabinvibrations.
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