Plath Planner With Tests

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This paper considers path following control for a robotic platform. The vehicle used for the experiments is a specially designed robotic platform for performing au- tonomous weed control. The platform is four-wheel steered and four-wheel driven. A diesel engine powers the wheels via a hydraulic transmission. The robot uses a Real Time Kinematic Differential Global Positioning System to deter- mine both position and orientation relative to the path. The deviation of the robot to the desired path is supplied to two high level controllers minimizing the orthogonal distance and orientation to the path. Wheel angle setpoints are de- termined from inversion of the kinematic model. At low level each wheel angle is controlled by a proportional con- troller combined with a Smith predictor. Results show the controller performance following different paths shapes in- cluding a step, a ramp, and a typical headland path
  Auton Robot (2010) 29: 85–97DOI 10.1007/s10514-010-9182-3 A path following algorithm for mobile robots Tijmen Bakker  · Kees van Asselt  · Jan Bontsema  · Joachim Müller  · Gerrit van Straten Received: 8 December 2008 / Accepted: 18 March 2010 / Published online: 21 April 2010© The Author(s) 2010. This article is published with open access at Abstract  This paper considers path following control fora robotic platform. The vehicle used for the experimentsis a specially designed robotic platform for performing au-tonomous weed control. The platform is four-wheel steeredand four-wheel driven. A diesel engine powers the wheelsvia a hydraulic transmission. The robot uses a Real TimeKinematic Differential Global Positioning System to deter-mine both position and orientation relative to the path. Thedeviation of the robot to the desired path is supplied to twohigh level controllers minimizing the orthogonal distanceand orientation to the path. Wheel angle setpoints are de-termined from inversion of the kinematic model. At lowlevel each wheel angle is controlled by a proportional con-troller combined with a Smith predictor. Results show thecontroller performance following different paths shapes in-cludingastep,aramp,andatypicalheadlandpath.Arefined T. Bakker (  )  ·  K. van Asselt  ·  G. van StratenSystems and Control Group, Wageningen University,P.O. Box 17, 6700 AA Wageningen, The Netherlandse-mail: Present address: T. BakkerTyker Technology, P.O. Box 507, 6700 AM Wageningen,The NetherlandsJ. BontsemaWageningen UR Greenhouse Horticulture, P.O. Box 644,6700 AP Wageningen, The NetherlandsJ. MüllerFarm Technology Group, Wageningen University, P.O. Box 17,6700 AA Wageningen, The Netherlands Present address: J. MüllerInstitute for Agricultural Engineering, University of Hohenheim,70593 Stuttgart, Germany tuning method calculatescontroller settingsthat let the robotdrive as much as possible along the same path to its setpoint,but also limit the gains at higher speeds to prevent the closedloop system to become unstable due to the time delay in thesystem. Mean, minimum and maximum orthogonal distanceerrors while following a straight path on a paving at a speedof 0.5 m/s are 0.0,  − 2.4 and 3.0 cm respectively and thestandard deviation is 1.2 cm. The control method for fourwheel steered vehicles presented in this paper has the uniquefeature that it enablescontrol of a user definableposition rel-ative to the robot frame and can deal with limitations on thewheel angles. The method is very well practical applicablefor a manufacturer: all parameters needed are known by themanufacturer or can be determined easily, user settings havean easy interpretation and the only complex part can be sup-plied as a generic software module. Keywords  Robot  ·  Path following  ·  4WS  ·  RTK-DGPS 1 Introduction In organic farming there is a need for weeding robots thatcan replace manual weeding. The required labour for handweeding is expensive and often difficult to obtain. In 1998in the Netherlands on average 73 hours of hand weedingwere spent on one hectare of sugar beet in organic farming(Van der Weide et al. 2002). In this paper a path followingcontrol system for a weeding robot is presented enabling therobot to navigate autonomously along a path.A common design for a control system for agriculturalvehicles is to split up the control system in a low level and ahigh level controller (Bendtsen et al. 2002; Bak and Jakob-sen 2004). The low level electro-hydraulic system is a sys-temwithdeadtime.Awellknownmethodtocompensatefor  86 Auton Robot (2010) 29: 85–97 time delays is the Smith predictor (Stephanopoulos 1984).Ge and Ayers (1991) applied this successfully to control anelectro-hydraulic system on a hydraulic test bench. We useda Smith Predictor to compensate for time delays in the ap-plication of an electro-hydraulic steering system in practice.The high level control system is partly inspired by work of Hague and Tillett (1996) and Bendtsen et al. (2002). Bendtsen et al. (2002) used a model for a four-wheel steeredvehicle derived from Campion et al. (1996) and presentedsimulation studies applying feedback linearization as a con-trol method. Hague and Tillett (1996) worked out a methodfor path following for a vehicle with two driven wheels andtwo free rolling wheels. For a simplified vehicle model theydevelopedacontroller.Fromtheoutputofthiscontrollerfol-low the wheel speed setpoints by inversion of the kinematicvehicle model. In this paper this method is worked out fora four wheel steered robot, using the kinematic model de-rived from Campion et al. (1996) resulting in wheel angleand wheel speed setpoints for the low level control system.A refined tuning method of the high level controller, adaptedfrom Skogestad (2003), lets the robot drive as much as pos-sible along the same path to its setpoint independent fromspeed, but also limit the gains at higher speeds to preventthe closed loop system from becoming unstable at higherspeeds because of the time delay. 2 Robotic platform 2.1 PlatformThe vehicle used for the experiments is a specially designedrobotic platform for performing autonomous weed control(Fig. 1). The design of the platform was described earlier byBakker et al. (2008). The platform is four-wheel steered andfour-wheel driven. There is no mechanical constraint on themaximum turning angle of a wheel around its vertical axis, Fig. 1  Robot platform but the wheel angles should be constrained to prevent twist-ing of the cables of the wheel speed sensors. Power is pro-vided by a diesel engine that powers the wheels via an hy-draulic transmission. The hydraulic transmission consists of a pump supplying oil to eight proportional valves, each con-nected to one fixed displacement hydraulic motor. Four hy-draulic motors are used to drive the wheels, the other four tosteer the wheels. Computer control of the valves is achievedusingpulsewidthmodulationviatwomicro-controllerscon-nected to a CAN bus. The pump/valves combination is a‘loadsensing’ system: the pressure drop overthevalvescon-trols the displacement of the pump via an hydraulic loadsensing connection and is limited to a small value, indepen-dent of load pressure. The platform is further equipped witha hitch that can be lifted hydraulically. A second hydraulicpump mounted in series with the first, supplies oil to twovalves: one for lifting the hitch, one for control of auxiliaryimplements. Computer control of the valves is achieved alsovia a micro controller connected to the CAN bus.2.2 ElectronicsThe weeding robot electronics consists of 9 embedded con-trollers connected by a CAN bus using the ISO 11783 proto-col. In the inside of every wheel rim a cogwheel is mountedfor wheel speed measurement. The two magneto resistivesensors per cogwheel are placed in such a way that thedirection of rotation can be resolved. The rotation of thewheels is measured by these sensors with a resolution of 100 pulses per wheel revolution. The wheel angle of eachwheel is measured by a Kverneland 180 degree sensor withan accuracy of one degree. Per wheel a micro controller ismounted transmitting wheel speed and wheel angle via theCAN bus. Two GPS antennas are used to measure both vehi-cle position and orientation. Both are connected to a Septen-trio PolaRx2eH RTK-DGPS receiver with a specified posi-tion accuracy of 1–2 cm and a specified orientation accuracyof 0.3 degrees (1 σ  ). The two GPS antennas are mountedon a metal plate to prevent multipath errors. A base stationwith a Septentrio PolaRx2e RTK-DGPS supplies the RTK-correction signals via a radio connection to the SeptentrioPolaRx2eH receiver. The position of the base station itself can be configured by a correction supplied by the service of a company called 06-GPS via GPRS. One embedded con-troller running a real time operating system (National Instru-ments PXI system) also connected to the CAN bus controlsthe vehicle. The GPS receiver, and a radio modem are con-nected to the PXI via RS232. The radio modem interfacesthe remote control used for manual control of the weedingrobot. The manual control is used for guaranteeing safetyduringfieldtrialsandfortransportationtoandfromthefield.Different colored lamps of the signal tower can be operatedvia a micro-controller to indicate the current status of the ro-bot platform. The platform is further equipped with sensors  Auton Robot (2010) 29: 85–97 87 measuring diesel level, hydraulic oil level, engine temper-ature and hitch height. The PXI system gathers wheel an-gles, wheel speeds, GPS data, remote control data and hitchheight and controls the vehicle by sending messages to thethree micro controllers connected to the hydraulic valves.A safety system consisting of four red emergency switchesat the corners of the vehicle and a remote switch, controlsthe valves to neutral position on activation, overruling thecomputer control. 3 Path following structure The vehicle control consists of two levels. At high level thewheel angle setpoints and wheel speed setpoints are deter-mined in order to decrease the deviation from the path andthe error in orientation. At low level, controllers are used torealize the wheel angles and wheel speeds determined by thehigh level control.The deviation and the orientation error of the robot froma path are determined by a specially designed orthogonalprojection on the path using the measured orientation andthe GPS position. The orthogonal projection is designed tocalculate the deviation and the orientation error relative to aline of positions  y(x) . 4 Low level control 4.1 Wheel angle process modelThe low level control realizes for each wheel the wheel an-gle and the wheel speed. The hydraulic valves used for steer-ing the wheels of the weeding robot have a certain reactiontime, resulting in a time delay of the steering. Furthermore,if a valve has a commanded open time of less than the deadtime, a control does not have any effect. So the wheel angleprocess can be represented by: ˙ β  = 0 for  t  open  < t  dead   (1) ˙ β  = K p  · u(t   − t  d  )  for  t  open  > t  dead   (2)and: u(t)  =− 1995 if   U <  2500 u(t)  = U   − 4495 if 2500 ≤ U   ≤ 4000 u(t)  = 0 if 4000  < U <  6000 u(t)  = U   − 5405 if 6000 ≤ U   ≤ 7500 u(t)  = 2095 if   U >  7500where: ˙ β  is the wheel steering angle speed  [ ◦ / s ] . K p  is the gain of the process and equals 0.0712. u  is the control corrected for the dead band. U   is the control  [ % U  DC  · 100 ] . U  DC  is the power supply voltage and equals about 12 [V]. t  d   is the delay of the system and equals 0.25 [s]. t  open  is the time generated by a counter counting the timethat the commanded control is in the active band (out-side the dead band where 4000  < U <  6000). It resetswhenthecommandedcontrolreturnsto the deadband. t  dead   is the dead zone of the system and equals 0.15 [s].The value of   t  dead   was determined from tests in which theopen time of a valve was varied,  K p ,  t  d   and the values thatrelate  u  to  U   followed from step responses of the system.4.2 Wheel angle controlTo compensate for the time delay a P controller withSmith predictor is used for the wheel steering control(Stephanopoulos 1984).The wheel angle control of the robot was tested by ap-plying setpoint changes to one wheel while the robot wasstanding still on a flat concrete floor. From some first mea-surements it appeared that at large setpoint changes the vari-able pump controlled by the load sensing system could notreactfastenoughforthechangeintheflowrequiredtomain-tain full pressure in the hydraulic system. Furthermore, if weimagine the robot driving over the field, the flow needed forsteering will require only small changes in the flow alreadypresent for driving. So to simulate the presence of a contin-uous oil flow for driving during the wheel angle control test,one wheel was lifted from the floor and a constant controlwas put on the valve controlling its speed.The average error of a series of 96 measurements on awheel angle setpoint change of 10 degrees decreased withinone second to zero plus or minus 2 degrees (see Fig. 2). Fig. 2  Performance of the wheel angle control, average of 96 mea-surements. The setpoint changes at  t   =  0 s from 10 to 0 degrees (---front left , — rear left,  · - ·  rear right,  ···  front right)  88 Auton Robot (2010) 29: 85–97 5 High level control 5.1 Vehicle modelThe point of the vehicle that should follow the path is the ve-hicle implement attached to the vehicle at a certain speed  v .Consider a path-relativecoordinate system  (x P  ,y P  )  as illus-trated in Fig. 3. The implement position is then completelydescribed by  ξ   = [ x y θ  ] T  where  x  denotes the distancealong the path,  y  the perpendicular offset from the path, and θ   the heading angle of the platform relative to the path (seeFig. 3).Consider a coordinate system ( x v ,y v ) fixed to the robotframe.Thepositionofawheelinthisvehiclecoordinatesys-temischaracterizedbytheangle γ  i  andthedistance l i  where i  is the wheel index. The orientation of a wheel relative to  x v is denoted  β i . The model assumes pure rolling and non-slipconditions and driving in a horizontal plane. Therefore themotion of the robot can always be viewed as an instanta-neous rotation around the instantaneous center of rotation(ICR). At each instant, the orientation of any wheel at anypoint of the robot frame must be orthogonal to the straightline joining its position and the ICR. The two-dimensionallocation of the ICR is specified by the angles of two wheels.For convenience a virtual front wheel  β f   and a virtual rearwheel  β r  is introduced with corresponding  γ  f  ,  l f  ,  γ  r  and  l f  ,respectively located right in between the front wheels andright in between the rear wheels. The motion of the vehicleimplement is described by the following state-space model Fig. 3  Robot with ICR derived from earlier work from Campion et al. (1996) andBendtsen et al. (2002): ˙ ξ   = R T  (θ)Σ(β i )η  (3)where  R(θ)  is the orthogonal rotation matrix: R(θ)  =  cos (θ)  sin (θ)  0 − sin (θ)  cos (θ)  00 0 1   (4)and: Σ(β i )  =  l f   cos (β r ) cos (β f   − γ  f  ) − l r  cos (β f  ) cos (β r  − γ  r )l f   sin (β r ) cos (β f   − γ  f  ) − l r  sin (β f  ) cos (β r  − γ  r ) sin (β f   − β r )  (5)The scalar  η  is a velocity input. The wheel orientation β 3  and  β 4  follow from  β f   and  β r  as described by Bendt-sen et al. (2002) and Sørensen (2002) and  β 1  and  β 2  can befound in a similar way: β 1  = arctan   L sin (β f  ) cos (β r )L cos (β f  ) cos (β r ) −  12 W   sin (β f   − β r )  β 2  = arctan   L cos (β f  ) sin (β r )L cos (β f  ) cos (β r ) −  12 W   sin (β f   − β r )  β 3  = arctan   L cos (β f  ) sin (β r )L cos (β f  ) cos (β r ) +  12 W   sin (β f   − β r )  β 4  = arctan   L sin (β f  ) cos (β r )L cos (β f  ) cos (β r ) +  12 W   sin (β f   − β r )  (6)where  L  is the distance between the front and rear wheelsand  W   the distance between the left and right wheels.The wheel angular speeds  ˙ φ  = [ ˙ φ 1 ,  ˙ φ 2 ,  ˙ φ 3 ,  ˙ φ 4 ] T  are con-trolled at low level, and follow from the vehicle model: ˙ φ  = J  − 12  J  1 (β i )Σ(β i )η(t)  (7)where: J  1 (β i ) =  cos (β 1 )  sin (β 1 ) l 1 sin (β 1  − γ  1 ) cos (β 2 )  sin (β 2 ) l 2 sin (β 2  − γ  2 ) cos (β 3 )  sin (β 3 ) l 3 sin (β 3  − γ  3 ) cos (β 4 )  sin (β 4 ) l 4 sin (β 4  − γ  4 )   (8) J  2  =  r 1  0 0 00  r 2  0 00 0  r 3  00 0 0  r 4  and  r 1 ,r 2 ,r 3 ,r 4  are the radii of the four wheels.
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