PATH TRACKING OF AN AUTONOMOUS LHD ARTICULATED VEHICLE. J. Z. Sasiadek and Y. Lu

Transcription

1 PAH RACKING OF AN AUONOMOUS LHD ARICULAED VEHICLE J. Z. Saiaek an Y. Lu Dept. of Mechanical & Aeropace Engineering, Carleton Univerity, Ottawa, Ontario, KS 5B6, Canaa Abtract: hi paper preent the path tracking an following for the Autonomou LHD (Loa-Haul-Dump) articulate vehicle. A fuzzy logic control ytem for the articulate vehicle wa evelope to control longituinal an lateral motion. he lateral control keep the vehicle along the planne path an the longituinal control make the vehicle running at the require pee. Fuzzy logic control metho i compare with claical metho to evaluate it performance an everal imulation experiment were performe to evelop the path following control ytem with 3-D motion animation. Different type of path were tete an analyze. Copyright@5 IFAC Keywor: Autonomou Sytem, Intelligent Control, Articulate Vehicle, Path racking, an Fuzzy Logic Controller.. INRODUCION Invetigation on the motion analyi of wheele vehicle preente in thi paper have been mainly focue on tability an controller performance. Automatic navigation for autonomou operation of thee vehicle ha become the focu of everal reearch work. he emphai i on autonomy an replacement of human river or telerobotic operator. hi problem ha been aree firt for the evelopment of Automatic Guie Vehicle (AGV) in the automotive inutry an later for the poition control of wheele mobile robot. he purpoe of thi paper i to evelop an avance, fuzzy logic bae controller that woul take into account complex nature of an articulate robot vehicle. he experimental reult were performe on the articulate robot vehicle evelope at Carleton Univerity. hi paper preent only imulation experiment performe to evelop an avance fuzzy logic controller (FLC). he articulate vehicle i compoe of two boie connecte by a kingpin hitch. Each boy ha a ingle axle an the wheel are all non-teerable. Path following control of articulate vehicle i more challenging than controlling car-type robot ue to the more complicate kinematic an ynamic eign. he teering action i performe on the joint, changing the angle between the front an rear part uing hyraulic actuator. he vehicle can teer in place i.e. the orientation of the vehicle change varying the teering angle alone an the with panne by the vehicle when turning i maller than the car-like vehicle. he articulate vehicle i an uneractuate rift-free nonlinear ytem with two input (propulion an teering torque), which are controllable (Laumon, 993). In the articulate vehicle cae, the error between the current an eire trajectorie can be eaily calculate in ome ituation. he main problem for the autonomou navigation of the articulate vehicle i to be able to follow the path while keeping a afe itance from the ege of the path. he problem of

2 the vehicle navigation can be moele a a path to follow an the proper criterion can be formulate for reucing the tracking error of the vehicle. Given the meaurement of the poition an the orientation ifference between the require an the actual value, a controller i expecte to et the teering angle to the appropriate magnitue to bring the vehicle to it eire path. he rate of change of the heaing for the front part of the robot i proportional to the teering angle, the pee of the machine, an the change rate of the teering angle. here are everal metho to chooe the navigation reference point of the articulate vehicle (Saiaek an Green, 996, DeSanti, 99, 997; Altafini an Gutman, 998; Altafini, 999; Polotki, ; Hemami an Polotki, 996; Petrov an Bigra, ; Sampei, et al., 995; Zhang, et al., 997). DeSanti (99, 997) preente a path following controller. An articulate robot wa moele by taking into account not only it baic kinematic, but alo ynamic. he ynamic equation take into account the inertial propertie of the ma of the robot. Uing thi moel, the error term are preente a lateral, heaing, an teering angle error, which are multiplie by proper gain an ae to the nominal turning rate to get the correcte turning rate for the robot. he navigation reference point i electe to be in the mile of the front axle of the vehicle. Analytically erive rule on how to etermine the gain o that the controller become table were hown in (DeSanti, 997). he propoe controller inclue a velocity controller. he articulate vehicle moel ue in thi paper i bae on reult obtaine in (DeSanti, 99, 997). he performance of the controller i verifie by imulation experiment with traight, circular, an mixe path where the poition an heaing error have been mall. he controller woul alo perform well when the error are large or the hape of the reference path i more complicate. he reult with the two ifferent controller are hown in Fig. to Fig... HE ARICULAED VEHICLE MODELS he eire articulate kinematic moel wa preente in (DeSanti, 99) x& = coθv y& = inθv inφv φ& = (+ co φω ) l θ& = ω () where [ x, y ] i the coorinate frame locate in the mile point of the front axle, vi the longituinal velocity, ω i the angular velocity, θ i the orientation of the front boy, an φ i the angle between the front an rear boie. Uner the lippage-free aumption an rear-wheel riving conition, the ynamic moel may be rewritten a: Nv [& ω &] = F + H[ f f ] () where [ fp, f ] i the propulion, teering force acte on the vehicle, an N, F, H are matrice with the ytem parameter. he moel ue the current poition [ x, y, θ, φ ] a the initial conition an integrate it to etermine the vehicle new poition after the input are change for next time tep. 3. PID CONROLLER he input to thi PID path-following controller require the knowlege of the longituinal an angular velocitie [ v ω ] p, an followe by a force block to elect the propulion an teering torque f p an f. he path following tak of the vehicle may be enure by applying the propulion an teering control a: (DeSanti, 997) f p u H F N = f u (3) where [ fp f ] i the propulion an teering torque acte on the riving wheel an the articulate joint. he vector [ u u ] i peuoacceleration. It may be et a: u v v K u = ω ω with K [ K K ] () = i the force component gain vector an [ v ω ] i the reference velocity vector. Coniering the characteritic of the articulate vehicle, velocitie are et a: (DeSanti, 997) v = v ω = ω K [ θ φ l ] o o o where K [ K K K ] 3 (5) = i the velocity component gain vector. θ, φ, l are the irection offet an lateral offet. o o o Uner the aumption that θ, l, an v / v are very o o o mall, the erivative of the path tracking offet can be written in form: θ& o = ω ω φ& o = φ& φ& l& = v θ o o (6)

3 Accoring to the ikhonov theorem on ingular perturbe ytem, if electe ynamic gain K i ufficiently large, it can be et a: v= v hen, it follow: ω = ω K [ θ φ l ] o o o inφv φ& o = (+ co φ)( ω K[ θo φo lo ]) l inφv + + (+ co φ) ω l θ& = K [ θ φ l ] o o o o We can regroup the above equation a: [ ] [ θ & φ & l& ] = A BK [ θ φ l ] (7) where o o o o o o v coφ A= + ω inφ l v B = + [ ( co φ ) ] he kinematic gain vector K = K, K, K 3 can be compute uing Lyapunov inirect metho o a to tabilize the matrix A. Following the pace BK tate claical controller eign proceure, K i aigne the eigenvalue of left-half complex plane. A BK locate in the For the election of the ynamic gain vector K, by the ikhonov theorem, it aigne a ( 3 ) K >> max p, p, p (8) where p, p, p 3 are the eigenvalue of the matrix A BK. where NB, NM, NS, ZE, PS, PM, an PB are linguitic value repreenting negative big, negative meium, an o on. Accoring to the initial conition an the ynamic gain given in the previou ection, the with of the premie an conequent of the rule for K an K are et a: [ ] [ ] K :.5,.5 [, 8] K : 8, 8 [ 8,8] he tructure of the control ytem inclue the conventional proportional control an the aaptive fuzzy logic control unit. he output from the control unit i converte to teering an propulion torque. 5. SIMULAION EXPERIMEN Figure i the flowchart of the control ytem. Firt, the initialization involve creating the vehicle an path for animation an placing the vehicle on the path. Next, the vehicle poition on the path i etermine an the value neee by the controller are calculate. With thee value known, the controller then calculate the neceary propulion an teering input to make the vehicle follow the path. hee input are ue in the ynamic moel to upate the vehicle poition an the animation i then upate to how the vehicle new location. hee tep are repeate until the en of the imulation i reache. Start Simulation Initialize Variable Determine the Poition of Vehicle on Path Calculate Lateral an Longituinal Deviation, Hea Angle an the Relation Angle Determine Propulion an Steering Input he PID controller tranfer function i given by: D () = KP( + + KD) (9) K I. FUZZY CONROLLER Upate the Poition, Poture En? Animation Save Data to file Prepare Data for iplay he fuzzy logic rule bae can be contructe in a ymmetric fahion with rule uch a:. IF e i NB hen u i PB. IF e i NM hen u i PM 3. IF e i NS hen u i PS. IF e i ZE hen u i ZE 5. IF e i PS hen u i NS 6. IF e i PM hen u i NM 7. IF e i PB hen u i NB Fig.. Flowchart for the MALAB Simulink Program he imulation experiment ha been performe with ix ifferent path, a traight line, a circle path, a mixe path that contain two traight egment, followe by a part of circle, a path with two ector (imilar to igit 8), an two other path with uen tuning corner. he path are efine in the [ x, y ]

4 global coorinate. he length an raiu can be efine by the ignal builer in Simulink. he mixe path can be efine uing a ignal generation toolbox. All the path initial parameter are put in the imulation initial m-file. With the vehicle following the path, the controller mut know where the car i locate an how it i oriente (Initial conition). In practice, there are enor on the front an rear of the vehicle that etect the current poition an compare it with the eire poition. In the imulation, the error etimation moel fin the itance between the path an the vehicle. Subequently, thi value i ent to the controller a a reference input. he input to the controller are the heaing angle, or, the angle between the car an the path. he eire value can be calculate uing the path parameter an contraint. he actual value can be calculate uing the vehicle velocity block an contraint. he next tep in the program i to etermine the teering an propulion input to move the vehicle along the path. he controller mut know the value for the linear an angular velocitie error. he movement i animate to provie a mean of viewing the vehicle motion. he firt tep of animation i to create the 3-D moel. he vehicle boy poition i efine uing the patch comman in MALAB. It can be locate an oriente anywhere on it path. In the experiment, the vehicle initial poition i et at [ x y ] = [ 3] an the initial heaing angle θ = π /, an the initial angle between the two part of the vehicle φ =. he vehicle geometry parameter are efine a: l = l = ma ; = m b= m m = m = kg; j = j = kg m In the cae of a line maneuver, we et the linear velocity v =.5 m/. For the circle tracking, the vehicle ha the ame linear velocity an the raiu i efine a R = m. Subequently, the angular velocity i ω =.65 ra /. Alo it i eay to get from the geometry that the angle between vehicle two component φ =. ra while the vehicle move on the circle trajectory. With the conition an parameter lite above, the matrix A an B in Equation 7 can be calculate a: ( forcircle following) ( for line following) A =.67 A.653 =.5.5 [.98 ] B [.3 ] B = = Uing the leat-quare metho, for the kinematic part of the controller, the real part of the eigenvalue of the matrix A BK are negative. It obtain: [ ] K =.8.3 ( for circle following) K = [.9.7 ] ( for line following) he matrix A BK become: or ( for circle following) ( for line following) K i et a K = [5 ].Figure to 5 are the reult from the traight line following cae imulation. Both PID an Fuzzy Logic controller perform initially the tak atifactorily. From Fig. an Fig.3, Y (Meter) rajectorie Deire rajectory X (Meter) Fig.. Enlarge Drawing (Line Following) Lo (Meter) 5 3 Lateral Offet - 3 v o ffet (Meter/Secon) Linear Velocity Offet.5.5 Fig. 3. Lateral an Linear Velocity Offet (Line Following) Hea Angle Offet(Raian) Hea Angle Offet Hea Angle (Raian) Hea Angle Fig.. Hea Angle & Hea Angle Offet (Line Following)

5 the Fuzzy Logic controller ha a maller lateral error than thi with PID.Accoring to the Figure an Figure 5, the orientation error of the vehicle two boie with the FLC are alo maller than thi with PID an converge to zero more quickly. -. Rear Component hea Offet.. Rear Component hea Rate Figure 9 to preent reult for an elliptical path following tak. Similarly, the FLC controller yiel more quickly converging an maller error than thoe with PID. For the rear boy orientation, both controller give imilar reult. It can be notice that the trajectory with the FLC controller i moother than the one with the PID controller. Rear Component hea Offet(Raian) Rear Component hea Rate (Raian) Hea Angle Offet(Raian) Hea Angle Offet Hea Angle (Raian).5 Hea Angle Fig. 5. Rear Component Hea Angle an Offet (Line Following) rajectorie Deire rajectory Fig. 8. Hea Angle an It Offet (Circle Following) rajectorie Deire rajectory Y (Meter) Y (Meter) X (Meter) - Fig. 6. Circle Following Performance (Enlarge Drawing) Lateral Offet Linear Velocity Offet X (Meter) Fig. 9. Oval path Following Performance rajectorie Lo (Meter) v o ffet (Meter/Secon). 6 Deire rajectory.5. Y (Meter) Fig.7. Lateral an Linear Velocity Offet (Circle Following) Figure 6 to 8 are the imulation reult from circle following tak. For thi path, the performance of the PID an FLC are imilar. he error for both controller are mall an eliminate quickly. he rear boy orientation error from FLC i maller an the error curve i moother than PID error X (Meter) Fig.. Oval path Following Performance (Enlarge Drawing)

6 6. CONCLUSION In thi paper, the articulate vehicle i tuie on ynamic level. he problem about the relation between the torque exerte on the vehicle an the acceleration, velocitie, poition, an orientation angle are preente an olution uggete. Bae on the cacae tructure, a controller i eigne to allow the articulate vehicle to follow the aigne path aequately with a given velocity. he tructure Lo (Meter).3.. Lateral Offet v o ffet (Meter/Secon).5..5 Linear Velocity Offet he imulation reult how that the performance of the controller are atifie an table for the ifferent aigne path. By analyzing the experimental reult for ifferent path, one can fin that the controller performance for the path, which curve have the continuing erivative, are better than thoe with noncontinuou erivative. he limitation or iavantage reult from the ynamic characteritic of the articulate vehicle, uch a the tructure limitation, contraint, inertia an other. he two controller, the conventional controller an the aaptive fuzzy controller, can control the vehicle to follow the ifferent path well, but the controller with the fuzzy logic unit perform the path following tak more accurately an more mooth. he lateral itance error an the error of velocitie in the fuzzy logic control proce are maller an converge more quickly Fig.. Lateral an Linear Velocity Offet (Oval path) Hea Angle Offet(Raian) Hea Angle Offet Hea Angle (Raian).5 Hea Angle.5 3 Fig.. Hea Angle an It Offet (Oval path) of thi controller i mae up of a velocity component, which provie the longituinal an angular velocitie that are require for aequate path following, followe by a force component, which provie the propulion an teering torque that are neceary to acquire thee velocitie. hi paper preent a Fuzzy Logic Controller (FLC) tranforming a knowlege bae into a nonlinear mapping, where the knowlege bae conit of a collection of fuzzy IF-HEN rule. An aaptive fuzzy proportional controller ha been evelope to control the vehicle following the precribe path. he fuzzy logic unit can ajut the force gain aaptively accoring to the velocitie error in realtime. REFERENCES Altafini, C., 999, A Path-racking Criterion for an LHD Articulate Vehicle, Int. Journal of Robotic Reearch, Vol.8, pp.35-. Altafini, C., P. -O. Gutman, 998, Path Following with Reuce Off-racking for the n-trailer Sytem, Proc. Of the 37th IEEE Conf. on Deciion & Control, Floria USA. DeSanti, R. M., 997, Moeling an Path-racking for a Loa-Haul-Dump Mining Vehicle, ran. of ASME, Vol.9/5. DeSanti, R. M., 99, Path racking for a ractor- railer-like Robot, Int. Journal of Robotic Reearch, Vol.3, pp Hemami, A., V. Polotki, 996, Problem Formulation for Path racking Autonomation of Low Spee Articulate Vehicle, Proc. of the IEEE Int. Conf. on Control Application, Dearborn, MI. Laumon, J. P., 993, Controllability of a Multiboy Mobile Robot, IEEE ran. on Robotic an Automation, Vol.9, No.6. Petrov, P., P. Bigra,, A practical Approach to Feeback Path Control for an Articulate Mining Vehicle, Proc. of the IEEE/RSJ Int. Polotki, V.,, New reference point for guiing an articulate vehicle, Proc. of the IEEE Int. Conf. on Control Application, pp Sampei, M.,. amura,. Kobayahi, N. Shibui, 995, Arbitrary Path racking Control of Articulate Vehicle uing Nonlinear Control heory, IEEE ran. of Control Sytem echnology, Vol.3, No.. Saiaek, J.Z., Green, D., 996, Path racking, Obtacle Avoiance an Dea Reckoning by an Autonomou Planetary Rover, Int. J. of Vehicle Deign, No..