Tracking Control of Combination Vehicles with Preview Feedforward Compensation

Transcription

1 Memois of the Facult of Engineeing, Okaama Univesit, Vol. 4, pp. -8, Janua 9 Tacking Contol of Combination Vehicles with Peview Feefowa Compensation Shuji MABUCHI, Masami KONISHI, Jun IMAI Dept. of Electical an Electonic Engineeing Okaama Univesit --, Tsushima-Naka Okaama, 7-85 (Receive Novembe, 8) A tacto-taile vehicle in the facto might move on the oute etemine befoehan. Howeve, automation of a tacto-taile vehicle is ifficult so that it is necessa to consie the nonlineait of a vehicle an a taile. In this eseach, the effective tacking contol metho of a tacto-taile vehicle is popose. The contol metho using time-state contol fom fo a peview contol is pesente, an be applie a tacto-taile vehicle with nonholonomic chaacteistics. Results of numeical expeiments ae pesente to check effectiveness of the popose contol metho. INTRODUCTION A tacto-taile vehicle is wiel use to ca mateials fom the supplie stock pat to the conveance pat in the facto. Because thee is an avantage that the lage amount of supplie mateials can be tanspote b a tacto-taile with low-cost. The opeation equies a lot of skill fo ive when a tactotaile vehicle moves along a cuve law an paks into a gaage. Because of such a ifficult opeation, the ive suppot of fou-wheele vehicle fo the paking suppot sstem is appeae in avance. Howeve, automation is not seen in the tacto-taile vehicle. The tacto-taile vehicle cannot be stabilize b a static feeback that is calle nonholonomic sstem. To eal with this poblem, using a time vaing contolle metho an a time-state contol fom metho, a new contol sstem is popose. The popose contol metho is applie to the tacking contol of the vehicle. At the same time, a metho using a geometic tansfomations is popose. In man cases, the tacto-taile caie in the facto moves on the oute mabuchi@cnt.elec.okaama-u.ac.jp ecie befoehan. Theefoe, it ma lea lage contibution in the automation of the iving a tactotaile vehicle if the new contol metho b which the vehicle follows the oute automaticall. Howeve, it is limite to a two-wheele o a thee-wheele vehicle that is examine befoehan, an the tacto-taile vehicle is not teate. In this eseach, the pupose is to cultivate tacking contol metho of the tactotaile vehicle. It assues a goo follow up capabilit using peview contol fo the poblem whee the taget oute is known befoehan. Moeove, because the vehicle is nonholonomic sstem, time-state contol fom is use in the application of the popose metho. The effectiveness of the poposal metho is examine though numeical expeiments an the laboato expeiments. TRACTOR-TRAILER VEHICLE MODEL The tacto-taile vehicle moel that teats b this eseach is shown in Fig.. This wok is subjecte to copight. All ights ae eseve b this autho/authos.

2 Shuji MABUCHI et al. MEM.FAC.ENG.OKA.UNI. Vol. 4 L Tacto ( x, ) ( ) x, v ψ θ L. Tacking Contol Metho It is assume that the objective oute is given as shown in Fig. b iscete sequence of points when the vehicle uns. Objective point ( ) x, v θ θ Taile Fig. Tacto-taile vehicle moel L an L ae length of the tacto an taile of the vehicle, (x, ) is cooinate of mipoint of the ea wheels of a tacto, v an v ae the spees of the vehicle. Assuming that the wheels on t ski, velocit constaints ae given the Eq. (). ẋ sin(θ + ψ) ẏ cos(θ + ψ) ẋ sin θ ẏ cos θ ẋ sin θ ẏ cos θ x () Moeove, the next expessions ae obtaine as angula velocit estaints. θ v L tan ψ θ v L sin(θ θ ) () Let inputs ae u v, u ψ. Eq. () an Eq. () give state equation as shown in Eq. () which is nonholonomic sstem with velocit an position constaint. t x θ θ ψ cos θ sin θ tan ψ L sin(θ θ ) L u u TRACKING CONTROL OF TRACTOR-TRAILER VEHICLE () This chapte escibes the metho when the vehicle follows the oute. In the following, elements an its functions of the popose metho will be pesente. ( p x, p ) Oigin shift Rotational cooinate tansfom φ Linea intepolation Fig. Objective path fo tacking contol metho Fist, it intepolates between the point an the point b linea intepolation an it is assume a new objective oute. Next, axis is matche to the taveling iection axis b the otational cooinate tansfom an oigin shift. As a esult, the eo fom the oute can be teate with axis. Moeove, the pupose of it is to evae that the input iveges at the singula point if angle of the vehicle becomes 9 egees when time-state contol fom is use. Afte that, the vehicle is stabilize to axis b state feeback contol fom pesent objective point to next objective point. Finall, it is possible to follow to the taget oute b epeating fom the otational cooinate tansfom to the state feeback contol.. Rotational Cooinate Tansfom an Oigin Shift The x cooinate sstem is convete into cooinate sstem b the otational cooinate tansfom an oigin shift. The elation between the x cooinate sstem an the cooinate sstem is as follows. cos φ sin φ sin φ cos φ x x p p x (4) (x p, p ) is cooinates of the objective point an φ is a otation angle of the cooinate sstem. As fo φ, axis is ecie to coespon with objective oute of linea intepolation. When the angle of the vehicle in

3 Janua 9 Tacking Contol of Combination Vehicles with Peview Feefowa Compensation cooinate sstem is assume to be ϕ an ϕ, the next expessions ae obtaine. ϕ π θ + φ ϕ π θ + φ (5) Rewiting Eq. () b Eqs. (4) an (5), Eq. (6) can be eive. φp + sin ϕ u ṙ φp + cos ϕ u ϕ φ v (6) L tan ψ ϕ φ v L sin(φ φ ) Eq. (6) is the state equation concening,, ϕ, ϕ, ψ assuming φ is constant because the objective oute is staight. t ϕ ϕ ψ cos ϕ sin ϕ tan ψ L sin(ϕ ϕ ) L u u (7). Convesion to Time-State Contol Fom The constaint like Eq. () is calle nonholonomic constaint which can t enote position an angle,, an the constaint that can t be integate. The sstem incluing such constaints ae calle nonholonomic sstem, an it is obseve as the sstem which it is ifficult to contol geneall. Thee is a time-state contol fom metho to nonholonomic sstem. This metho is suitable fo the contol of the vehicle. Because it is eas fo iving the vehicle fowa an back to be eas, an to appl the linea contol theo. This chapte explains the contol metho that uses a time-state contol fom. Eq. (7) is convete into a time-state contol fom. Let tansfome state vaiable an input ae z, z, z, z 4, z 5 T an µ, µ T, espectivel. As a cooinate tansfom, consie the following equation. z z sec ϕ sec ϕ tan ψ L L + sec ϕ sec ϕ sin(ϕ ϕ ) L {cos ϕ + sin ϕ sin(ϕ ϕ )} z sec ϕ sec ϕ sin(ϕ ϕ ) L z 4 tan ϕ + sec ϕ sec ϕ sin(ϕ ϕ ) z 5 L ln ( ) + sin ϕ sin ϕ (8) As a input tansfom, consie the following equation. µ cos ϕ u µ tan ψ sin ϕ L L cos 5 cos ϕ sin(ϕ ϕ ) ϕ cos ϕ L cos ϕ cos 4 ϕ + tan ψ cos ϕ cos ϕ L L cos4 ϕ cos ϕ + tan ψ cos ϕ sin ϕ sin(ϕ ϕ ) L L cos4 ϕ cos ϕ 4 cos ϕ sin ϕ sin (ϕ ϕ ) + sin (ϕ ϕ ) L cos ϕ cos 4 ϕ sec ψ L L cos 4 u ϕ cos ϕ u Eq. (8) an Eq. (9) give a time-state contol fom. : ż µ : z 5 z 5 () z 4 z z z 4 z + µ z z The expession above is compose of subsstem Σ an Σ. The singula point is geneate b convesion when the angle of the vehicle an the steeing angle at 9 egees. This poblem is solve b the otational cooinate tansfom an oigin shift intouce b subsection.. It is the featue of this sstem that subsstem Σ is iffeentiate b z, usuall tempoal iffeentiation. Theefoe, the linea contol theo can be applie b monotonousl inceasing ( o eceasing ) z just like time, subsstem Σ comes to linea sstem..4 Peview Contol The tacto-taile vehicle is able to tack the objective oute b otational cooinate tansfom an oigin shift, an evasion of singula point. The eo inceases ve much at the otational cooinate tansfom in the next objective point, because it is the settling contol to the taveling iection axis between the objective point an next objective point. The vehicula swept path becomes not smooth because of this poblem. Moeove, even if the contolle paamete is ajuste to pevent this poblem, the eo fom the objective oute inceases. Theefoe, the opeation befoehan of cutting the steeing wheel befoe the cuve is ae to the vehicle. To achieve such an opeation, a peview contol4 is taken. When the vehicle unestans the efeence value in the futue, the peview (9) 4

4 Shuji MABUCHI et al. MEM.FAC.ENG.OKA.UNI. Vol. 4 feefowa compensation that as the input coesponing to it is esigne as shown in Fig.. Objective value Peview compensation + + Steeing spee Spee Vehicle moels Contolle sevo sstem Vehicle position, angle,steeing angle Fig. Block iagam of peview contol sstem Application to the eal sstem is foeseen an it esigns in the iscete-time sstem. Because it is constant that the input of subsstem Σ is contolle, the peview feefowa compensation like Fig. is esigne fo subsstem Σ. The expane sstem that as eviation to the state vaiable is efine, an the peview feefowa compensation is esigne to that sstem..4. Design of Peview Contol Feefowa Compensation It is assume that the taget value fom M R step ahea is alea-known as of pesent. The evaluation function incluing the eo an the input is efine. J Z T (k)qz (k) + µ T (k)h µ (k) (5) k M + The eason of summing fom k M + is that the input is change M R step ago. The efeence value is change as shown in Fig. 4. The evaluation function is minimize b contol input as follows. value R(k) Input u(k).4. Sevo Sstem The sstem that iscetizes the subsstem Σ b k is efine as follows. Z(k + ) A Z(k) + B µ (k) (k) C Z(k) () Hee, it is Z z 5, z 4, z, z T. The eviation e is efine as follows. e(k) R(k) (k) () R(k) : vecto of objective value The expane sstem is given b the fist-oe eviation e an iffeence of. e(k + ) I C A e(k) Z(k + ) A Z(k) C B I + µ (k) + R(k + ) () B is the fist-oe eviation smbol. Define Z e(k), Z(k) T, the expane sstem is shown as follows. Z (k + ) Φ Z (k) + G µ (k) + G R R(k + ) (4) The feeback contolle esigns the optimal sevo sstem that settles z 5 in the efeence value. The eason of not consieing the angle of vehicle as efeence value is that the angle of vehicle tacks to it natuall. Because the angle of vehicle is ajuste b tuning a hanle in oe to egulate the eviation on axis to zeo. M R + M R Peview step Fig. 4 Peview contol time M R µ (k) F Z (k) + F R (j) R(k + j) (6) j Hee, F R (j) is tem which is ae b pevision. It is the contol sstem that the feefowa compensation is ae to the optimal sevo sstem. The peview feefowa coefficient F R (j) is obtaine to minimize the evaluation function of Eq. (5). F R () F R (j) H + G T P G G T (Φ + GF T ) j P G R (j ) Eq. (7) is the peview feefowa compensation which as the input of M R step ahea in Fig.. P is positive efinite solution which given b the iccati equation as follows. P Q + Φ T P Φ Φ T P GH + G T P G G T P Φ (8).4. To Decie Value Fig. 5 shows how to take the peview efeence value. The setup peview efeence value is obtaine fom (7) 5

5 Janua 9 Tacking Contol of Combination Vehicles with Peview Feefowa Compensation pesent vehicle position in tansfome sstem. In this eseach, It is use to a peview contol that the efeence value is obtaine b z 5 of Eq. (8), because the value of axis is egae onl. colo to the bo, an pocessing the image of the USB camea. The contol signal geneate with PC b the Bluetooth communication is tansmitte to the vehicle. Peview steps Objective oute USB camea Eo of vetical iection Vehicles value Distance of taveling iection Fig. 5 Pospective efeence Infomation is acquie fom the image The contol signal 4 VERIFICATION BY EXPERIMENT 4. Numeical Expeimental Result Vehicle Fig. Expeimental sstem PC The effect of peview contol is confime b the efeence oute as shown in Fig. 6. The paamete of the vehicle is L.m an L.9m in simulation. The peview feefowa compensation was esigne b M.m. The contolle of Fig. is esigne b section.4.. The esults of the contol sstem without the peview feefowa compensation is shown in Fig. 6 an 7, an the esults of contol sstem with the peview feefowa compensation is shown in Fig. 8 an 9. The eo aea an peak inpu ae shown in Table.. Table. Effect of peview contol without peview with peview eo aea cm peak inputa/s. -.8 As against the contol sstem without peview, the eo aea is ecease at 7, an the peak input is ecease at 7 in the contol sstem with peview. The effectiveness of peview contol follows fom these esults. 4. Expeiment of Real Sstem The expeimental sstem is shown in Fig.. The vehicle of the same size as simulation is configue b LEGO MINDSTORMS NXT. The contol sstem at the velocit an the angula velocit of the vehicle use the PI contol. The position an the angle of the vehicle ae measue b pasting seals with iffeent The esult of contol sstem without peview is shown in Fig. an the esult of contol sstem with peview is shown in Fig.. In Fig., the vehicle tuns in oe to coect eo, an it slie off the oute. Because a lage input entes the vehicle as shown in Fig. 7, afte that, it cannot etun the steeing wheel. The vehicle using a peview contol follows without sliing off the oute. Both Fig. an Fig. show eviations fom simulation esults, because the velocit contol sstem of the vehicle i not show ieal esponse. 5 CONCLUSION In this eseach, tacking contol of the tacto-taile vehicle fo the case of a peetemine efeence oute was stuie. The contol esign technique was popose to combine a time-state contol fom an a peview contol. The effectiveness of the popose metho was confime b expeiments. In the futue tasks, consieing the qualit of the vehicle s velocit contol sstem is to be consiee an the contol sstem is to be esigne to check the pefomance with constaints. REFERENCES C. Samson, IEEE Tans. on Automatic Contol, Vol. 4, No., pp (995). 6

6 - Shuji MABUCHI et al. MEM.FAC.ENG.OKA.UNI. Vol um/sec ua/sec um/sec ua/sec Input of angula Input velocit of angula velocit timesec timesec Fig. 7 Input without peview contol Fig. 6 Swept path of vehicle without peview contol um/sec ua/sec um/sec ua/sec Input of angula Input velocit of angula velocit timesec timesec Fig. 9 Input with peview contol Fig. 8 Swept path of vehicle with peview contol Fig. Swept path of vehicle without peview contol Fig. Swept path of vehicle with peview contol 7

7 Janua 9 Tacking Contol of Combination Vehicles with Peview Feefowa Compensation Mitsuji Sampei, Jounal of the Societ of Instument an Contol Enginees, Vol. 6, No.6, pp.96-4 (997). 4 Tsuchia Takeshi an Taashi Egami, Digital Peview Contol Sangoutosho(99). Taashi Egami, Jounal of the Societ of Instument an Contol Enginees, Vol. 45, No.7, pp (6). 8