Geometric Path Planning for Automatic Parallel Parking in Tiny Spots

Transcription

1 13th IC Smposium on Contro in Transportation Sstems The Internationa ederation of utomatic Contro September 1-14, 01. Sofia, ugaria eometric Path Panning for utomatic Parae Parking in Tin Spots Héène Vorobieva Sébastien aser Nicoeta Minoiu-nache Saïd Mammar Renaut, 1 avenue du of, 7888 uancourt, rance (e-mai: heene.vorobieva@renaut.com, nicoeta.minoiu-enache@renaut.com) ISTTR, IVIC, 14 route de a Minière, âtiment 84, Sator, Versaies, rance (e-mai: sebastien.gaser@ifsttar.fr) IISC, Université d vr Va d ssonne, 9105 vr, (e-mai: said.mammar@ifsttar.fr) bstract: This paper deas with path panning for car-ike vehice in parae parking probems. Our path panning method uses simpe geometr of the vehice kinematic mode. The presented strateg consists in retrieving the vehice from the parking spot and reversing the obtained path to park the vehice. Two methods for parking in tin spaces, where parking in one tria is not possibe, are proposed. In these cases, the number of needed trias to park the vehice can be cacuated from a simpe formua or from an iterative agorithm. The proposed panning methods are independent of the initia position and the orientation of the vehice. Reference trajectories are generated so that the vehice can park b foowing them. Simuations are provided for both methods. Kewords: utomatic Parking Sstem, Trajector Panning, Path Panning, Parae Parking, eometric pproaches, utonomous vehices. 1. INTROUCTION Since parking spots have become ver narrow in big cities, drivers need to be eperimented and ver attentive when maneuvering the vehice. This often eads to minor scratches on the car and increases traffic jam b mutipe repositioning. Therefore, automatic parking is a soution to reduce stress and increase comfort and securit of the driver. Here we consider parae parking probem, which is particuar demanding for the driver. There are man methods to tacke the trajector generation for the parking probem: - Methods based on the use of reference functions. ee et a. (1999) present a method using apunov function to stabiize the vehice to a ine corresponding to the parking spot. Paromtchik et a. (1997) propose the optimization of two vaues to find steering anges and durations of commands to eecute the parking maneuvers. These methods strong depend on gains and parameters chosen for the functions, which can be difficut to adjust and not certain eads to correct parking maneuvers. - Methods based on fuzz ogic (for eempe Zhao and Coins (005)) or neura network (for eampe enkins and Yuhas (1993)) to earn human technique. The are imited to human eperts knowedge and are difficut to generaize. - Methods based on two phases path panning, for eampe acobs et a. (1991) and aumond et a. (1994): creation of coision-free path b a ower-eve geometric panner that ignores the motion constraints and subdivisions of this path to create an admissibe path. n optimization routine can reduce the ength of the path. - eometric methods based on admissibe coision-free circuar arcs, which ead the vehice in the parking spot in one tria (o et a. (003), upta and ivekar (010), Choi et a. (011)). The trajectories created with these methods invove eas geometrica equations. Whereas in the method proposed b upta and ivekar (010) the minimum parking spot depends on the initia position of the vehice, o et a. (003) and Choi et a. (011) propose a minima parking spot, which on depends on the characteristics of the vehice. These methods propose on parking in one tria (without ongitudina veocit sign changing); possibe on if the parking spot is sufficient ong. Here a geometric approach is considered based on retrieving a vehice from parking spot and reversing this procedure to sove the parking probem (o et a. (003), Choi et a. (011)). If the vehice is parked and the parking spot is ong enough to retrieving in one tria, a human driver steers the front whee to maimum ange toward the outside of the parking spot and moves forward unti the vehice is retrieved. Then, he steers on the opposite side to position correct the vehice aong the road. This procedure being reversibe, we app it in parae parking probem to form a simpe path composed b two circes connected b a tangent point. Whereas in o et a. (003) and Choi et a. (011) the authors proposed two identica circes, here on the circe with the minimum radius for the second part of the maneuver is retained. This aows to have vehices /1/$ IC /

2 September 1-14, 01. Sofia, ugaria with a different maima steering ange for the right direction than for the eft direction. t the beginning of the maneuver, this aso avoids steering at zero speed unti the maimum steering ange is reached, which is demanding to the eectric machine of the steering coumn. In Choi et a. (011), the initia position of the vehice has to be parae to the parking spot, on a minimum circe path. In the present soution, if the vehice is sufficient far from the spot, its initia position can be everwhere, with an orientation. In consequence, a new path can be cacuated at ever moment of the maneuver if the rea position of the vehice differs with respect to the panned path. Two generaizations are aso proposed to park the vehice in severa trias when the parking spot is not ong enough to park the vehice in one tria. The parking maneuvers are presented assuming that the width of the parking is at east equa to the width of the vehice. The stud concerns the parking on the right side, but it can be easi generaized for the eft side parking. Reeds and Shepp (1990) showed that optima paths for a vehice going forwards and backwards can be obtained with circes of minima radius. In consenquece, this paper puts forward paths constituated when possibe b arcs of circe of minima admissibe radius. In the net section, the path panning in one tria is depicted. Sections 3 and 4 are devoted to the generaization for parking in severa trias. In section 5, the differences between these two generaization methods are discussed. In section 6, the generation of reference trajectories is outined. ina, section 7 contains the concusion.. PTH PNNIN IN ON TRI.1 Mode of the vehice In this paper front whees steering vehices are studied. The mode chosen for simuations is the Renaut uence Z. Tabe 1, Tabe, ig. 1, and ig. show the notations and the vaues used here. The vehice is represented b its bounding rectange, incuding the outside rear-view mirrors. The front track is approimated to have the same ength as the rear track. Tabe 1. Mode of used vehice Parameters Notation Vaue Wheebase a 701 mm Track b 1537 mm ront overhang d front 908 mm Rear overhang d rear 1114 mm istance from the eft, right whee to the eft, right side of the vehice (eterior mirrors foded) d,d r 136 mm Maimum eft, right steering ange δ ma,δ rma 38 degree. eometric properties The parking maneuver is a ow-speed movement. Consequent the ckerman steering is considered with the four whees roing without sipping, around the instantaneous center of rotation. ifferent turning radius are cacuated. d rear d d r b d front ig. 1. Vehice in goba (,, )-coordinates R R ig.. Vehice in a parking spot With and being respective the center of the rear and the front track, it ieds: a R = a/ tan δ, R = a/ sin δ (1) or eampe, to have R, we take δ. Minimum radius is obtained with the maimum steering ange. With,, being respective the right front, the right rear and the eft rear etremities of the vehice, apping the Pthagorean Theorem it resuts: R = (R + b + d r ) + (a + d front ).3 Strateg R r = (R r + b + d ) + d rear R r = (R r b d r ) + d rear To park the vehice in one tria, the probem is taken in the reversed wa b retrieving the vehice from the parking spot ike in o et a. (003) and Choi et a. (011). tria or a maneuver is defined as a sequence without veocit sign changing. It is assumed that the parking space is ong enough for the one tria parking and that the distance between the rear vehice and the used vehice, when it is δ Tabe. Notations Meaning Notation istance between points (e:, ) d Circe of center C and radius R C(C, R) eft, right instantaneous center of rotation, istance between a point (e: ) and, R, R r ength of the parking spot bsoute vaue of the steering ange (right, eft) δ (δ r, δ ) Orientation of the vehice δ () 37

3 September 1-14, 01. Sofia, ugaria initia position (mm) (mm) d C init α init δ r R min (mm) (mm) R min fina position R init r R init r ig. 3. Strateg for parae parking in one tria parked, is sma. In these conditions, to eit the parking spot, a human driver steers the front whees to maimum ange toward the outside of the parking spot and moves forward unti the vehice is retrieved. This creates a path in two arcs of circe, connected b a tangentia point. Choi et a. (011) propose to take a both circes of minimum radius. This means, that at the beginning of the maneuver the car has to be perfect positioned and parae to the parking spot. One of the objectives in this stud is to aow the vehice an initia position and orientation and to give it the possibiit to cacuate a new maneuver at an moment without need to repace itsef at initia position, for instance if the path of the vehice differs from the cacuated one. or that, the second arc of circe (of the retrieving) must have the possibiit to be of higher radius than the minima radius. See ig. 3 for visuaization of one tria path and its concerned vaues. The first arc of circe (which is the second in the parking maneuver) is of minimum radius, but the second has to connect the first circe with the rea initia point of the car. s the first arc of circe is of minimum radius and as the position where the car has to park is known, and R min can be deduced. uring this arc of circe, the point goes a over C(, R min ). or the second arc of circe, we search C(, R initr), which aows to go from its initia position to C(, R min). s and the initia position of are known, d C init is cacuated. Having the initia orientation of the vehice, the ange α = C init is deduced from: d C = R min + R initr and from the ines ( init ) and ( init ) beeing penpedicuar. pping the -Kashi Theorem to the triange init (see ig. 3) it ieds: dc R initr = init R min R min + d init cos α.4 Conditions for feasabiit (3) δ r = arctan (a/r initr) (4) The one tria parking without coision is possibe under two conditions: the ength of the parking has to be bigger than a minima ength and the circe C(, R initr) has to be admissibe for the vehice. ig. 4. Simuations of parking in 1 tria with different initia orientations of the vehice ike in Choi et a. (011) the minima ength is: min = d rear + R min (a/(tan δ ) b d ) (5) Unti the end of the section, = init is taken. This means, that the vehice is considered at the start position. or C(, R r ) being admissibe, R r R rmin is necessar. Taking R r = R rmin, the circe is admissibe when d C d C min. rom (3) it is deduced: d C min = R rmin cos α + (Rrmin cos α) + R min + R rminr min If d C < d C min the vehice has to move forward unti d C min. d C.5 Simuation Simuation on Matab were performed with the cinematic vehice mode with the parameters indicated in section.1. With this mode the minima ength of the parking is 630 mm. The ig. 4 shows two simuations with different initia orientations. It can be noticed, that the second arc of circe of the path beongs to the same circe in both cases. On the ength of these arcs of circe differs. In these simuations the initia position of the vehice is not parae to the parking spot, in consequence these resuts cannot be obtained with the method of Choi et a. (011). 3. PRKIN IN N PR TRIS When the ength of the parking spot is smaer than min, but sti bigger than the ength of the vehice, it is possibe to park the vehice in severa trias. One soution is to go to the nearest parae position from the parking spot using the one tria method. Then the vehice eecutes series of forward and backward moves to park in the spot. t the end of each of these moves, the vehice has to be parae to the spot. 3.1 Position at the end of the first tria uring the first tria, the vehice parks in the nearest parae position to the parking spot. This means that for the second arc of circe of the parking maneuver we have d C parae = d C = R min. Having this arc of circe, the first one is cacuated ike in section. However, to know the second arc of circe, the nearest parae position has to be cacuated, defined b the distance d between the parking spot and the vehice in nearest parae position (see ig. 5 for visuaization of the vehice in the nearest parae position). (6) 38

4 September 1-14, 01. Sofia, ugaria d ig. 5. Vehice at the nearest parae position to the parking spot after the first tria Δ d beginning of forward move Δ/ R / R R R end of forward move ig. 6. orward move during parking in n parae trias Cacuation of d: If the vehice was parked in the spot and it tried to retrieve with maimum steering ange, there woud have be the point 1 : intersection of the circe C(, R min ) and the ine = (see ig. 5 and ig. 8). In these conditions, 1 can be easi cacuated. Then d is deduced: d = Number of trias uring the series of forward and backward moves, the vehice covers ongitudina the avaiabe distance, where = (a + d front + d rear ). uring the first / distance, the steering is to eft, and during the second / distance, the steering is to right (see ig. 6). To be awas parae to the parking spot at the end of each move and to maimize atera dispacement, eft and right steering anges must be maima and equa: δ = min(δ ma, δ rma ). R = a/ tan δ is deduced. The atera dispacement during a forward or backward move is: ( ) = R R /4 (7) The tota number of trias is deduced (first tria pus series of forward and backward moves): nb trias = IntegerP art(d/ ) + (8) ig. 7. Simuation of parking in 5 trias b parae method 3.3 Simuation Simuation on Matab was performed with the same mode as in section.5. The ig. 7 shows a case with parking in 5 trias. 4. PRKIN IN N OPTIM TRIS To find the parking path with fewest maneuvers, the retrieving path ike a human driver woud do is searched and then reversed. The retrieving b a human is composed b forward and backward moves unti the vehice can retrieve. forward move is steering to maimum ange toward eft and moving forward unti approaching the front obstace. backward move is steering to maimum ange toward right and moving backward unti approaching the rear or atera obstace. When a forward move aows the vehice to retrieve without coision, the concerned arc of circe is considered, which wi be the second arc of circe of the parking path. Then, the feasibe arc of circe is searched, which connects b a tangentia point the rea initia position of the vehice to the second arc of circe. 4.1 gorithm for a trias ecept for the first one Here is defined the agorithm, which finds the arcs of circe for forward and backward moves, ecuding the first tria, b retrieving the vehice. The first and second arcs of circe composing the first tria are determined in the net section. (1) The ast position of the vehice when it is parked is known: it is the goa position.,, R min, R rmin for this position are cacuated (see ig. 8). () If R min d C go to (3), ese go to (4). (3) The vehice can now retrieve in one tria. Having current, the method detaied in net section is appied to find the path of the first tria. (4) It is not possibe to retrieve in one tria. The vehice moves forward with maimum eft steering ange. The created arc of circe C(, R min) cross the ine = to a point 1 (see ig. 8). t the end of this arc of circe, the point of the vehice is in 1. (5) Having, 1, and before step (4), the rotation ange θ is deduced. (6) is the instantaneous center of rotation during the maimum steering; consequent, it can be considered that it beongs to the vehice. When the vehice moves aong the arc of circe determined in steps (4) and (5), a points of the vehice, incuding, are transformed b the rotation of center and ange θ : (, ) vehice (9) 39

5 September 1-14, 01. Sofia, ugaria initia position θ R θ R min d C init α init δ r R min 1 H position at the end of the second arc of circe R init r R init r θ r R r ig. 10. The two arcs of circe of the first tria during parking in n optima trias ig. 8. Vehice moving forward with maimum eft steering ange θ 1 vehice moves aong the arc of circe determined in steps (7), (8), and (9), a the points of the vehice, incuding, are transformed b the rotation of center and ange θ r : (, ) vehice (10) ( ) ( new cos θr sin θ = C r + r new sin θ r cos θ r ) (( ) ) rom now on, on points after this rotation are considered. (11) New d C is cacuated and then go to step (). 1 θ θ r r ig. 9. Vehice moving backward with maimum right steering ange ( ) ( ) (( ) ) new cos θ sin θ = C + C new sin θ cos θ rom now on, on points after this rotation are considered. (7) The vehice moves backward with maimum right steering ange. The created arc of circe C(, R rmin) crosses the ine = to a point 1 (see ig. 9). Moreover, the created arc of circe C(, R rmin) crosses the ine = to a point. (8) Having current,,, and cacuated 1 and, the rotation anges θ r and θ r are deduced. (9) The vehice must move backward as far as possibe, but without coision. Consequent, it must stop when is in or when is in 1. The corresponding arc of circe is defined b θ r = min(θ r, θ r ). (10) is the instantaneous center of rotation during the eft maimum steering; consequent, it can be considered that it beongs to the vehice. When the 4. irst tria t the end of the agorithm above, the current step is the step (3) and current position of the vehice, incuding C, is known. This is the position, in which the vehice wi be at the end of the first tria (after the two arcs of circe of the first tria) in maneuver of entering the parking spot. This means that during the second arc of circe, the point goes a over C(, R min). ike in section, C(, R initr), which aows to go from its initia position to C(, R min), is searched (see ig. 10). ike in section, d C init, ange α = C init are cacuated and then b apping the -Kashi Theorem, R initr (3) and corresponding δ r (4) are obtained. Remark about the feasibiit: The condition of non-coision with the aread parked vehices is satisfied b positioning the vehice on a noncoision path during the agorithm detaied in the previous section. The condition of feasibiit of the first arc of circe is the same as in section. The rotation anges θ r and θ for the two arcs of circe of the first tria coud be determined as foows: s the initia positions of, and and their positions at the end of the second arc of circe are known (ig. 10), the anges θ r = init and θ = C for the first and second arc of circe are deduced. 40

6 September 1-14, 01. Sofia, ugaria (mm) (mm) ig. 11. Simuation of parking in 3 trias b otima agorithm 4.3 Simuation Simuation on Matab was performed with the same mode as in section.5. The ig. 11 shows a case with parking in 3 trias. 5. ISCUSSION Sections 3 and 4 presented two methods for parae parking in n trias, when parking in one tria is not possibe. The method in n parae trias is ver simpe to impement and the number of trias is easi cacuated from the geometr of the vehice and of the parking spot. Nevertheess, the condition of dispacement with beginning and end of each move being parae to the parking spot is ver restraining: the tota number of trias increases ver fast when decreases (see Tabe 3), what can be unsatisfactor if a passenger or a driver is present in the vehice. The method in n optima trias is instinctive, when reversing the human retrieving maneuver. With this method, the number of trias can be cacuated on from an iterative agorithm. eometric transformations and formuas invoved in the agorithm are more compe in comparison with the method in n parae trias. It is not eas to visuaize for a human which trajector the vehice wi generate. Nevertheess the number of trias impied in this method is much ess considerabe than in the method with n parae trias (see Tabe 3 for comparison of these methods using mode of the section.1 for a spot of standard width,5 m). In consequence, this method is to be used in priorit with passengers on board or to reduce the time of the parking maneuver. Tabe 3. Comparison between the two methods in severa trias ength of the spot (cm) Parae method (trias) Optima method (trias) NRTION O RRNC TRCTORIS To make the vehice foow the generated path, time contro commands of the steering ange δ and ongitudina veocit v need to be buit. s each path presented in this paper can be divided in arcs of circe, a genera contro approach used for each arc of circe is presented. ongitudina veocit (mm/s) steering ange (degree) time (s) time (s) ig. 1. Simuation of contro commands for the parking in 3 optima trias ach arc of circe is defined b a radius R and an ange θ. The contro commands are generated for the midde of front track, in consequence each radius concerned in the arcs of circes can be cacuated b R = a/ sin δ. or each arc of circe of the path, the used θ and δ have been presented in sections 3 and 4. In consequence, δ known for each arc of circe and the ength of each arc of circe arc = θr are deduced. To generate ongitudina veocit, the same method as in Choi et a. (011) is used. The needed time to reach a desired maima veocit or to reach nu veocit from the maima veocit v ma is t 1 = v ma /γ des, where γ des is the desired acceeration for the parking maneuver. uring t 1, the vehice goes over the distance d 1 = γ des t1 /. Then, the vehice stas at constant speed v ma during the time t = ( arc d 1 )/v ma. The contro commands are: t (0, t 1 + t ) δ(t) = k δ δ known t [0, t 1 + t ] (11) { kv γ des t t [0, t 1 ] v(t) = k v v ma t [t 1, t 1 + t ] (1) k v γ des t t [t 1 + t, t 1 + t ] where k δ = ±1 corresponds to a eft (+1) or right steering (-1), k v = ±1 corresponds to forward (+1) or backward (-1) motion. simuated eampe is shown in ig. 1. These commands are open-oop in the (,, )-coordinates. The steering coumn and the engine are controed to eecute these commands to provide the desired path and orientation of the vehice. Possibe errors can be compensated b recacuating the geometric path at an moment. 7. CONCUSION as geometric path panning for parae parking was proposed. Two methods for the parking in severa trias, when the parking in one tria is not possibe, were desribed. The methods are independent of initia position and orientation of the vehice. The parking is possibe as ong as the ength of the parking spot is onger than the ength of the vehice. The methods aow to generate a new path, 41

7 September 1-14, 01. Sofia, ugaria without repacing the vehice at an initia position, if the path of the vehice differs from the first generated one. Simuations were performed and in future, these methods wi be tested on rea vehices. The methods wi be generaized for the forward parae parking and for the other parking configurations. RRNCS Choi, S., oussard, C., and d ndrea Nove,. (011). as path panning and robust contro for automatic parae parking. In Proc. of the 18th IC Word Congress. Miano, Ita. upta,. and ivekar, R. (010). utonomous parae parking methodoog for ckerman configured vehices. In Proc. of Int. Conf. on Contro, Communication and Power ngineering. Chennai, India. acobs, P., aumond,.p., and Tai, M. (1991). fficient motion panners for nonhoonomic mobie robots. In Proc. of I/RS Int. Work. on Inteigent Robots and Sstems. Osaka, apan. enkins, R.. and Yuhas, H.P. (1993). simpified neura network soution through probem decomposition: the case of the truck backer-upper. I Trans. on Neura Network, 4(4). aumond,.p., acobs, P.., Tai, M., and Murra, R.M. (1994). motion panner for nonhoonomic mobie robots. I Trans. on Robotics and utomation, 10(5). ee, S., Kim, M., Youm, Y., and Chung, W. (1999). Contro of a car-ike mobie robot for parking probem. In Proc. I Int. Conf. Robotics and utomation. etroit, Michigan, US. o, Y.K., Rad,.., Wong, C.W., and Ho, M.. (003). utomatic parae parking. In Proc. I Conf. Inteigent Transportation Sstems. Shanghai, China. Paromtchik, I.., arnier, P., and augier, C. (1997). utonomous maneuvers of a nonhonoomic vehcice. In Proc. of the Int. Smp. on perimenta Robotics. arceona, Spain. Reeds,.. and Shepp, R.. (1990). Optma paths for a car that goes both forwards and backwards. Pacific ourna of Mathematics, 145(), Zhao, Y. and Coins,.. (005). Robust automatic parae parking in tight spaces via fuzz ogic. Robotics and utonomous Sstems, 51(-3). 4