Automati veie perpendiuar parking design using saturated ontro Pamen Petrov, Fawzi Nasasibi To ite tis version: Pamen Petrov, Fawzi Nasasibi. Automati veie perpendiuar parking design using saturated ontro. 5 IEEE Jordan Conferene on Appied Eetria Engineering and Computing Tenoogies (AEECT), Nov 5, Amman, Jordan. IEEE, 5, <ttp://ew.ieee.org/onf/aeet/cfp.tm>. <.9/AEECT.5.736566>. <a33> HAL Id: a33 ttps://a.inria.fr/a33 Submitted on 9 May 6 HAL is a muti-disipinary open aess arive for te deposit and dissemination of sientifi resear douments, weter tey are pubised or not. Te douments may ome from teaing and resear institutions in Frane or abroad, or from pubi or private resear enters. L arive ouverte puridisipinaire HAL, est destinée au dépôt et à a diffusion de douments sientifiques de niveau reere, pubiés ou non, émanant des étabissements d enseignement et de reere français ou étrangers, des aboratoires pubis ou privés.
Automati Veie Perpendiuar Parking Design Using Saturated Contro Pamen Petrov Fauty of Meania Engineering Tenia University of Sofia Sofia, Bugaria ppetrov@tu-sofia.bg Fawzi Nasasibi Robotis & Inteigent Transportation Systems (RITS) INRIA Roquenourt Domaine de Voueau, Frane fawzi.nasasibi@inria.fr Abstrat Tis paper onsiders te perpendiuar reverse parking probem of front wee steering veies. Reationsips between te widts of te parking aise and te parking pae, as we as te parameters and te starting position of te veie for panning a oision-free reverse perpendiuar parking in one maneuver are presented. A noninear saturated (tan-type) feedbak steering ontroer for straigt-ine traking is proposed and evauated. It is demonstrated tat te saturated ontroer, wi is ontinuous, aieves quik steering an be suessfuy used in soving parking probems. Simuation resuts and first experimenta tests onfirm te effetiveness of te proposed ontro seme. Keywords automati parking; perpendiuar parking; pat panning; saturated ontro I. INTRODUCTION Te perpendiuar parking is effiient and eonomia beause it aommodates te most veies per inear meter []. However, due to te speia onstraint environments, mu attention and driving experiene is needed to ontro te veie, and tis parking maneuver may be a diffiut task. For tis reason, automated operation attrats signifiant attention from resear view point, as we, and from te automobie industry. One of te diffiuties in aieving automati parking is te narrow operating pae for oision-free motion of te veie, and panning of optima trajetories is often used in te appiations. In [], an optima stopping agoritm was designed for parking using an approa ombining an oupany grid wit panning optima trajetories for oision avoidane. Te geometry of te perfet parae parking maneuver is presented in [3]. In [4], a pratia reverse parking maneuver panner is given. A trajetory panning metod based on forward pat generation and bakward traking agoritm, espeiay suitabe for bakward parking situations is reported in [5]. A ar parking ontro using trajetory traking ontroer is presented in [6]. In [7], a saturated feedbak ontro for an automated parae parking assist system is reported. In reent years, automati parking systems ave been aso deveoped by severa automobie manufaturers [9, ]. In tis paper, we dea wit te probems of geometri oision-free pat panning and feedbak steering ontro for automati perpendiuar reverse parking. An important ondition to aieve effetive oision-free parking maneuver is to determine a onvenient start position of te veie depending on te widts of te parking aise and te parking pae, as we as te dimensions and te steering apabiities of te ar. Geometri pat panning based on admissibe iruar ars witin te avaiabe spot is presented in order to steer te veie in te diretion of te parking pae in one maneuver. A saturated (tan-type) feedbak ontro is proposed wi aieves quik steering in tiny spots and stabiizes (pratia stabiization) te veie in te parking pae. Te parking probem is seen as an extension of te traking probem, sine te veie as to trak a straigt ine wi passes troug te goa position in te parking spae and is parae to te parking pae wit varying veoity depending of te position of te veie wit respet to te goa position. Te rest of te paper is organized as foows: In Setion II, geometri onsiderations for panning perpendiuar reverse parking in one maneuver are presented. In Setion III, a saturated feedbak steering ontroer is desribed. Simuation resuts and first experimenta tests are reported in Setion IV. Setion V onudes te paper. II. COLLISION-FREE PATH PLANNING WITH CONSTANT TURNING RADIUS A. Veie Mode In tis paper, front-wee steering veies are onsidered and a retanguar mode of te veie is assumed. Fig.. Te automati CyCab veie deveoped at INRIA equiped wit odometers, DGPS, and aser range finder
To iustrate te resuts presented in tis paper, we ondut simuation and experiments using a CyCab veie deveoped at INRIA, wi is sown in Fig.. Te veie parameters wi affet te parking maneuver, as we as te parameter vaues used in te simuations, are presented in Tabe I. TABLE I. VEHICLE PARAMETERS Veie parameters Notation Vaue Longitudina veie base.m Wee base b.m Distane between te front axe and te front bumper.35m Distane between te rear axe and te rear bumper.35m Maximum steering ange α max π/6rad B. Geometri Considerations for Coision-Free Perpendiuar Parking in One Maneuver Te geometry of te reverse perpendiuar parking in one oision free maneuver is sown in Fig.. y G 3 y p Parking aise x F x G p A D pr Parking ot L Fig.. Geometry of te oision-free perpendiuar parking maneuver B 4 B 3 r B4 Two inertia oordinate systems are assigned to te parking pae: Te frame Axy wit enter A wi is paed in te midde between te borders of te parking spae and wi as its Fy-axis aigned wit te boundary of parking ot L; te frame Gxy wit enter G paed at te goa position of te parking pae wi as its Gx-axis aigned wit te Ax-axis of Axy, as sown in Fig.. C C O Parking ot L r B s P B B b ) Feasibiity onditions for oision-free perpendiuar parking in one maneuver In te perpendiuar parking senario, te veie starts to move bakward from an initia position in te parking aise, wit onstant steering ange α wi is smaer or equa to te maximum steering ange α max and as to enter in te parking pae witout oiding wit te boundary of parking ot L and boundaries, and 3 of parking ot L. Wen a parae orientation wit respet to te parking pae is attained, te veie ontinues to move bakward in a straigt ine into te parking pae unti it reaes te fina position (Fig. ). Point P paed in te midde of te rear wee axe is assigned as te referene point of te veie (Fig. ). Assuming a veie iruar motion wit enter O and turning radius ( = OP) auated from te formua =. () tanα Te boundaries of te turning pat during perpendiuar parking are determined by te dimensions of te traes (iruar ars) formed by te eft orner of te front bumper B wit radius r B, te eft orner of te rear bumper B 4 wit radius r B4, and te end of te rear wee axe C, as sown in Fig.. Sine te veie motion is rotation in te pane, te trajetories of tese points form ars of onentri ires. From te ΔOC B, appying te Pytagorean teorem, we obtain an expression for te radius r B of te iruar ar traed by te eft orner of te front bumper B in terms of te veie parameters,, b (Tabe I), and te onstant turning radius, as foows b B = OB = ( + ) + +. () r Simiary, from te ΔOC B 4, we determine te radius r B4, of te iruar ar traed by te eft orner of te rear bumper B 4 b 4 4 rb = OB = + +. (3) Let O denotes te enter of rotation (te Instantaneous Center of Rotation (ICR)) of te veie wen it starts te parking maneuver wit onstant front-wee steering ange α. Depending on te sign of te x-oordinate of ICR O wit respet to oordinate frame Axy, i.e., te offset s (Fig. ), different formuas an be derived in order to determine te required widt p of te parking pae and te widt of te parking aise (te orridor), in order to ensure oision-free perpendiuar parking in one maneuver. Due to a imited spae, in tis paper, we onsider te ase wen s is negative and beongs to te interva b s,, wi is te ase wen te dimension of te parking aises and te parking pae
are not too arge. Te ower vaue orresponds to te ase wen te rigt side of te veie B B 3 (Fig.) ies on te boundary ine of parking ot L. In order to avoid oision between te eft orner B of te front bumper wit te boundary of L (Fig. ), using (), we obtain an expression for te widt of te parking aise, as foows = r B s = b. (4) ( + ) + + s Te funtion = f(s) is inear in s, positive and monotoniay inreasing in te above-mentioned osed interva for s. Terefore, it takes its minimum and maximum vaues at te ends of tis interva. To avoid oision between te rigt point C of te rear axe wit vertex A of te obstae L, from te ΔOAD, appying te Pytagorean Teorem, te distane OD is determined as foows speified in advaned. Suppose tat te widts of te parking aise and te parking pae are set as = d and p = pd, respetivey, and aso tat d < r B. In tis ase, from () and (4), it foows tat s = d r. (8) max B From (3) and (6), we obtain a formua for te minimum vaue of s as foows ( r ) b s = min. (9) B4 pd It soud be noted tat, wen s = - s min te veie wi park witout oision, but asymmetriay (not entered in te midde) in te parking spae wit respet to its boundaries, i.e., p pr (Fig.). Wen te veie is parae to te parking spae, te distanes between te ar and te parking spae boundaries p and pr are determined as foows b = s. (5) OD In order to avoid oision between te eft orner B 4 of te rear bumper wit te edge 3 of te parking pae, using (3) and (5), te foowing expression for te widt of te parking spae p, is obtained p = rb 4 OD = + b b + s. (6) Consider te funtion p = f(s), wi is ontinuous on te osed interva of s mentioned above. Te funtion is differentiabe on te open interva b s,, and its derivative is p s = s b s. (7) < Terefore p = f(s) is strity dereasing on te osed interva [-( b/), ]. Te maximum and minimum vaues of p an be found by repaing te imit vaues s = -( b/) and s = of te interva, respetivey, in (6). ) Coision-free parking wit assigned vaues for te widts of te parking aise and te parking pae From a pratia point of view, it is important to determine te starting positions of te veie for automati parking witout oision in one maneuver, in te ase, wen te widts and p of te parking aise and te parking spae, are pr b b s, () = p pd pr = b. () 3) Symmetria oision-free parking wit assigned vaues for te widts of te parking aise and te parking pae From a pratia view point, it is better to park te ar symmetriay wit respet to te boundaries of te parking pae, sine it is not very wide. For tis end, we auate te minimum vaue of te offset s = s m, in order to park symmetriay te veie in te enter of te parking spae (Fig. ). We set ps pd b : = pr = p =. () From te ΔOAD (Fig. 3), te distane OD is determined as b s m OD =. (3) Sine te turning radius an be expressed as b = + pr + OD, (4) and substituting pr from () and OD from (3) into (4), we arrive to an expression for s m, as foows
b pd s m =. (5) Te new offset - s m is bigger tan minimum offset - s min given by (9), (- s m > - s min ). In genera, it must be eked weter te new offset - s m is smaer tan - s max given by (8). If it is te ase, te ar an park symmetriay witout oision in reverse wit a minimum offset s = - s m. In tis ase, owever, te boundary 3 of te parking pae wi not be tangent to te ar of ire traed by point B 4 of te eft orner of te rear bumper; neverteess, point A (vertex A of obstae L) wi ie again on te ar of ire traed by point C of te rear veie axe. Terefore, given speified vaues = d and p = pd, te offset s an take vaues in te osed interva - s [- s m, - s max ], were - s m and - s max are determined by formuas (5) and (8), respetivey. Hene, in order to perform reverse perpendiuar parking witout oision in one maneuver and to pae te veie symmetriay into te parking pae wit assigned onstant steering ange α wi is smaer or equa to te maximum steering ange α max, te starting position, i.e., te referene point P of te veie as to be on any one of te ars of ires wit radius of enter O(x O, y O ) wit respet to te oordinate system Fxy attaed to te parking pae (Fig.), were x O beongs to te interva [- s m, - s max ] and y O = -. Te veie initia orientation as to be tangent to tis ar. In partiuar, if te orientation of te veie is (amost) perpendiuar to te parking pae, te initia oordinates of te referene point P wit respet to te frame Fxy ave to be x P () = - s, were - s [- s m, - s max ] and y P () = -. Te referene pat of te parking maneuver onsists of two parts. Te first one is a iruar ar wit enter O onneting te staring position of te veie and te Fx-axis of Fxy, wi is tangent to te iruar ar. At te tangent point T wit oordinates (- s, ), te ar wi be parae to te parking pae. Te seond part of te referene pat is a straigt ine wit distane TG aong te Fyaxis of te oordinate frame Fxy between te tangent point T and te goa position G of te parking pae, sine point G ies on te Fx-axis of Fxy. III. STEERING CONTROL For a ow speed motion, wi is te ase of te parking maneuver, we assume tat te wees of te veie ro witout siding, and te veoity vetors are in te diretion of te orientation of te wees. We onsider a simpified (biye mode) of te veie, were te front and rear wees are repaed by two virtua wees, paed at te ongitudina axis of te veie. Te inertia oordinate system Gxy is attaed to te parking pae in te goa position G (Fig. ). Te oordinates of te referene point P in Gxy are denoted by (x P, y P ). Te orientation of te veie θ is defined as an ange between te x-axis of Gxy and te ongitudina veie base. Information from te enoders mounted on te wees was used for estimation te position of te veie in te parking area. Te steering ange of te front virtua wee is denoted by α. Te equations of motion of te veie in te pane ave te form [7] x P = v y P = v vp θ = P P osθ sinθ, (6) tanα were v P is te veoity of point P. We onsider a pratia stabiization of te veie in te parking pae. Our approa is based on ontroing te motion of te veie aong a straigt ine (te Gx-axis of Gxy) passing troug te goa point G in te parking pae and aigned wit te orientation of te pae. Te veoity profie is dependent of te distane between te veie and te goa position [7]. Sine te referene pat for te first part of te parking maneuver is a iruar ar, a bang-bang feedbak ontroer [8], were te front wee steering ange is onstrained by magnitude and takes ony two onstant vaues may be a soution, sine te veie trajetories wi represent iruar ars. However, in pratie, due to te disontinuity of te bang-bang ontro aw, an undesirabe beavior of te system (attering) wi our. In order to avoid te attering, in tis paper, a saturated ontro based on yperboi tangent funtion is proposed, wi is aso onstrained by magnitude, but te ontro funtion is ontinuous. A. Saturated Contro In tis paper, we propose a differentiabe saturation in te form of yperboi tangent (tan(.)) onstraint. Tis funtion is bounded by ±. Aso tan(x) if x, and tan(x) < if x <. Tan(x) is ose to te signum funtion, wen in tan(k t x) te gain K t is arge, as sown in Fig. 3. tan(x).8.6.4. -. -.4 -.6 -.8-8 -6-4 - 4 6 8 x Fig. 3. Te funtion tan(k tx) for K t =, 3, and. Te proposed bounded ontroer as te form were [ u tan( K v) ] α = a tan t, (7) v = K( θ a yp ), (8) tanα u =, (9)
and K t, K and a are positive onstants. Te proposed ontroer aieves asymptoti stabiity of te osed oop system omposed of equations (6)-(9). Te proof is simiar to tose presented in [7] and is omitted ere due to spae imitation. It soud be noted tat te onstant vaue α for te front wee steering ange an be smaer or equa of its maximum vaue α max. If te starting position of te veie does not satisfy te onditions mentioned in Setion II for parking in one maneuver, mutipe maneuvers ave to be performed in order to rea te goa position in te parking pae. IV. SIMULATION RESULTS Simuation resuts using MATALAB are presented to iustrate te effetiveness of te geometria pat panning proedure for oision-free perpendiuar parking in one maneuver, as we as te performane of te proposed saturated steering ontroer for pratia stabiization of te veie into te parking pae. Te parameters of te test veie (CyCab) are given in Tabe I. A. Pat Panning Simuation In order to demonstrate te proposed pat panning proedure for oision-free perpendiuar parking in one maneuver presented in Setion II, in te first simuation, we present te reationsips between te widts and p of te parking aise and te parking spae, as funtions of te offset s wi beongs to te interva [-( b/), ] were s using parameters of te test veie wit α = α max, ( = min ). Te assigned vaues of and p were osen to be: d = 3m and pd = m. [m], p[m] 3.5.5.5 = f(s).4. -.8 -.6 -.4 -. s[m] Fig. 4. Coision-free interva for s = f(s), p = f(s) d = 3m pmin =.m - s min =.36m pd = m p = f(s) - s max = -.946m As seen from Fig.4, te funtion p = f(s) (te soid bue ine) dereases in te interva and onverges to b=.m (te red dotted ine), wi is exaty te engt of te wee base of te veie. Meanwie, te grap intersets te orizonta ine for te assigned vaue of pd = m (te bue dotted ine) at s = - s min =.36m, wi is te minimum vaue of s obtained from (9) for oision-free parking. In order to park te veie in one maneuver for s = - s min =.36m, from (4), te required minimum widt of te parking aise as to be =.793m wi is ess tan te speified vaue of d = 3m. Te funtion = f(s) (te green soid ine) inreases ineary in tis interva and te grap intersets te orizonta ine for te assigned vaue of d = 3m (te green dotted ine) at s = - s max = -.946m, wi is te maximum vaue of s, obtained from (8). For s = - s max = -.946m, from (6), te required minimum widt p of te parking pae as to be p =.58m, wi is ess tan te speified vaue of pd = m. Terefore, given speified vaues = d = 3m and p = pd = m for te parking aise and te parking spae, respetivey, for oision-free parking, te offset s an take vaues in te interva [- s min, - s max ] = [.36, -.946], were te boundary vaues are determined by (9) and (8), respetivey. For s = - s min, te veie wi park witout oision but not entered in te midde of te parking pae. In tis ase, Using () and (), simuation resuts indiate tat te distanes between te ar and boundaries of te parking pae are pr =.777 m and p =.8m for pd = m. In order to park symmetriay in te parking pae ( pr = p =.4m), by using (5), te minimum offset - s m was obtained to be - s m = -.3m, and tis vaue is bigger tan - s min =.36, (- s m > - s min ), obtained from (9), but smaer tan - s max = -.946m. Terefore, given speified vaues d = 3m and pd = m, for entered parking into te parking pae, te offset s an take vaues in te osed interva - s [- s m, - s max ] = [.3, -.946]. B. Simuation of Perpendiuar Parking using Saturated Contro Simuation resuts are presented to iustrate te performane of te proposed saturated steering ontroer for perpendiuar reverse parking. Using (), for α = α max, for te minimum turning radius = min is obtained min =.785m. Te dimensions of te parking aise and te parking pae were te same used before: d = 3m and pd = m. ) Simuations for perpendiuar parking in one maneuver Te initia oordinates of te veie referene point P wit respet to te inertia frame Gxy attaed to te goa position were osen to be (x P (), y P ()) = (3m, -) =(3m, -.7m) were y P () = - min. In tis ase te offset - s =.785m [- s m, - s max ] = [.36, -.946]. Te initia orientation of te veie is osen to be θ() = -π/rad. Terefore, te veie is abe to park in one maneuver. Te maximum vaue of te timevariant veie veoity was v P =.3m/s. Te vaues of te saturated tan-type ontroer were K t = 8, K = 5.85, a =.7. Te panar pat of te veie and an animation of te perpendiuar reverse parking in one maneuver are presented in Fig. 5.Evoution in time of te veie veoity and te front-wee steering ange are presented in Fig. 6. Sine te saturation ontro is ontinuous, tere is not attering wen te position of te veie is in te viinity of te traking ine and in addition, te orientation error is sma too. As seen from te simuation resuts (Fig. 6,b), te saturated ontro an be used instead of bang-bang ontro in order to steer te veie into te parking pae witout oision, sine it is ose to te signum funtion wen te gain K t is arge enoug.
-.5.5.5.5 3.5 -.5.5 - -.5 Veie pat in te pane - -3-4 Perpendiuar parking - Veie motion in te pane -5 3 4 5 6 Fig. 5. Perpendiuar parking in one maneuver: Panar pats of te veie and animation of te parking maneuver vx[m/s] -.5 -. -.5 -. -.5 -.3 Veie veoity "vx" -.35 5 5 5 3 afa[rad]. -. -. -.3 -.4 -.5 -.6 -.7 -.8 Steering ange "afa" -.9 5 5 5 3 Fig. 6. Perpendiuar parking in one maneuver: Evoution in time of te veie veoity and te front-wee steering ange ) Simuations for perpendiuar parking in mutipe maneuvers In tis ase, te initia oordinates of te veie referene point P wit respet to te inertia frame Gxy attaed to te goa position were osen to be (x P (), y P ()) = (3m, -(+.5)) = (3m,.57m). Te offset - s =.785m beongs to te interva [- s min, - s max ] = [.36, -.946], but y P () - min and te veie annot park in one maneuver. Te initia orientation of te veie was osen to be θ() = -π/rad. In tis ase, te veie annot park in one maneuver and mutipe maneuvers are neessary. Te vaues of te ontroer gains were ike in te first simuation for automati parking in one maneuver. Te panar pat of te veie and an animation of te perpendiuar reverse parking in tree maneuvers are presented in Fig. 7. Te overa parking maneuver was aieved by tree onseutive maneuvers (bakward-forwardbakward)..4. -. -.4 -.6 -.8..4 Veie pat in te pane.6.5.5.5 3 3.5 - -3-4 Perpendiuar parking - Veie motion in te pane -5 3 4 5 6 Fig. 7. Perpendiuar parking in tree maneuvers: Panar pats of te veie and animation of te parking maneuver Evoution in time of te veie veoity and te frontwee steering ange are presented in Fig. 8. Te saturated ontroer as been impemented on an experimenta automati eetri veie CyCab and initia tests of perpendiuar reverse parking in one maneuver ave been initiaized (Fig. ). In te first tests, ony information from te enoders mounted on te wees were used for estimation of te te position and oaisation of te veie in te parking area. First experiments onfirm te effetiveness of te proposed ontroer. vx[m/s].5..5 -.5 -. -.5 -. -.5 Veie veoity "vx" -.3 4 6 8 afa[rad].6.4. -. -.4 -.6 Steering ange "afa" -.8 4 6 8 Fig. 8. Perpendiuar parking in tree maneuvers: Evoution in time of te veie veoity and te front-wee steering ange V. CONCLUSION In tis paper, te probem of perpendiuar oision-free reverse parking of front wee steering veies as been onsidered. Geometri onsiderations for oision-free perpendiuar parking in one reverse maneuver ave been first presented, were te sape of te veie and te parking environment ave been expressed as poygons. Reationsips between te widts of te parking aise and parking pae, as we as te parameters and te initia position of te veie ave been given, in order to pan a oision-free maneuver, in te ase, wen te ar as to be entered into te parking pae. A tan-type steering ontroer for straigt-ine traking as been proposed and evauated. It was demonstrated tat tis ontroer wi is ontinuous, was abe to aieve quik steering avoiding attering and an be suessfuy used in soving parking probems. Simuation resuts, as we as te first experimenta tests wit an automati CyCab veie onfirm te effetiveness of te proposed ontro seme. REFERENCES [] USAF - LANDSCAPEDESIGN GUIDE, avaiabe at: ttp://www.ttap. mtu.edu/pubiations/7/parkingdesignconsiderations.pdf. [] B. Gutjar and M. Wering, Automati oision avoiding during parking maneuver an optima ontro approa, In Pro. 4 IEEE Inte.. V. Symposium, 4, pp. 636-64. [3] S. Bakburn, Te geometry of perfet parking, Avaiabe at: ttp://persona.ru.a.uk/ua/58/perfet_parking.pdf. [4] C. Pradaier, S. Vaussier and P. Corke, Pat panning for parking assistane system : Impementation and experimantation, In Pro. Austr. Conf. Rob. Automation, 5. [5] J. Moon, I. Bae, J. Ca, and S. Kim, A trajetory panning metod based on forward pat generation and bakward traking agoritm for automati parking systems, In Pro. IEEE Int. Conf. Inte. Transp. Systems, 4, pp. 79-74. [6] K. Lee, D. Kim, W. Cung, H. Cang, and P. Yoon, Car parking ontro using a trajetory traking ontroer, In Pro SICE_ICASE Int. J. Conferene, 6, pp. 58-63. [7] P. Petrov and F. Nasasibi, Saturated feedbak ontro for an automated parae parking assist system, In Pro. IEEE Conf. Contr. Autom. Rob. Vision, 4, pp. 577-58. [8] P. Petrov, C. Boussard, S. Ammoun, and F. Nasasibi, A ybrid ontro for automati doking of eetri veies for rearging, In Pro. IEEE Int. Conf. Rob. Automation,, pp. 966-97. [9] Avaabe at: ttps:// www.youtube.om/wat?v=ybrwurxfybq [] Avaabe at: ttps://www.youtube.om/wat?v=b_m8dqtole