Planning Strategy for the Omni-directional Movement Path of the Four-wheel Differential Robot

Transcription

1 Planning Stratgy for th mni-dirctional Movmnt Path of th Four-whl Diffrntial Robot Wang u, a, Ma Wi,b and Chn iran,c School of Elctrical & Information, Chongqing Univrsity of Scinc & Tchnology, Chongqing 433, P. R. China School of Mchanic Enginring Institut, Chongqing Univrsity, Chongqing,444,P.R.China a saltmakr@63.com, b maiwi3@63.com, cbnyah@63.com Kywords: four-whl diffrntial; kinmatics; dynamics; tmporary path. Abstract. Th four-whl diffrntial omni-dirctional mobil robot has shown a strong rsarch valu and application prospct. This papr xplord th kinmatics and dynamics modl of th omni-dirctional mobil robot by formr diffrntial and rar diffrntial faturs, with a particular focus on its path planning stratgy. Th dsign includs path planning rfrnc point, whilst analysis was mad on th tmporary path dsign stratgy for linar and circular two faturs, with th rsulting tmporary path bing optimal and most valid. This papr concluds with th rsarch rsults that ar availabl for providing th ncssary prliminary studis and thortical basis for th control of th mobil robot of this typ. Introduction Among th mobil robots, Whld Mobil Robot (WMR) is prfct for fficint opration in a smooth, flat ground, du to its fast, stabl, simpl dsign, and thrfor th WMR has bn listd in on of th main dirctions of mobil robot tchnology dvlopmnt. According to th driv and string, th WMR can b roughly dividd into th following typs [~3]: two-whl driv, thr-whl driv, four-whl driv and omni-dirctional driv. 4-whl diffrntial omni-dirctional mobil robot (4WDR) is an omni-dirctional mobil robot, which consists of two th diffrntial robots, with th formd diffrntial unit abl to b rotatd around th robot body. 4WDR faturs high flxibility and controllability, abl to complt th translational motion, rotational motion, and th combination of movmnt in any dirction, with grat valu and a good prospct. Kinmatics and dynamics analysis of 4DWR 4WDR driv part is composd of four castrs and two diffrntial units: th castr is distributd in th four cornrs of th robot, only playing a scondary support rol; diffrntial unit is composd of two coaxial driving whls that ar indpndntly drivn by a motor. Diffrntial units ar connctd to th robot body via a rvolut pair, which is locatd at th midpoint of th two whls. Du to this, th diffrntial units ar availabl for 36 rotation around th robot body. Th two sts of diffrntial units work as th powr sourc for th robot movmnt. Blow is th 4WDR diagrammatic illustration: ωl Diffrntial unit Rvolut pair ωr Castr θ Robot body Fig. Diagrammatic Illustration of 4WDR Fig. Kinmatic Modl of th Diffrntial Unit 333

2 Th castr only plays a scondary support rol on 4WDR, not acting a binding rol in th robot plan. This maks it possibl for kinmatic analysis to ignor th castr, mrly focus on two diffrntial units. During th kinmatic analysis, w mak th following assumptions: ①Robot movs in th plan, and always in th point contact with th whl and th motion plan; ②Whls only hav pur rolling on a flat surfac without slipping; ③Th robot body, th diffrntial unit, whls and running surfac ar all th rigid bodis. 4WDR is drivn by two idntical diffrntial units, so this sction is first to stablish th kinmatic modl of th diffrntial unit. Th diffrntial unit is providd with th kinmatic modl shown in Figur, hr, - Th world coordinat systm, - solid joint coordinat systm, is th midpoint of th two cntr connction, i.. projction point of th rotation pair in th plan, ωl and ωr rspctivly, th angular vlocity of th lft and right driving whl, θ - hading angl of th diffrntial unit, i.. th angl btwn and. Lt in th absolut coordinat systm has th coordinats (x, y), thn th position and postur of th diffrntial unit in th absolut coordinat systm can b xprssd as (x, y, θ). According to th point spd dcomposition thorm, w hav: x = x + t v(t ) cos θ (t )dt t t () y = y + t v(t ) sin θ (t )dt θ = θ + t ω (t )dt t In(),(x, y, θ) is th position and postur of th diffrntial unit at th momnt t. Stratgy for path planning Slcting th path rfrnc point.during th robot path planning and control, th usual practic is to slct th gomtric cntr of th robot as a rfrnc point [4~8]. Du to th particularity of th robot structur in this study, whr th two diffrntial units work indpndntly of ach othr, th choic of th rfrnc point can b considrd to hav th gomtric cntr of th diffrntial unit as a rfrnc point. In this papr, th dual rfrnc point is st to achiv th robot path planning and th follow-up control, i.. th gomtric cntr of th front and rar diffrntial units ar takn qually as th robot path planning point of rfrnc, nabling th two points to follow th sam path. In th path following, th rar rfrnc point should hav a crtain tim of dlay than th front rfrnc point. Such an approach will gratly rduc th complxity of th control systm, making it possibl to hav a simpl control ovr th robot. Assum that th path quation y=f(x), thn at th momnt t, th position and postur quation of th robot body (with gomtric cntr of th robot body as a rfrnc point) as follows: () x = ( xf + xr ) / y = ( f ( x ) + f ( x )) / f r θ = arctan( f ( xf ) f ( xr ) ) xf xr In(), xf and xr ar, rspctivly, th coordinat x of th front and rar rfrnc points at th momnt t, and both mting th following rlationship: (3) (f ( xf ) f ( xr )) + ( xf xr ) = L L is th distanc btwn th front and rar rfrnc points. ovisional path planning.during th path following control, th starting point of th robot is not st in th dsird path to b followd. Whn th starting point of th robot is away from th dsird path, no mattr what control mthod might b mployd, all asily lad to xcssiv control, failing to achiv th idal ffct. Thrfor, it is rquird to dsign a tmporary transition path from a starting point to th dsird path. 334

3 At prsnt, th gomtric mthod is most common to gnrat a tmporary path. In this papr, two rfrnc points ar combind with th manual driving ida to propos a nw mthod that is usd to gnrat a tmporary path, whr two cass ar includd to covr th straight lin and circular dsird paths (th angl btwn th fixd joint coordinat systms of th front and rar is rstrictd to a rang of [-π/, π/]). a)expctd rout as a straight lin L L 5 3 C P θ 6 C C P 4 7 Fig. 3 Tmporary paths for Path No. ~4 P 8 Fig. 4 Tmporary paths for Path No. 5~8 As shown in Fig.3, and ar, rspctivly, th slctd rfrnc points. With as th origin of th coordinats, th dirctd lin sgmnt as th axis, w stablishd th coordinat systm. It is dfind that th straight lin to b followd is crossd with th axis at th coordinat (x, ), with th axis at th coordinat (, y); As th path vctor has a positiv projction on th axis, th dirctivity of th dfind path is positiv, othrwis it is ngativ; If th path is paralll to th axis, th projction of th path vctor on th axis is thn usd to dfin th dirctivity. Basd on th crossing point btwn th dsird path and th axis and th axis, as wll as th path dirctivity, th dsird path can b dividd into ight catgoris, as shown in Tabl, whr path ~4 ar indicatd in Fig.3, and path 5~8 ar shown in Fig.4. Tabl Catgoris of th dsird path as straight lin Coordinats Dirctivity Cod Positiv x< Ngativ Positiv 3 x> Ngativ 4 x N/A, y> Ngativz Positiv 5 6 x N/A, y< Ngativ Positiv 7 8 ①Gnratd tmporary paths for th dsird path ~4: Tak path for analysis, and th tmporary path is gnratd as indicatd in Fig.3: first is to lt th rfrnc point travl along a priod of countrclockwis arc to th straight lin vrtically pointing at th dsird path L. And thn along th lin, it coms towards th point P, which has a distanc R from th straight lin L. Finally, it is followd by /4 countrclockwis arc to rach that is tangnt to th dsird path. St th coordinats of th points in th figur as: (xr,yr), (xf,yf), (x,y), P(x,y), (x3,y3), C(xC,yC), C(xC,yC). It is known that th first sgmnt of th arc has th qualtion as: x = R cos α R ( α θ ) (3) y = Rsinα Th quation of th straight lin sgmnts that conncts th first and th scond sgmnts of th arc is: y = cot θ x + R(cscθ cot θ ), x < x < x (4) Th scond sgmnt of th arc has th quation: x = Rcosα + xc π (θ π < α θ ) (5) y = Rsinα + yc Manwhil, as th rar rfrnc point lags bhind th front on, tmporary path should b addd with th lin sgmnt, that is, th ultimatly gnratd tmporary path is ( P ). 335

4 As for th path ~4, path and path ar symmtrical against th lin P, path 3 and path against th axis, path 4 and path against th axis. Thus, path ~4 can also b obtaind asily. ②Th dsird path 5~8 with th gnratd tmporary path Tak path 5 for analysis, and th Tmporary path is gnratd as indicatd in Fig.4: first is to lt th rfrnc point travl to P, followd by rotating countr-clockwis by a π/ arc to rach P that is tangnt to th dsird path. Th straight lin sgmnt Pc has th quation as x = ( < y y R) (6) Th arc sgmnt quation: x = R cos α R π ( < α ) (7) y = Rsinα + y R Lik path,it is rquird to add a straight lin sgmnt,that is, th ultimatly gnratd tmporary path is( P). b)dsird rout as a circl As shown in Fig.5, and ar, rspctivly, th slctd rfrnc points. With as th origin of th coordinats, th dirctd lin sgmnt as th axis, w stablishd th coordinat systm. Basd on th positional rlationship btwn th starting point and dsird circl, th circl following can b dividd into two cass: th first on is that th starting point is outsid th dsird circl, th scond, insid. Th following discussion will b givn sparatly. ①Th starting point outsid th dsird circl C R C R C P C P C Fig. 5 Path of th following circl round C Fig. 6 Tmporary path of th following As shown in Fig. 5,assum that th dsird circl has its cntr C in th scond quadrant of th coordinat systm, with th radius RC and th dsird path in countrclockwis rotation. At this momnt, th plannd tmporary path it starts by rotating countrclockwis th rfrnc point to,which rquirs th tangnt lin that crosss th point to b paralll to C;And thn, it movs along a straight lin to P;Finally, it clockwis rotats to. Hr, th scond sgmnt arc has th cntr with a distanc R from P,and RC+R from th cntr C. So th ultimat tmporary path is ( P ). As th dsird path rotats clockwis,th cntr of th scond arc is positiond blow th straight lin P,subjct to th sam calculation mthod. ②Th starting point insid th dsird circl As shown in Fig.6,assum that th dsird circl has its cntr C in th fourth quadrant of th coordinat systm, with th radius RC and th dsird path in countrclockwis rotation. At this momnt, th plannd tmporary path:it starts by rotating countrclockwis th rfrnc point to (lik th cas of th starting point outsid),which rquirs th tangnt lin that crosss th point to b paralll to C;And thn, it movs along a straight lin to P;Finally, it clockwis rotats to. Undr th cas whr th starting point is outsid th circl, hr th scond sgmnt arc has th cntr with a distanc R from P,and RC-R from th cntr C. So th ultimat tmporary path is 336

5 ( P ). As th dsird path rotats countrclockwis,th cntr of th scond arc is positiond blow th straight lin P,subjct to th sam calculation mthod. Similarly, whn th dsird cntr is in th othr quadrants, rgardlss of th starting point that is insid or outsid th circl, th tmporary path can b gnratd according to th principl of th mirror. Conclusions This papr mad th kinmatics and dynamics analysis of four-diffrntial omni-dirctional mobil robot and providd its motion modl. Basd on th modl, th robot path tracking problm is xplord by dsigning th path rfrnc points. Targting at th dsird rout as a straight lin and a circl, planning mthods and stratgis ar proposd to nabl th robot to movs along an optimum tmporary path with th highst timlinss. Acknowldgmnts This work was financially supportd by th Rsarch Foundation of Chongqing Univrsity of Scinc & Tchnology, CKB7. Rfrncs [] ulin Zhang,Ja H. Chung,Stvn A.Vlinsky.Variabl structur control of a diffrntially strd whld mobil robot[j].journal of Intllignt and Robotic Systms,3,(36):3-34 [] Chih-Lyang Hwang,Song-u Han,uan-Shng u.a ntwork-basd fuzzy dcntralizd slding-mod control for car-lik mobil robots[c].ieee Intrnational Confrnc n Fuzzy Systms,5,6-66 [3] Grgor Klancar,rago Matko,aso Blazic.obil robot control on a rfrnc path[c].rocdings of th 3th Mditrranan Confrnc on Control and Automation,5, [4] Khald Blarbi,aouzi Titl.ntic Algorithm for th Dsign of a Class of Fuzzy Controllrs An Altrnativ Approach[J].EEE Transactions on Fuzzy Systms,7,(4): [5] Eduardo Frir,Todiano Bastos-Fiho,Mario Sarcinlli-Filho,t A nw mobil robot control approach via fusion of control signals[j].ieee Transactions on Systms,Man and Cybmtics-Part B:Cybrntics,4,34():49-49 [6] Sunghwan Ahn,Nakju Ltt Doh,KyoungMin L,t..Incrmntal and robust construction gnralizd voronoi graph (GVG) for mobil guid robot[c].ocdings of th 3 IEEE/RSJ Intrnational Confrnc on Intllignt Robots and Systms,3, [7] T.L.L,C.J.Wu.Fuzzy motion planning of mobil robots in unknown nvironmnts[j].journal of Intllignt and Robotic Systms,3,37():77-9 [8] Sungh L, Jong Hyon Park.Dynamic Path-Following Using Tmporary Path Gnrator for Mobil Robots with Nonholonomic Constraints[C].IEEE SMC 99 Confrnc ocdings, 999,VI,