Wllams et al., Fnal anuscrpt, IEEE TRANSACTIONS ON ROBOTICS AND AUTOATION, arch 22 Dynamc odel wth Slp for Wheeled Omn-Drectonal Robots Robert L. Wllams II and Bran E. Carter Oho Unversty, Athens, Oho Paolo Gallna and Gulo Rosat Unversty of Padova, Padova, Italy IEEE Transactons on Robotcs and Automaton Vol. 8, No., pp. 285-29 22 Correspondng author: Robert L. Wllams II, ember IEEE Assocate Professor Department of echancal Engneerng 257 Stocker Center Oho Unversty Athens, OH 457-2979 Phone: (74) 59-96 Fax: (74) 59-476 E-mal: wllar4@oho.edu URL: http://www.ent.ohou.edu/~bobw
Wllams et al., Fnal anuscrpt, IEEE TRANSACTIONS ON ROBOTICS AND AUTOATION, arch 22 2 Dynamc odel wth Slp for Wheeled Omn-Drectonal Robots Robert L. Wllams II, Bran E. Carter 2, Paolo Gallna, and Gulo Rosat 4 Abstract--A dynamc model s presented for omn-drectonal wheeled moble robots, ncludng wheel/moton surface slp. We derve the dynamcs model, expermentally measure frcton coeffcents, and measure the force to cause slp (to valdate our frcton model). Dynamc smulaton examples are presented to demonstrate omn-drectonal moton wth slp. After developng an mproved frcton model, compared to our ntal model, the smulaton results agree well wth expermentallymeasured trajectory data wth slp. Intally we thought that only hgh robot velocty and acceleraton governed the resultng slppng moton. However, we learned that the rgd materal exstng n the dscontnutes between omn-drectonal wheel rollers plays an equally mportant role n determnng omndrectonal moble robot dynamc slp moton, even at low rates and acceleratons. Index Terms-- Dynamc model, sldng frcton model, omndrectonal moble robot, wheel slp. I. INTRODUCTION Research nterest n moble robots has been tremendous n the past few years, as evdenced by revew artcles (e.g. [] and [2]). Some researchers have consdered slppng moton between the wheels and moton surface n moble robots and vehcles. Cho and Sreenvasan have desgned artculated wheeled vehcles wth varable-length axles to elmnate knematc wheel-surface slppng []. Hamdy and Badreddn developed a tenth-order nonlnear dynamc model for a wheeled moble robot that ncludes slp between the drven wheels and the ground [4]. Rajagopalan developed an expresson for the angular velocty of wheel slp for wheeled moble robots wth dfferent combnatons of steerng and drvng wheels, consderng knematcs only [5]. Shekhar derves a dynamc model for moble robots wth wheel slp usng accessblty and controllablty n nonlnear control theory [6]. Balakrshna and Ghosal present a tracton model accountng for slp n nonholonomc wheeled moble robots [7]. Schedng et al. present expermental evaluaton of a navgaton system that handles autonomous vehcle wheel slp va mult-sensor feedback [8]. Several research groups are developng omn-drectonal moble robots and vehcles due to nherent aglty benefts. Jung et al. developed an omn-drectonal moble robot base for the RoboCup competton [9]. RoboCup (www.robocup.org) s an nternatonal competton wheren teams of autonomous moble robots compete n the game of soccer. oore et al. present a control algorthm for an omn-drectonal sx-wheeled vehcle; each wheel s steered and drven ndependently []. Watanabe et al. present a controller for an autonomous omn-drectonal moble robot for servce applcatons []. Wtus nvestgates the moblty of a 6-wheeled omn-drectonal vehcle wth tre nflaton control [2]. A recent artcle n these transactons presented a clever desgn plus expermental results for a sphercal rollng robot []; however, ths moble robot s not omn-drectonal and a no-slp condton was assumed. Our lterature search revealed only two papers whch mentoned slp n omn-drectonal wheeled robots. or et al. clam that ther vehcle avods tre slppage by desgn snce ther omn-drectonal moton base decouples steerng and drvng [4]. Dckerson and Lapn present a controller for omn-drectonal ecanum-wheeled vehcles, that ncludes wheel slp detecton and compensaton [5]. The current paper presents a dynamc model for omndrectonal moble robots that ncludes slppng between the wheels and the moton surface. Ths paper was motvated by a need n the Oho Unversty cross-dscplnary RoboCup team: n prelmnary hardware testng of our omn-drectonal three-wheeled player robot, sgnfcant slppng occurred whch necesstated development of a dynamc model wth slp. Though our work s motvated by RoboCup, the result s a general dynamc model for omn-drectonal wheeled vehcles ncludng slp. Our model ncludes both frcton n the wheel rollng drecton and n the transverse drecton (normal to the frst). One mportant ssue turned out to be dfferng frctonal characterstcs due to the rgd materal dscontnutes between rollers n the omn-drectonal wheels. Ths artcle does not focus on real-tme control snce our objectve was to model and understand the sldng dynamcs problem n smulaton and expermentally. Based on our dynamc model, we wll develop real-tme control n the future, ncludng a means to measure the slppng for feedback control. Ths paper frst presents our omn-drectonal robot desgn, followed by dynamc modelng ncludng slp, a method to expermentally determne the coeffcents of frcton and valdate our frcton model, and then smulaton and expermental results to demonstrate omn-drectonal robot dynamcs consderng slp. II. ONI-DIRECTIONAL ROBOT ODEL In early evaluaton of our three-wheeled omn-drectonal moble robot hardware, slppng was encountered between the wheels and the carpet playng feld when the robot was n moton. Ths unexpected behavor motvated the development of a dynamc model ncludng slp. Ths model s presented n the next secton; the current secton descrbes the omn-drectonal robot hardware and model. Assocate Professor, Oho Unversty, Athens, OH; ember, IEEE 2 Graduate Research Assstant, Oho Unversty, Athens, OH Assstant Professor, Unversty of Padova, Padova, Italy; on sabbatcal at Oho Unversty durng the work reported 4 Graduate Research Assstant, Unversty of Padova, Padova, Italy
Wllams et al., Fnal anuscrpt, IEEE TRANSACTIONS ON ROBOTICS AND AUTOATION, arch 22 Fgure shows the CAD model for the three-wheeled omndrectonal moble robot and Fg. 2 shows a photograph of the prototype hardware. Fgure shows the top vew of our general three-wheeled omndrectonal moble robot model. The varables, used n the dynamc model of the next secton, are explaned below. The nertally-fxed frame s {} and the movng Cartesan reference frame s {}. The rear wheel s algned n the X drecton; the front two wheels are symmetrcally-placed, algned by constant angle δ from the Y axs (shown only for the left wheel n Fg. ). We assume the center of mass for the robot s located at the center of the robot crcle, whch s the orgn of {}. Ths was one of our gudng prncples n desgn. The robot mass s m and the robot mass moment of nerta about the Z axs through the center of mass s I. Each wheel center poston s gven by poston vector r, from the orgn of {} to the center of the wheel. The unt vector n ths drecton s also the drecton of each wheel s angular velocty vector (.e. s the axle drecton). The unt vector ŝ s normal to, gvng the nstantaneous drecton of each wheel. The Cartesan varables for omn-drectonal moton are X = { x y φ} T. As seen n Fg., the translatonal vector gvng the poston of the orgn of {} wth respect to the orgn of {} s { x y} T (expressed n the coordnates of {}); also, the angle φ gves the orentaton of the robot wth respect to the nertal frame horzontal drecton X. note that these wheels were not ntended for omn-drectonal moble robots; rather, they were developed for materal handlng applcatons. For a good dscusson on omn-drectonal wheels for moble robots, see [6]. Our applcaton dctated economcal, commercally-avalable wheels, whch led to our choce of wheel. As seen n Fg. 4, the axle s mounted normal to the wheel s crcle as n a standard wheel. However, the contact wth the ground s va rollers that are free to spn about an axs n-lne wth the crcle crcumference, normal to the wheel axle. Ths enables omndrectonal moton. III. ONI-DIRECTIONAL ROBOT DYNAICS ODELING Ths secton presents omn-drectonal moble robot modelng wth slp ncluded between the wheels and moton surface. The frst subsecton presents the model, plus the frcton model and expermental measurement of the frcton coeffcents; the second subsecton presents a method to expermentally valdate our theoretcal frcton model and measured frcton coeffcents. A. Dynamcs odel wth Slp The dynamcs model s developed n ths subsecton for a threewheeled omn-drectonal robot, but t apples to any omndrectonal robot wth three or more wheels. The dynamc model s shown n the top vew of Fg. above, and s descrbed n Secton 2. Fgure 5 shows modelng detals for the th wheel from a sde vew. Fgure 5 shows that our omn-drectonal wheel contans n=8 rollers; further, the fxed angle θ covers each roller sector and the fxed angle θ covers each sector between rollers. θ' θ" 2π n r^ θ Fgure. CAD odel Y δ 2 X x,y Y m,i s^ r Fgure 2. Hardware Photo φ ^r X Fgure. Omn-Drectonal Robot odel, Top Vew The omn-drectonal moton s enabled va specal wheels. Fgure 4 shows a commercal omn-drectonal wheel (kornylak.com) used n our moble robot desgns. It s mportant to Fgure 4. Commercal Wheel ρ P v r Fgure 5. Wheel Detal As seen n Fg. 5, we denote P (=,2,) as the contact pont between the th wheel and the ground. Instantaneously, P belongs to the ground and the wheel, but we consder that P s on the wheel. The velocty vector v for pont P s: v = v + ω r + v () G v G s the vehcle center of mass translatonal velocty, ω s the vehcle rotatonal velocty (both translatonal and rotatonal velocty vectors are expressed wth respect to the nertally-fxed frame {}), r s the poston vector gvng the wheel center poston wth respect to the movng frame {}, expressed n the nertal frame, and v r s the perpheral wheel speed wth respect to the movng frame, expressed n the nertal frame. Note that when v s null, there s no slppng moton. We can express v as a functon of the wheel angular velocty vector θ & and the wheel radus vector r r ρ : r s^ v = θ & ρ (2)
Wllams et al., Fnal anuscrpt, IEEE TRANSACTIONS ON ROBOTICS AND AUTOATION, arch 22 4 The wheel angular velocty vector θ & = & θ s the scalar wheel speed & θ n the unt drecton, and the wheel radus poston vector ρ s scalar wheel radus ρ from the wheel center to pont P. Note the result of (2) s scalar ρ & θ n the unt wheel drecton ŝ (normal to ). The next two subsectons present our ntal frcton model and expermental measurement of frcton coeffcents. ) Intal Frcton odel There are two drectons of wheel/ground frcton to consder: the frst s frcton n the drecton of the wheel rotaton, ŝ, and the second s transverse to ths drecton,. Intally our model only ncluded the former case, but ntal trals wth the omn-drectonal moton base hardware ndcated that we must also nclude the latter frcton case. For use wth frcton n the drecton of the wheel rotaton, the sldng velocty component v W n the th wheel s obtaned by dottng the total pont P velocty from () nto the ŝ unt drecton: vw G ( ω r ) ˆ + ρ & θ = v = v + s () To convert wheel postons and unt drectons n the movng frame {} ( r and ŝ ) to the nertal frame {} ( r and ŝ ), use (4): R R = r R r = (4) s the orthonormal rotaton matrx gvng the orentaton of the movng frame {} wth respect to the nertal frame {}: cosφ snφ R = (5) snφ cosφ Therefore, () becomes: ( ) vw = vg R + ω R r R + ρ & θ On the other hand, the transverse sldng velocty component v n the th wheel, along the wheel axle drecton s: T vt ( ω R r ) R vg R + (6) = (7) If we assume the vehcle weght s equally dstrbuted on each wheel, the frcton force exerted on wheel by the moton surface through pont P s gven by: mg F ( W ( vw ) R = µ + µ T ( vt ) R ) (8) where µ W () s a functon representng the frcton coeffcent versus the sldng velocty n the drecton of the wheel rotaton and µ T () s the frcton coeffcent for the transverse wheel drecton. The dynamc equatons are: m& x = x ˆ F m y& = y ˆ F = = xˆ, yˆ, zˆ where & I & φ = zˆ ( r F ) & (9) = are the unt drecton vectors of the nertal frame. The X & = f X, X& : nonlnear dynamc equatons are of the form ( ) g ˆ ( ( ) ˆ x µ W vw R s + µ T ( vt ) R ) & = x () g & = ˆ ( ( ) ˆ y y µ W vw R s + µ T ( vt ) R ) = & φ mg [ ( ( ) ( ) )] ˆ ˆ z R r µ W vw R s + µ T vt R I = In our smulaton we use the followng smplfed formulas for coeffcent of frcton: 2 µ W ( vw ) = µ W max atan( kvw ) π () 2 µ T ( vt ) = µ T max atan( kvt ) π where µ W max, µ T max, and k are constants, and v W and v T are the sldng velocty magntudes n the wheel rotaton and transverse drectons, respectvely. Notce that n our frcton model, the dynamc frcton coeffcent s assumed to be constant and equal to the constant statc frcton coeffcent; we assume that ths smplfed model wll be suffcent to match expermental results. Equatons () are artfcal functons to convenently represent the frcton coeffcents stably n smulaton, avodng algorthmc problems that may arse when usng a dscontnuous functon at zero sldng velocty. Ths s a common approach; for example see [7] whch presents a parametrc model and expermental results for tre-road frcton coeffcents for automotve applcatons wth dfferent road condtons. The k constant governs the steepness of the change between postve and negatve µ W max and µ T max about zero sldng velocty. We chose k= by eye and to ensure numercal stablty n Smulnk. Note that () defnes postve frcton coeffcent to correspond wth postve sldng velocty; the opposte sgn behavor (Coulomb frcton acts opposte to the sldng drecton) s taken nto account n (8). Also, µ T max s much less than µ W max due to the desgn of the omn-drectonal wheels used, wth smaller frcton n the transverse drecton than the prmary drvng drecton, owng to the passve rollng cylnders (see Fg. 4). Our smple frcton model s ntended to capture gross realworld frcton characterstcs. Improvements are certanly possble by consderng stcton and dfferent coeffcents for the statc and dynamc frcton coeffcents. For an mproved frcton model, see [8]; these authors present a tre/road frcton model usng the LuGre dry frcton model and ncludng tre dynamcs. Our wheel and rollng surface materals are not smlar to the tre/road problem. 2) Expermental Frcton Coeffcent easurement We measured expermental values for µ W max and µ T max for use n the dynamc smulaton, for two moton surfaces: paper and carpet. In order to estmate µ W max, we bult a specal vehcle n whch all the wheels were algned along a common drecton. Each wheel angle was fxed n such a way that only the rollers were n contact wth the moton surface. The surface was made up of a rgd board covered wth paper or carpet. The square board was pvoted on one edge. The vehcle was placed so that all wheel axes were parallel to the pvotng edge. Then we gradually lfted the board untl the specal vehcle sld. µ W max was determned as the tangent of the angle between the lfted board and the horzontal plane. To measure µ T max, we repeated the above procedure, placng the specal vehcle so that all wheel axes were perpendcular to the pvotng edge. Agan n ths case only the rollers were n contact wth the moton surface. The results are: µ. 26 and W max =
Wllams et al., Fnal anuscrpt, IEEE TRANSACTIONS ON ROBOTICS AND AUTOATION, arch 22 5 µ T max =.9 for the paper moton surface, and µ W max =. 25 and µ T max =.5 for the carpet moton surface. These results were averaged over several trals. For each surface, the transverse roller frcton ( µ T max ) s much smaller than that n the wheel rotaton drecton ( µ W max ). The wheel rotaton drecton frcton coeffcents are nearly the same for paper and carpet. B. Intal Smulaton and Expermental Results Ths secton presents smulated and expermental results to demonstrate omn-drectonal moble robot moton consderng slp. Smulaton results are presented frst usng the ntal frcton model; next, the expermental procedure and results are presented and compared wth the smulaton results. For both of the followng subsectons (smulaton and expermental results), the same moton condton s used: we command straght-lne moton from ntal pont X = {.. } T to fnal pont X F = {.4. }T (m) n a specfed tme of t F =. 5 sec. We consder only X moton snce, due to robot symmetry, Y moton s nherently less affected by slp. φ moton could have sgnfcant slp behavor; ths wll be the subject of future expermental work. Fgure shows our robot o hardware geometry (our desgn has δ =5 : we were drven to ths o choce by RoboCup sze constrants; δ = s preferable for robot symmetry). The moton s commanded n the nertal frame, {} n Fg.. Robot orentaton φ s also mportant n slp dynamcs, but the pure X moton wll also demonstrate (unwanted) orentaton slp moton. Robot orentaton s commanded as zero for the moton example. Snce we wsh to demonstrate slppng, we make no attempt to smooth the commanded velocty moton from rest or endng at rest. Hence, the smulated commanded wheel acceleraton s nfnte at the start and the deceleraton s nfnte at the end of the tme perod. Of course, nether the real or smulated robot can acheve nfnte acceleraton or deceleraton, but the hgh acceleratons at the start and end are suffcent to cause slp. Constant velocty s commanded n between the start and end. Clearly for omndrectonal moble robot applcatons we need smoother trajectory generaton, perhaps usng 5 th -order polynomals for wheel dsplacements. For ths moton example, the requred constant wheel angular speeds are & θ = & θ2 = +. 6 and & θ = 4. 5 (rad/s); note the wheel numberng conventon s gven n Fg.. Wth ths moton example, we consder two moton surfaces to nclude dfferent levels of frcton: a smooth paper surface and a rough carpet surface. The expermentally-measured frcton coeffcents for use n smulaton were presented n Secton.A.2. ) Intal Smulaton Results We developed a atlab Smulnk model to smulate omndrectonal moble robot dynamc moton consderng slp. In ths subsecton we present smulated dynamcs results usng the ntal frcton model of Secton.A.. The smulated moton condton, surfaces, and frcton coeffcents are descrbed above. To save space, we only show the smulated case wth the paper moton surface. Fgure 6a shows the Cartesan dsplacements and Fg. 6b shows the assocated sldng speeds n the wheel drectons for each wheel, for the smulated moton. In Fg. 6b, the smulated sldng speeds for wheels one and two are dentcal due to symmetry. As seen n Fg. 6b, slppng s encountered at the start and end of moton (due to the nfnte commanded acceleraton and deceleraton), but not n the mddle. The effect, seen n Fg. 6a, s that x falls short by 4 mm ts goal of.4 m, whle y drfts -9 mm from the desred zero. φ drfts from ts commanded value of zero by -.6 rad at the end; n the mddle of moton, the φ drft s larger. From approxmately t =.25 sec to t = t F =. 5 sec, the x moton s lnear, whch means constant velocty has been acheved and the smulaton predcts no unwanted slppng n ths range..4..2. -. -.2 -. -.4 -.5.5.5 -.5 φ x.5.5 2 2.5.5 4 t (sec). -. Fgure 6a. Cartesan Dsplacements (m and rad),2.5.5 2 2.5.5 4 t (sec) Fgure 6b. Wheel Sldng Speeds v W, v W 2, v W (m/s) 2) Expermental Procedure and Results Experments were performed to valdate the results of our smulaton work, usng both paper and carpet moton surfaces. Our moble robot was tethered for the experments; eventually our moble robots wll be free, the on-board PCs communcatng wth the host PC va wreless Ethernet. To control the robot durng the expermental trals, WnCon. n conjuncton wth Smulnk was used. Ths enabled us to use a Quanser ult-q board to control the motor angular veloctes through a feedback loop. The expermental robot was shown earler n Fg. 2. Please note that the robot cables must be held manually to avod constranng the robot moton. Expermental trajectores were traced by attachng a lghtweght pencl to the robot center of mass, for the paper surface. Ths was not feasble for the carpet surface, so only the end ponts and fnal orentatons were recorded n the carpet cases. Another way to present the smulaton result from Fg. 6a s gven n Fgs. 7a and b, plottng y vs. x for the paper and carpet moton surfaces. The expermental data s ncluded for comparson. y,2
Wllams et al., Fnal anuscrpt, IEEE TRANSACTIONS ON ROBOTICS AND AUTOATION, arch 22 6 Fgure 7a. Smulated (dashed) and Expermental (sold) Results for Paper oton Surface Fgure 7b. Smulated (dashed) and Expermental (O) Results for Carpet oton Surface In Fg. 7a, the dashed curve s the smulated result usng the ntal frcton model wth the paper moton surface; ths curve was obtaned by plottng y vs. x (rather than vs. tme t) from Fg. 6a. The four sold curves are the results of the four expermental trals for the same moton case, wth the paper moton surface. In Fg. 7b, the three sngle O ponts are the endng ponts for the three expermental trals wth the carpet moton surface (these cases also tred to obtan pure X moton from to.4 m; no trajectory s avalable as explaned above). The dashed curve s the smulated result usng the ntal frcton model and the carpet moton surface (not prevously shown). IV. IPROVED ODEL Clearly from Fgs. 7, the ntal smulaton results do not agree well wth the expermental results, when usng the ntal frcton model wth ether moton surface, paper or carpet. Ths poor result motvated the need for an mproved frcton model. We use the same type of smple frcton model, but augmented for the specal nature of our omn-drectonal wheels geometry. Ths secton presents our mproved frcton model, valdaton of our frcton coeffcent measurements, and mproved smulaton results, compared wth the precedng expermental results. A. Improved Frcton odel We notced that, for our choce of wheel, the frcton coeffcent s a functon of the wheel angle θ. When the rgd wheel materal between two rollers s n contact wth the moton surface (see Fg. 5), the frcton coeffcents change. Ths undesrable behavor cannot be blamed on the wheel manufacturer snce the wheels were not made for use n omn-drectonal moble robots. We account for ths phenomenon by ntroducng nonlnear frcton coeffcents as a functon of the sldng speeds and wheel angle. Let n be the number of rollers n the wheel (n=8 n Fgs. 4 and 5). Each roller and rgd porton s wthn angular sector 2 π n. Each sector can be splt nto two dfferent (roller and rgd) portons wth dfferent frcton coeffcents: θ + θ = 2π n as seen n Fg. 5. Therefore, we have dfferent frcton coeffcents accordng to whch part of the sector s n contact wth the moton surface at a gven tme. In our hardware wheel, θ (the roller) accounts for 9% of each angular sector 2 π n and θ" (the rgd materal) accounts for %. To summarze, our mproved frcton model s the same as (), but we use roller values for frcton coeffcent ( µ and ' W max ' µ T max ) when the wheel angle s wthn the θ sector and we use " " rgd materal values ( µ W max and µ T max ) when the wheel angle s wthn θ". ' ' We measured expermental values for µ W max, µ T max, " W max " T µ, and µ max for both paper and carpet moton surfaces. ' W max ' T µ and µ max were already measured n Secton.A.2; now we have added a sngle prme superscrpt to ndcate roller. " W max " T µ and µ max were measured n the same way, but n these cases the wheel angle was fxed so that only the wheel sector between two consecutve rollers was n contact wth the moton surface. The double-prme superscrpt ndcates the rgd materal between rollers. The results, averaged over several trals, are shown n Table I below (the frst two columns are the same as the prevous results n Secton.A.2). Table I. Expermental Frcton Coeffcents Surface ' ' " " µ W max µ T max µ W max µ T max Paper.26.9.47.47 Carpet.25.5.56.56 ' T max Agan, for each surface, the transverse roller frcton ( µ ) s ' much smaller than that n the wheel rotaton drecton ( µ W max ). ' The wheel rotaton drecton roller frcton coeffcents ( µ W max ) are nearly the same for paper and carpet. For the materal between the rollers (double-prme), the carpet frcton coeffcent value s hgher than that of the paper surface. As expected, the wheel and transverse coeffcents of frcton ( µ max and µ max ) are dentcal for the materal between the rollers (for a gven moton surface). B. Valdaton of Frcton Coeffcent easurements In attempt to valdate both our expermental statc frcton coeffcents and our mproved frcton model, we now derve and measure the maxmum allowable force yeldng statc equlbrum for the omn-drectonal robot. The frcton force on each wheel s: " W " T
Wllams et al., Fnal anuscrpt, IEEE TRANSACTIONS ON ROBOTICS AND AUTOATION, arch 22 7 F = F ˆ + F ˆ (2) Ws Tr Where F W and F T are respectvely the frcton force magntudes along the wheel moton drecton and along the wheel axs drecton (transverse). Suppose we apply an external force F = { } T at the robot mass center. What s the maxmum force e Fe F emax we can apply at the mass center and stll mantan statc equlbrum (avod sldng moton)? The external force we apply s ressted by the frcton forces and moment: F e = = = = ( rx y ry x ) FW + ( rx y ry x ) F T = = x FW + x FT = y FW + y FT = = () whch s rewrtten as: F e = [ AW FW AT ] FT (4) where: x AW = y r x y r y x x AT = y r x y r y x 2x 2 y r2 x 2y r2 y 2x 2x 2 y r2 x 2y r2 y 2x x y r x y r y x x y r x y r y x F = { F F } T F = { F F } T W FW W 2 W T FT T 2 T Notce that the elements of A W, A T are expressed n the nertal frame, so they are functons of robot orentaton φ. We can express A W, A T as a product of two matrces: A W = RB W A T = RBT (5) where B W, B T are dentcal to A W, A T, except all vector components are expressed n the movng frame {}, and: cφ sφ R = R = sφ cφ Notce that wth ths notaton, B W, B T are constant, whle R s a functon of the angular poston. Therefore: FW F e = R [ BW BT ] (6) FT The set of frcton forces that can be exerted by the moton surface contact pont on the robot s gven by: Σ = mg mg mg mg F F µ W max FW µ W max, µ T max FT µ max T (7) where: FW F = F T = { F F F F F } T FW W 2 W T T 2 T Equaton (7) represents the domnon of the lnear transformaton gven by (6). The mage of Σ F s a polytope that represents the maxmum force avalable along x, y, and φ. Our am s to calculate the maxmum value of F e that satsfes (6) and at the same tme belongs to the mage of Σ F. There are several standard ways to solve ths problem usng polytope theory [9]. For our partcular problem we can turn the problem nto a typcal constraned maxmzaton problem and solve t wth numercal software lke athematca. If we splt the matrx R [ B W B T ] n two sub-matrces, we have: B F = R [ B W B T ] e = BF (8) B 2 = B2F therefore F = { F F = B F, = B F, F Σ } e max e max F s a functon of φ. ' W max e e 2 ' T F. Notce that Usng µ max and µ max determned expermentally above, we can plot the maxmum force before sldng, F e max, versus moble robot orentaton φ, usng the theory of ths secton. Fe max can also be measured expermentally wth the followng procedure. When we apply an external force to the robot along the X drecton, f the sum of the three frcton force components along X s hgh enough, the robot does not move. We ncrease the external force untl the robot moves. The mnmum external force to move the robot s recorded as F e max. The procedure s repeated wth dfferent angular orentatons φ. An expermental plot of F e max vs. φ s shown n Fg. 8. Note ths fgure gves results for the omn-drectonal robot (Fgs. 2 and ), not the specal vehcle constructed for frcton coeffcent measurement n Sectons.A.2 and 4.A. Due to robot symmetry, plottng results from to π / 2 rad n robot orentaton s suffcent. Three seres of data have been collected. Consderng statc condtons, the expermental data (O) and the theoretcal result (sold curve, solved va athematca) compare reasonably well, whch serves to valdate both our frcton coeffcents and our frcton model. Though the agreement s reasonable, the expermental repeatablty s very low. Femax (N) 2.5 2.5.5 2 4 5 φ (deg) 6 7 8 9 F e max Fgure 8. Expermental (O) and Theoretcal (sold)
Wllams et al., Fnal anuscrpt, IEEE TRANSACTIONS ON ROBOTICS AND AUTOATION, arch 22 8 C. Improved Smulaton Results In an attempt to mprove the poor smulaton/expermental agreement of Fgs. 7, the mproved frcton model of ths secton s mplemented n smulaton n ths subsecton, and then compared wth the exstng expermental data. The mproved frcton model accounts for the rgd materal n the dscontnutes between wheel rollers (see Fg. 5). Fgures 9 show smulated results for the same moton nput case as for Fgs. 6; Fgs. 9 nclude the real-world effect of the rgd materal between the wheel rollers. Agan, to save space, the smulated results are shown only for the paper moton surface n Fgs. 9. Fgure 9a shows the Cartesan dsplacements and Fg. 9b shows the assocated sldng speeds v W for each wheel, for the smulated pure X moton on the paper surface. Agan n Fg. 9b, the smulated sldng speeds for wheels one and two are dentcal due to symmetry. As seen n Fg. 9b, slppng agan s encountered at the start and end of moton. In addton, wheel three experences sgnfcant slp durng the mddle of the moton; ths behavor was not predcted by the ntal frcton model. The effect, seen n Fg. 9a, s that x falls short by 5 mm ts goal of.4 m, whle y drfts -22 mm from the desred zero. φ drfts from ts commanded value of zero by -. rad n the worst case at the end. All three Cartesan drfts are much larger than predcted n Fg. 6a. Fgure a. Improved Smulated (dashed) and Expermental (sold) Results for Paper oton Surface.4..2 x. y -. -.2 φ -. -.4 -.5.5.5 -.5.5.5 2 2.5.5 4 t (sec). -. Fgure 9a. Cartesan Dsplacements (m and rad),2.5.5 2 2.5.5 4 t (sec) Fgure 9b. Wheel Sldng Speeds v W, v W 2, v W (m/s) Fgures a and b compare the precedng expermental data wth ths new, mproved frcton model smulaton, plottng y vs. x for paper and carpet moton surfaces, respectvely. The expermental data of Fgs. s dentcal to that of Fgs. 7.,2 Fgure b. Improved Smulated (dashed) and Expermental (O) Results for Carpet oton Surface In Fg. a, the dashed curve s the smulated result usng the mproved frcton model and the paper moton surface; ths curve was obtaned by plottng y vs. x from Fg. 9a. The four sold curves are the prevously-presented expermental results for the same moton case, wth the paper moton surface. The three sngle O ponts n Fg. 9b are the same expermental endng ponts wth the carpet moton surface as shown n Fg. 7b. The dashed curve s the smulated result usng the mproved frcton model and the carpet moton surface (not prevously shown). Clearly, the smulaton/ expermental agreement obtaned by the mproved frcton model (shown n Fgs. a and b) s much better than that dsplayed n Fgs. 7a and b whch used the ntal frcton model. Fgures 7 and gnore the moble robot orentaton φ. For the paper moton surface, the smulated (mproved frcton model) endng value of φ s -. rad. No expermental data s avalable for ths case, snce the four expermental trals all ended wth very small φ, close to the angular measurement precson. Even so, the agreement s good qualtatvely snce the smulated endng angle s also small. For the carpet moton surface, the smulated endng value of φ s.558 rad, whch compares favorably wth the measured expermental values of.524,.56, and.489 rad (leftto-rght for the expermental Os of Fg. b). As mentoned earler, we assumed the dynamc frcton coeffcent s equal to the statc frcton coeffcent. Perhaps better
Wllams et al., Fnal anuscrpt, IEEE TRANSACTIONS ON ROBOTICS AND AUTOATION, arch 22 9 smulaton/expermental agreement would be obtaned by use of a combned frcton coeffcent model where the dynamc frcton s less. Ths s dffcult to measure, and we are satsfed wth the agreement shown n Fg., usng statc coeffcents of frcton only. V. CONCLUSION Ths paper presents a dynamc model for omn-drectonal wheeled moble robots and vehcles, consderng slppng between the wheels and moton surface. We derved the dynamcs model, expermentally measured the frcton coeffcents, and valdated our frcton model by expermentally measurng the maxmum force causng slp at varous robot orentatons. Smulaton examples were presented to demonstrate slppng moton; the ntal frcton model results dd not agree wth expermental trajectory data. Therefore, an mproved frcton model was developed, consderng the rgd materal n the dscontnutes between omn-drectonal wheel rollers. Wth ths mproved frcton model the smulaton agreed well wth the expermental data. Two moton surfaces (paper and carpet) were used n smulaton and experments, wth dfferent frcton propertes. A pure X translatonal moton was commanded n smulaton and experment; smulatons show that slppng for Y translatonal motons are not as severe, due to robot symmetry. Wth zero commanded rotatonal moton, the robot experenced undesrable slp n rotatonal moton. In the future we wll study slppng n commanded rotatonal motons. Durng our ntal modelng and expermental work, we thought that omn-drectonal robot slp dynamcs would be lmted by hgh veloctes and acceleratons. Ths s stll true; however, we learned that, for our robot desgn, an equally sgnfcant factor n slp dynamcs s the rgd materal between rollers, even at low moton rates and acceleratons. Our development team response was to fle away as much of that materal as possble to avod contact n these sectors (after the experments). However, ths artcle s pertnent to any omn-drectonal moble robot desgn wth or wthout dscontnuty between rollers. Our work demonstrated reasonable smulaton/expermental agreement and we feel that we have captured the slp dynamcs behavor of our desgn. A future mprovement s to use statc and dynamc coeffcents of frcton; due to our demonstrated agreement, we conclude that the statc coeffcents of frcton are adequate. For dfferent omn-drectonal robot desgns, our modelng and smulaton work wll apply, but sgnfcant expermental work s stll requred to measure the varous frcton coeffcents and to fully understand the dynamc slp behavor. Snce our objectve was to model and understand the sldng dynamcs problem, ths artcle does not focus on real-tme control. We wll develop real-tme control n the future based on our dynamc model, ncludng measurement of varables for feedback control to overcome slppng dynamcs. [6] S. 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