Control of Motorcycle Steering Instabilities

Transcription

1 Control of Motorcycle Steerng Instabltes SIMOS EVANGELOU, DAVID J.N. LIMEBEER, ROBIN S. SHARP, and MALCOLM C. SMITH PASSIVE MECHANICAL COMPENSATORS INCORPORATING INERTERS Advances n modelng bcycle and motorcycle dynamcs are provdng mproved understandng of the prncpal modes of moton under straght-runnng and steadystate cornerng condtons; see, for example, [1] and the references theren. These studes show that under certan operatng condtons, some of the machne s modes can be lghtly damped or even unstable. The most mportant of these modes are wobble and weave. Wobble s a steerng oscllaton that s remnscent of the caster shmmy that occurs n the front wheels of a supermarket cart, whle weave s a fshtalng-type moton nvolvng roll and yaw. The frequency of the wobble mode s of the order 8 Hz, whle the weave frequency s about 3 Hz, where the exact fgures depend on the speed and machne parameter values. Modern hgh-performance motorcycles often employ a steerng damper producng a moment that opposes the angular velocty of the steerng assembly relatve to the man frame. or machnes wth a stff front frame, a steerng damper s requred to stablze the wobble mode at hgh speeds, whle older, more flexble machnes may requre a steerng damper at ntermedate speeds. Despte ts benefts for wobblemode performance, a steerng damper has a destablzng effect on the weave mode. As a result of ths conflct, only a narrow range of damper coeffcent values may be usable [2], [3]. Establshng damper settngs that provde an optmal compromse between wobble- and weave-mode dampng s an ssue of consderable nterest and s a focus of the present artcle. HARLEY-DAVIDSON 21 DIGITAL PRESS KIT 78 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER X/6/$2. 26IEEE

2 In partcular, we replace the conventonal steerng damper wth a network of nterconnected mechancal components that retans the vrtue of the damper, whle mprovng the weave-mode performance. The mproved performance s due to the fact that the network ntroduces phase compensaton between the relatve angular velocty of the steerng system and the resultng steerng torque. The approach descrbed here s underpnned by classcal passvty deas from crcut theory [4] as well as analoges between electrcal and mechancal networks [5]; see Passve Crcut Synthess. In the standard electrcalmechancal current-force analogy, an nductor corresponds to a sprng, whle a resstor represents a damper. To complete ths analogy, a mechancal component s needed to Passve Crcut Synthess One of the prncpal motvatons for the ntroducton of the nerter n [5] s the synthess of passve mechancal networks. The fact that the mass element, together wth the sprng and damper, s nsuffcent to realze the totalty of passve mechancal mpedances can be seen usng the force-current analogy THEOREM between mechancal and electrcal crcuts. In ths analogy, force and current are the through varables and velocty and voltage are the across varables. Moreover, the termnals of mechancal and electrcal elements are n one-to-one correspondence. or the mechancal elements, the sprng and damper have two ndependently movable termnals, whereas the termnals of the mass are ts center of mass and a fxed pont n an nertal frame (mechancal ground). The mass s therefore analogous to a grounded capactor. In contrast, the nerter s a two-termnal devce, analagous to an ungrounded capactor, wth both termnals freely and ndependently movable. gure A shows a table of element correspondences n the force-current analogy wth the nerter replacng the mass element. The admttance Y (s) s the rato of through to across quanttes, where s s the standard Laplace transform varable. or mechancal networks n rotatonal form, the through and across varables are torque and angular velocty, respectvely. It should be mentoned that the Mechancal recprocal defnton of admttance for mechancal elements s commonly found n the lterature, for example n [7], [8], and [25]. or further background on network analoges, see [5], d [24], and [26]. The theory of passve crcuts has been wdely studed n the electrcal engneerng lterature [4], [27]. A lnear tme-nvarant two-termnal network that possesses a real ratonal mpedance or admttance functon s passve f and only f the mpedance or admttance s postve real; see [4] and [27]. A celebrated result n electrcal crcut synthess, proved by Bott and Duffn, says that a ratonal, postve real functon can be realzed as the drvngpont mpedance or admttance of a network comprsed of only resstors, capactors, and nductors [28]. The result can be translated over to the mechancal settng as follows [5]. Let Y (s) be a postve-real ratonal functon. Then there exsts a two-termnal mechancal network that conssts of a fnte nterconnecton of sprngs, dampers, and nerters and whose admttance equals Y (s). Ths result allows the optmzaton of system propertes wthout fxng the network structure n advance. or example, t can be shown [3] that the bquadratc real ratonal functon Y (s) = a 2s 2 + a 1 s + a d 2 s 2 + d 1 s + d, where a 2, a 1, a, d 2, d 1, d are nonnegatve and at least one of d, d 1, d 2 s postve, s postve real f and only f ( a 1 d 1 a d 2 ) 2 d a 2. Sprng dt = k(v 2 v 1 ) d (v 2 v 1 ) dt = 1 L = b d(v 2 v 1 ) dt = c (v 2 v 1 ) Y(s) = k s Y(s) = bs Inerter Y(s) = c Damper Electrcal Inductor Capactor Resstor IGURE A Electrcal and mechancal crcut symbols and correspondences. In the forcecurrent analogy, forces substtute for currents, and veloctes substtute for voltages. The admttance Y(s) maps velocty and voltage nto force and current, respectvely. = C d(v 2 v 1 ) dt = 1 (v 2 v 1 ) R Y(s) = 1 Ls Y(s) = Cs Y(s) = 1 R OCTOBER 26 «IEEE CONTROL SYSTEMS MAGAZINE 79

3 represent a capactor. Although a mass s analogous to a capactor wth one termnal grounded, a new devce s requred to represent a capactor n general. The sutable masslke element s the nerter [5], whch can be ether a translatonal or rotatonal devce. In ts rotatonal form, the nerter generates a resstng moment M between two hnged bodes that s proportonal to the relatve angular acceleraton between them. Mathematcally, Vertcal Dsplacement (m) p7 M = b( ω 1 ω 2 ), p Longtudnal Dsplacement (m) IGURE 1 Schematc of the GSX-R1 motorcycle model. The scaled motorcycle and rder model shows the machne layout wth each mass depcted as a proportonally scaled red dsk. The motorcycle s shown n ts nomnal confguraton wth key ponts used n the multbody descrpton labeled as p. Swng_Arm; Rotate y Rear_Wheel; Rotate y p9 p13 p1 p8 p11 p22 p19 p21 p2 p1 Inertal rame; n p3 p2 Man; Translate (x, y, z); Rotate (z, y, x) Upper_Body; Rotate x Lnk 1; Rotate y No-Movement Lnk 2; Rotate y where ω 1 and ω 2 are the angular veloctes of the bodes, and b s the nertance n kg-m 2. As n [5], ths use of the term nertance s consstent wth ts usage n acoustcs [6] but s the recprocal of the tradtonal usage n mechancal vbratons [7], [8], where t represents a transfer functon from force to acceleraton and s synonymous wth the term accelerance n that feld. In control systems terms the nerter s used to produce phase lead. The effect of the nerter, whch contans a flywheel of modest mass and dmensons, s amplfed by hgh-rato gearng. Ths gearng can be realzed by an epcyclc arrangement [9] or by a harmonc drve [1]. It s possble to generate hgh nertance values wth relatvely low-mass components. More nformaton about the mechancal synthess of nerters can be found n Inerter. Twst Axs p5 p4 Steer Axs IGURE 2 Model parent-chld knematc dependences [15]. The machne and rder multbody herarchy show the relevant bodes and ther degrees of freedom. Reference axs drectons accord wth SAE vehcle dynamcs standards [16]. p6 ff_twst; Rotate x ff_steer; Rotate z ff_sus; Translate z ront_wheel; Rotate y MOTORCYCLE MODEL The motorcycle model used n ths artcle s based on the Suzuk GSX-R1, a stff-framed hgh-performance producton vehcle. Ths motorcycle s ftted wth a standard telescopc front fork suspenson system and a swngngarm-based rear suspenson that ncorporates a sngle sprng-damper unt lnked mechancally to the swng-arm, a monoshock suspenson; see gure 1. In gure 1, each consttuent mass s represented by a red dsk of dameter proportonal to the correspondng mass. The model has a tree structure, except for one knematcally closed loop, whch s assocated wth the monoshock descrpton. The freedom assocated wth each body s shown n gure 2. The symbolc multbody software system AUTOSIM [11] s used to assst the model-buldng process. A detaled descrpton of the model and ts parameter set can be found n [12]; only a summary s gven here. The man frame, whch s allowed unrestrcted moton, s pn connected to the steerng system, the rder s upper body, and the rear swng-arm. The torsonal complance of the frame near the steerng head s modeled by ncludng a twst degree of freedom; ths twst occurs about an axs that s perpendcular to the steer axs. The lower part of the front forks and the front wheel are free to translate along the fork lne. 8 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 26

4 Inerter The nerter s a two-termnal devce wth the property that an equal and opposte force appled at the termnals s proportonal to the relatve acceleraton between them. Mathematcally, the nerter obeys the force-velocty law = b( v 1 v 2 ), where the constant of proportonalty b s called the nertance and has the unts of klograms. In ts rotatonal form, the nerter obeys a moment-angular velocty law M = b( ω 1 ω 2 ); n ths case the nertance s measured n kg-m 2. To be practcally useful, the devce must have a small mass. Also, the devce must have reasonable overall dmensons and must be able to functon n all spatal orentatons. One way n whch a translatonal nerter can be constructed s llustrated n gure B. The devce comprses a rack-and-pnon mechansm, wth the rack constraned to translate relatve to the housng. or such devces the value b of the nertance s easy to compute n terms of the gear ratos and the flywheel s moment Pnons Rack Termnal 2 lywheel Termnal 1 IGURE B Schematc of the nerter prncple. A rack and pnon gearng arrangement drves a rotatng flywheel; the devce s nertance s a functon of the gear rato and the flywheel nerta. of nerta [5]. In general, f the devce gves rse to a flywheel rotaton of α radans per meter of relatve dsplacement between the t ermnals, then b = Jα 2, where J s the flywheel s moment of nerta and the remanng nertal effects are neglected. or a rotatonal nerter n whch there s a gear rato of n between rotatons of the termnals and a flywheel wth moment of nerta J, t follows that b = Jn 2. Several prototype devces accordng to [9] have been bult and tested n the Engneerng Department of Cambrdge Unversty; see for example fgures C and D. IGURE C A prototype nerter. Ths translatonal nerter employs a ballscrew to convert the lnear moton of the plunger nto rotatonal moton of the flywheel (not shown). The devce has a mass of approxmately 1 kg and an nertance n the range 6 2 kg dependng on the sze of the flywheel. The devce was desgned by N.E. Houghton and manufactured n the Cambrdge Unversty Engneerng Department. IGURE D Prototype motorcycle steerng compensator. Ths rotatonal devce employs an epcyclc gear box connected to a flywheel through a flud couplng to realze a seres nerter-damper. The mass s approxmately 1.5 kg, whle the nertance and dampng are approxmately.25 kg-m 2 and 1 N-m-s/rad, respectvely. The devce was desgned by N.E. Houghton and manufactured n the Cambrdge Unversty Engneerng Department. Each road wheel s axsymmetrc and s allowed to spn. Each tre s represented as havng a fnte wdth, and the ground contact ponts are allowed to mgrate both crcumferentally and laterally. Ths mgraton process s modeled by vewng the lowest pont on each tre as the center of the road-tre contact patch. The moton states of the tre contact centers, the compresson of the tres from ther nomnal statc equlbrum states, and the wheel camber angles are used n magc formula tre models [1], [12], [13] to compute the forces and algnng moments generated by the tres. Magc formula methods comprse emprcal formulas and parameters that descrbe the forces and moments developed, as measured n laboratores, as functons of the operatng condtons. Important operatng OCTOBER 26 «IEEE CONTROL SYSTEMS MAGAZINE 81

5 Wobble s a steerng oscllaton that s remnscent of the caster shmmy that occurs n the front wheels of a supermarket cart, whle weave s a fshtalngtype moton nvolvng roll and yaw. varables nclude the normal load, longtudnal and lateral slp, and wheel camber (or nclnaton) angle. As a result of the lateral complance of the tres carcasses, lateral force and moment varatons are not generated nstantaneously. The tme constants assocated wth the force and moment producton process or, more precsely, the relaxaton lengths, are functons of speed [14]. The suspenson sprngs and dampers are treated as lnear, although suspenson and steerng lmt stops are ncluded. Two feedback controllers are used n the model. The frst s a fxed-gan proportonal-plus-ntegral speed controller that generates the rear-wheel drve torque. The second s a speed-adaptve proportonal-ntegral-dervatve steerng torque controller that responds to the lean angle error. Wth the ad of these controllers, the machne can track prescrbed speed and lean-angle trajectores and, n partcular, can be run to any feasble trm state to determne equlbrum confguratons. The controller gans are set to zero for the uncontrolled motorcycle. The AUTOSIM model fle can be confgured n a lnear or nonlnear format. In the nonlnear confguraton, a numercal smulaton program for studyng transent machne behavor, or evaluatng the machne s trm states, s obtaned. Once a trm condton s reached, dsturbances such as road proflng can be ntroduced. When the model s confgured n ts lnear mode, the system s symbolcally lnearzed for small perturbatons about a general trm condton. The lnearzed models generated by AUTOSIM take a state-space form and a MATLAB M fle s produced. Each of the state-space matrces s parameterzed to correspond to the trm condton beng studed. The symbolc MAT- LAB fle can be used to generate standard plots such as Nyqust and root locus dagrams. When modelng the motorcycle n a conventonal manner, the steerng damper s ncluded as an ntegral part of the machne. Ths normally low-profle component generates a steerng torque that s proportonal to steerng velocty. To allow a more general relatonshp between the steerng torque and the steerng velocty, the steerng damper s separated from the remander of the system and represented as an external feedback loop as shown n gure 3 [2], [3], [17]. The nomnal motorcycle-damper combnaton wll be referred to as the standard machne, whle the motorcycle wthout a steerng damper wll be referred to as the basc machne. A conventonal steerng damper s represented as a pure gan n gure 3. More general mechancal.15.1 d(s) P (s) δ(s).5.5 A C B.1 T s (s) K(s) sδ(s) IGURE 3 Lnearzed motorcycle model and steerng compensator. The lnearzed motorcycle model s denoted P(s), whle the steerng compensator s shown as K (s) n the feedback loop. The nput d(s) represents vertcal road-dsplacement forcng, whle the output δ(s) s the steerng angle and T s (s) s the steer torque. The road forcng sgnal, whch s a scalar varable, s appled to both wheels. The rear-wheel forcng sgnal s a delayed verson of that at the front; the tme delay s gven by τ = w/v n whch w s the wheelbase and v s the forward speed. The steerng compensator K (s) s constant n the case of a conventonal steerng damper. IGURE 4 Nyqust dagram of the straght-runnng basc machne. The dagram corresponds to the sngle-loop system wth steerng torque as nput and wth steerng system angular velocty as output; the machne s forward speed s 75 m/s. The frequency assocated wth crossng pont A, the wobble-mode resonant frequency, s 47.6 rad/s. The frequency assocated wth the cusp B s 33.8 rad/s, and that at crossng pont C, the weave-mode resonant frequency, s 28.4 rad/s. If the damper coeffcent s selected so that the pont 1/c s at A, the motorcycle oscllates at the wobble-mode frequency. If the damper coeffcent s selected so that the pont 1/c s at C, the machne oscllates at the weave-mode frequency. 82 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 26

6 networks are represented by K(s) n whch s s the Laplace varable. The transfer functon K(s) from angular velocty to torque s called the admttance functon of a mechancal network followng the conventon of [24, pp. 45, 46, 326] but n contrast to other uses n mechancal engneerng [6, p. 238] and [8, p. 27]. CHARACTERISTICS O THE NEAR-STANDARD MACHINE gure 4 shows a Nyqust dagram of the lnearzed model of the basc machne n a straghtrunnng equlbrum condton at 75 m/s. Ths dagram can be used to study the effect of changng the steerng damper coeffcent. The model has two rght-half-plane poles for ths operatng condton because the wobble mode s unstable. Snce the steerng damper s represented as a pure gan n the feedback loop of gure 3, t follows from the Nyqust crteron that stablty of the standard machne requres two counterclockwse encrclements of the pont 1/c, where c s the steerng damper constant. If the steerng dampng s set at a low value such that the pont 1/c s located at A, the machne s on the stablty boundary and thus oscllates at 47.6 rad/s, whch s the wobble-mode frequency. If the steerng dampng s now ncreased, two counterclockwse encrclements of the pont 1/c result, and the motorcycle s stable. If the steerng dampng s ncreased further so that the pont 1/c concdes wth C, the machne oscllates at 28.4 rad/s and the weave mode s on the stablty boundary. urther ncreases n the steerng dampng render the machne unstable snce the pont 1/c s not encrcled. The nomnal steerng damper coeffcent s N-m-s/rad, thereby locatng the pont 1/c at.144, whch s approxmately mdway between A and C. When the speed ncreases from 75 m/s to 85 m/s, the nterval on the negatve-real axs assocated wth stablzng damper parameter values moves to the left [3]. Ths shft moves the weave-mode crossng pont toward the pont 1/c assocated wth the nomnal dampng coeffcent, and the wobble-mode crossng pont moves away from t. As a result, ths speed ncrease reduces the weave-mode dampng factor and ncreases the wobble-mode dampng factor. gure 5 shows straght-runnng root loc where speed s the vared parameter. When the steerng damper coeffcent s reduced from ts nomnal (standard machne) value, the wobble mode becomes unstable at hgh speed, whle the hgh-speed weave-mode dampng ncreases. The damper has almost no nfluence on the frequency (magnary part) assocated wth ether of these modes. Replacng the damper wth an nerter, for whch K(s) = bs, leads to the root locus dagram of gure IGURE 5 Influence of the steerng damper on the root loc of the straght-runnng motorcycle. Speed, whch s the vared parameter, s ncreased from 5 m/s ( ) to 75 m/s ( ). The locus s assocated wth the nomnal machne dampng value of N-m-s/rad, the locus s assocated wth a steerng dampng value of 3.94 N-m-s/rad, and the + locus wth a steerng dampng value of.94 N-ms/rad Wobble Weave Wobble Weave IGURE 6 Influence of a steerng nerter on the root loc of the straght-runnng motorcycle. Speed, whch s the vared parameter, s ncreased from 5 m/s ( ) to 75 m/s ( ). The locus s assocated wth the basc machne, the locus represents the basc machne ftted wth a steerng nertance of.1 kg-m 2, and the + locus represents the basc machne ftted wth a steerng nertance of.2 kg-m 2. OCTOBER 26 «IEEE CONTROL SYSTEMS MAGAZINE 83

7 The dynamc characterstcs of hgh-performance motorcycles can be mproved by replacng the conventonal steerng damper wth a passve mechancal steerng compensator. Although the nerter stablzes the weave mode at hgh speeds, t has a detrmental effect on the wobble-mode dampng. The reducton n the wobble-mode frequency s caused by the effectve ncrease n the steerng system s moment of nerta. Wth regard to the dampng of the weave and wobble modes, the nerter and damper ntroduce opposte trends. COMPENSATION NETWORK c b IGURE 7 Seres resonant flter. Ths seres connecton of a damper, nerter, and sprng, consttutng a resonant flter, provdes frequency-localzed dampng nfluence. k Desred Network Propertes The results of the prevous secton suggest that a mechancal network that s damper-lke over the wobble-mode frequency range (5 9 Hz) and nerter-lke at the lower frequences assocated wth the weave mode (2-3 Hz) mght be benefcal. Over the stll lower frequency range used by the rder (.5 Hz) [19], [2], for balancng and path-followng control, the network must be hgh pass to allow unhndered rder steerng acton. Sprng-lke propertes are undesrable as are hgh values of steady-state gan. It can be shown that the seres connecton of an nerter and damper has the admttance functon scb/(sb + c), where b and c are the nertance and dampng coeffcent respectvely. The frequency response of ths admttance functon s smlar to that of an nerter (respectvely, damper) at frequences below (respectvely, above) the break frequency c/b. Ths frequency response functon can be benefcal for smultaneous control of wobble and weave. A related network s the resonant flter consstng of the seres connecton of a damper, an nerter and a sprng (see gure 7). Ths network has the admttance s Y(s) = s 2 + sk/c + k/b, (1) where b, c, and k denote the nertance, dampng coeffcent, and sprng stffness, respectvely. Phase ( ) Magntude (db) ζ = requency (rad/s) (b) IGURE 8 requency-response characterstcs of the seres resonant flter network wth the resonant frequency normalzed to ω n = 1. Three values of dampng rato ζ are llustrated. (a) IGURE 9 Nyqust dagram of the straght-runnng motorcycle wth a forward speed of 75 m/s. The sold lne represents the standard machne, whle the dashed lne corresponds to the compensated system usng the seres resonant flter shown n gure 7 wth desgn values ω n = 5 rad/s, ζ =.4, and k = 5 N-m/rad. 84 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 26

8 requency Response Desgn It s nstructve to develop ntal desgn gudelnes for the network parameters usng classcal frequency-response deas. The network admttance (1) can be rewrtten n the form s Y(s) = k s 2 + 2ζω n s + ωn 2, n whch ω n = bk k/b, ζ = 2c. (2) It can be seen from the frequency-response characterstcs of the network shown n gure 8 that ths network acts lke an nerter at low frequences and ntroduces dampng n the vcnty of ω n, whch must be tuned to the wobblemode frequency. The dampng rato ζ s a desgn parameter that determnes the sharpness of the magntude peak and the rate of change of phase wth frequency. Smaller values of ζ gve a more rapd transton from nerterlke to damperlke behavor. Larger values of ζ provde a wder range 7 of frequences where damperlke behavor perssts. The peak value of the magntude characterstc s acheved at ω n and takes the 6 value of c, the damper constant. It can be seen 5 n (2) that ω n and ζ fully specfy the phase characterstcs of the network. 4 or an ntal tral desgn the parameters ω n = 5 rad/s and ζ =.4 are selected. Pror 3 to fxng a value of the sprng constant, t s observed that k = 32 N-m/rad places the 2 pont 1 n the mddle of the stable k-value range n the 75 m/s straght-runnng condton. However, snce ths value of k does not 1 produce adequate wobble-mode dampng performance at hgh lean angles, the sprng stffness s ncreased to k = 5. Ths change mproves the wobble-mode dampng globally but at the expense of the weave mode. Back substtuton gves nerter and damper parameter values of b =.2 kg-m 2 and c = 1 N-m-s/rad, respectvely. The nfluence of ths partcular choce of parameters s llustrated n gure 9. It may be observed that the network moves the negatve-axs crossng pont assocated wth weave-mode nstablty toward the orgn and the crossng pont lnked to wobble to the left of the dagram. The network opens up the nterval over whch two counterclockwse encrclements can be acheved. The root locus plot that results from ths mechancal network wth parameter values b =.2 kg-m 2, c = 1 N-m-s/rad, and k = 5 N-m/rad s shown n gure 1. Although the desgn s based on a sngle hgh-speed straght-runnng lnearzed model, compared wth the standard machne, substantal mprovements n the dampng of both the wobble and the weave modes under all operatng condtons are acheved. The mprovement n the hgh-roll angle (45 ) case s worthy of partcular note [3]. OPTIMIZATION We now optmze the parameters of the seres resonant flter. The chosen performance crteron reflects the role played by road-dsplacement dsturbances n stabltyrelated road traffc accdents [21] n the form of an H response measure, together wth a penalty on the closeness of approach of the open-loop Nyqust locus to the pont 1. The objectve functon consdered s where J f = max {max{j 1,γJ 2 }}, (3) Wobble Weave IGURE 1 Root loc for the compensated motorcycle. our values of roll angle are llustrated: straght runnng ( ), 15 ( ), 3 (+), and 45 ( ). The speed s vared from 7 m/s ( ) to 75 m/s ( ). The machne s ftted wth the seres resonant network wth the parameter values b =.2, c = 1, and k = 5. TABLE 1 Desgn parameters, objectve functons, and worst-case confguratons obtaned by optmzng the frequency-doman ndex (3) for the conventonal steerng damper and the seres resonant flter. Maxmum Compensator Parameters J f deg m/s rad/s Conventonal c = damper Seres resonant k = flter c = b = OCTOBER 26 «IEEE CONTROL SYSTEMS MAGAZINE 85

9 and db J 1 = max P 11 ( jω ) ω 1 K( jω )P 22 ( jω ) (4) J 2 = max ω 1 1 K( jω )P 22 ( jω ). (5) Wobble Weave IGURE 11 Root loc for the motorcycle compensated by the optmzed seres resonant flter wth parameters as n Table 1. our values of roll angle are llustrated, namely, straght runnng ( ), 15 ( ), 3 (+), and 45 ( ). The speed s vared between 7 m/s ( ) and 75 m/s ( ) Speed (m/s) Roll Angle ( ) 3 IGURE 12 Road-forcng gan as a functon of operatng condton. The speed s vared between 7 and 75 m/s, and the roll angle between 3 and 45. The motorcycle s ftted wth the frequency-response-optmzed seres resonant compensaton network. The set of lnear motorcycle models used n (3) contans lnearzed models correspondng to trm roll angles of, 3, 6,..., 45 and trm speeds of 7, 9, 11,..., 75 m/s. The subndex J 1 n (4) s remnscent of the L -norm of the transfer functon between road-dsplacement forcng and steer angle (see the Motorcycle Model secton and gure 3), whle J 2 n (5) s the recprocal of the dstance of closest approach between the Nyqust locus and the pont 1. J 2 effectvely penalzes hgh values of the classcal senstvty functon [22]. The subndex J 2 s weghted n J f by the constant γ, set by tral to a value 16. When evaluatng J f, a 1-pont angular frequency sequence n geometrc progresson, from ω = rad/s to ω = rad/s s used; ths range ncludes all of the peaks n J f. The MATLAB sequental quadratc programmng algorthm fmncon [23] s used for optmzaton. The algorthm s ntalzed usng the parameters obtaned from the results descrbed n the requency Response Desgn secton. RESULTS The results of optmzng wth the frequency response ndex (3) are gven n Table 1 for the conventonal damper and the seres resonant flter. In addton to the optmal parameter values, the table provdes the mnmum values acheved for J f, the trm condton at whch the mnmum s acheved, and the worst-case frequences assocated wth the lmtng value of the ndex. The wobble mode dctates the lowest achevable value of J f. or the flter, the closest approach between the Nyqust plot and the pont 1 exceeds 16/4.576 =.394, where 16 s the value of γ used n (3). gure 11 shows the root locus of the motorcycle s key modes for a wde range of speeds and roll angles usng the optmzed seres resonant flter network. The network acheves mproved dampng ratos for each of the machne s lghtly damped modes. gure 12 shows the road-forcng response J 1 n (4) wth the optmzed seres resonant flter network nstalled; the trm state ranges over the motorcycle s operatng regme. As expected, under straght-runnng condtons, the road-forcng response s zero. In common wth each of the other networks, the hghest gan values, whch occur at low speeds and hgh roll angles, correspond to the exctaton of the wobble mode. Hgh values of road-forcng-to-steer-angle gan also occur under hgh-speed, low-roll- 86 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 26

10 Magntude (db) requency (rad/s) Magntude (db) requency (rad/s) IGURE 13 Bode magntude plot of steer angle response to roaddsplacement forcng ( db = 1 rad/m). The machne s operatng at a forward speed of 15 m/s and a roll angle of 45. The sold lne represents the standard machne, whle the dashed lne represents the machne wth the optmzed seres resonant flter network compensator. IGURE 14 Bode magntude plot of steer angle response to roaddsplacement forcng ( db = 1 rad/m). The machne s operatng at a forward speed of 75 m/s and a roll angle of 15. The sold lne represents the standard machne, whle the dashed lne represents the machne wth the optmzed seres resonant flter network compensator. angle condtons (see 75 m/s and 15 roll angle), whch correspond to exctaton of the weave mode. The road-forcng characterstcs of the machne are also llustrated by the frequency-response plots shown n gures 13 and 14. As s evdent from the frst of these fgures, the seres resonant flter network decreases the peak wobble-mode road-forcng gan by approxmately 15 db. or the trm condton consdered n gure 13, the wobble mode s partcularly vulnerable to road-dsplacement forcng. gure 14 corresponds to a hgh-speed trm condton n whch the weave mode s correspondngly challenged. In ths case, the network reduces the weave-mode peak by approxmately 3 db as compared to the standard machne. CONCLUSIONS Ths artcle shows that the dynamc characterstcs of hghperformance motorcycles can be mproved by replacng the conventonal steerng damper wth a passve mechancal steerng compensator. The steerng compensators are mechancal networks comprsng sprngs, dampers, and nerters. The compensators show the potental to sgnfcantly mprove the dampng of both wobble and weave modes smultaneously. These networks explot the nerter s ablty to provde phase advance. The compensator s role can be nterpreted as that of a feedback element n a control systems structure, despte ts consstng of passve mechancal elements. The desgn methodology adopted uses Nyqust frequency response deas, root-locus analyss, and loopshapng desgn to obtan a prelmnary choce of parameters, whch are then refned by numercal optmzaton. The results show substantal global performance mprovements as compared wth conventonal steerng dampers. The broad ssue of the practcal mplementaton of passve mechancal compensators s the subject of ongong research, ncludng the fabrcaton of ntegrated mechancal networks as llustrated n gure 7. Addtonal ssues such as the selecton of optmal gear ratos for the nerters and the correct dmensonng of the components so that they have suffcent workng lfe are also mportant consderatons. rom a packagng perspectve, a steerng compensator needs to be acceptably small and lght. Ideally, the compensator wll ft n the space normally occuped by a conventonal steerng damper. AUTHOR INORMATION Smos Evangelou receved the B.A./M.Eng. degree n electrcal and nformaton scences from the Unversty of Cambrdge, Unted Kngdom, n 1999 and the Ph.D. degree n control engneerng from Imperal College London, n 24. In 26 he was apponted a lecturer at Imperal College London, for the departments of mechancal and electrcal and electronc engneerng. Prevously he worked as a research assocate from 24 to 25. Hs research nterests nclude the modelng, stablty analyss and control of sngle-track vehcles, the applcaton of control theory to mechancal systems, dynamc system modelng, and control systems theory. Davd J.N. Lmebeer (d.lmebeer@mperal.ac) receved the B.Sc. degree n electrcal engneerng from the Unversty of Wtwatersrand, Johannesburg, n 1974, the M.Sc. and Ph.D. degrees n electrcal engneerng from the Unversty OCTOBER 26 «IEEE CONTROL SYSTEMS MAGAZINE 87

11 of Natal, Durban, South Afrca, n 1977 and 198, respectvely, and the D.Sc. degree from the Unversty of London n He has been wth Imperal College London, snce 1984, where he s currently the head of the Department of Electrcal and Electronc Engneerng. He has publshed over 1 papers and a textbook on robust control theory. Three of hs papers have been awarded przes, ncludng the 1983 O. Hugo Schuck Award. He s a past edtor of Automatca and a past assocate edtor of Systems and Control Letters and the Internatonal Journal of Robust and Nonlnear Control. He s a ellow of the IEEE, the IEE, the Royal Academy of Engneerng, and the Cty and Gulds Insttute. Hs research nterests nclude control system desgn, frequency response methods, H-nfnty optmzaton, and mechancal systems. He s qualfed as an IAM senor motorcycle nstructor and receved a RoSPA certfcate for advanced motorcyclng. He can be contacted at Imperal College London, Department of Electrcal and Electronc Engneerng, Exhbton Road, London SW7 2AZ U.K. Robn S. Sharp s professoral research fellow n the Department of Electrcal and Electronc Engneerng at Imperal College London. He s a member of the Dynamcal Systems and Mechatroncs Workng Group of the Internatonal Unon of Theoretcal and Appled Mechancs, the edtoral board of Vehcle System Dynamcs, and the edtoral advsory board of Multbody System Dynamcs. He was frst vce-presdent and secretary general of the Internatonal Assocaton for Vehcle System Dynamcs, edtoral panel member and book revew edtor for The Proceedngs of the Insttuton of Mechancal Engneers, Journal of Mechancal Engneerng Scence, Part C, and edtoral panel member of The Proceedngs of the Insttuton of Mechancal Engneers, Part D, Journal of Automoble Engneerng. rom , he was professor of automotve product engneerng at Cranfeld Unversty. He was a vstng assocate research scentst at the Unversty of Mchgan Transportaton Research Insttute, Ann Arbor. Hs research covers topcs n automotve dynamcs and control as well as n control and stablty of sngle-track vehcles, unmanned ar vehcles, and the applcaton of optmal prevew and learnng control to road vehcle drvng/rdng. Malcolm C. Smth receved the B.A. (M.A.) degree n mathematcs, the M.Phl degree n control engneerng and operatonal research, and the Ph.D. degree n control engneerng, all from the Unversty of Cambrdge, U.K. He was subsequently a research fellow at the German Aerospace Centre, a vstng assstant professor and research fellow wth the Department of Electrcal Engneerng at McGll Unversty, and an assstant professor wth the Department of Electrcal Engneerng at the Oho State Unversty. In 199 he returned to the Unversty of Cambrdge where he s currently a professor. He s a fellow of Gonvlle and Caus College. Hs research nterests nclude control system desgn, frequency response methods, H optmzaton, nonlnear systems, actve suspenson, and mechancal systems. He was a corecpent of the 1992 and 1999 George Axelby Outstandng Paper Awards, both tmes for jont work wth Dr Tryphon T. Georgou. He s a ellow of the IEEE. REERENCES [1] D.J.N. Lmebeer and R.S. Sharp, Bcycles, motorcycles, and models, IEEE Control Syst. Mag., vol. 26, no. 5, pp , 26. [2] S. Evangelou, D.J.N. Lmebeer, R.S. Sharp, and M.C. Smth, Steerng compensaton for hgh-performance motorcycles, n Proc. 43rd CDC, Paradse Island, Bahamas, Dec. 24, pp [3] S. Evangelou, D.J.N. Lmebeer, R.S. Sharp, and M.C. Smth, Mechancal steerng compensators for hgh-performance motorcycles, J. Appl. Mech., to be publshed. [4] R.W. Newcomb, Lnear Multport Synthess. New York: McGraw Hll, [5] M.C. Smth, Synthess of mechancal networks: The nerter, IEEE Trans. Automat. Contr., vol. 47, no. 1, pp , 22. [6] C.L. Morfey, Dctonary of Acoustcs. New York: Academc, 21. [7] D.E. Newland, Mechancal Vbraton Analyss and Computaton. Whte Plans, NY: Longman, [8] D.J. Inman, Engneerng Vbraton, 2nd ed. Englewood Clffs, NJ: Prentce Hall, 21. [9] M.C. Smth, orce-controllng mechancal devce, patent pendng, Int. pub. WO 3/5142, 4 July 21, [Onlne]. Avalable: com/textdoc?db= EPODOC&IDX=WO35142&= [1] Harmonc drve. [Onlne]. Avalable: [11] Autosm 2.5+ Reference Manual, Mechancal Smulaton Corp., Ann Arbor, MI, 1998 [Onlne]. Avalable: [12] R.S. Sharp, S. Evangelou, and D.J.N. Lmebeer, Advances n the modellng of motorcycle dynamcs, Multbody Syst. Dynamcs, vol. 12, no. 3, pp , 24. [13] H.B. Pacejka, Tyre and Vehcle Dynamcs. Oxford, U.K.: Butterworth Henemann, 22. [14] E.J.H. de Vres and H.B. Pacejka, The effect of tyre modelng on the stablty analyss of a motorcycle, n Proc. AVEC 98, Nagoya: SAE Japan, 1998, pp [15] R.S. Sharp, S. Evangelou, and D.J.N. Lmebeer, Multbody aspects of motorcycle modellng wth specal reference to autosm, n Advances n Computatonal Multbody Systems, J.G. Ambróso, Ed. Dordrecht, The Netherlands: Sprnger-Verlag, 25, pp [16] Socety of Automotve Engneers, Vehcle dynamcs termnology, n Proc. SAE J67e, Warrendale, PA, [17] C. Papageorgou and M.C. Smth, Postve real synthess usng matrx nequaltes for mechancal networks: Applcaton to vehcle suspenson, n IEEE Trans. on Contr. Syst. Tech, vol. 14, no. 3, pp , 26. [18] R.C. Dorf and R.H. Bshop, Modern Control Systems. Englewood Clffs, NJ: Prentce Hall, 21. [19] A. Aok, Expermental study on motorcycle steerng performance, SAE paper 79265, [2] M. Sugzak and A. Hasegawa, Expermental analyss of transent response of motorcycle rder systems, SAE paper , [21] D.J.N. Lmebeer, R.S. Sharp, and S. Evangelou, Motorcycle steerng oscllatons due to road proflng, J Appl. Mech., vol. 69, no. 6, pp , 22. [22] M. Green and D.J.N. Lmebeer, Lnear Robust Control. Englewood Clffs, NJ: Prentce Hall, [23] The Mathworks Inc., MATLAB 6 Reference Manual, 2 [Onlne]. Avalable: [24] J.L. Shearer, A.T. Murphy, and H.H. Rchardson, Introducton to System Dynamcs. Readng, MA: Addson-Wesley, [25] E.L. Hxson, Mechancal mpedance, n Shock and Vbraton Handbook, 2nd ed., C.M. Harrs and C.E. Crede, Eds. New York: McGraw-Hll, 1976, chap. 1. [26] A.G.J. Macarlane, Dynamcal System Models. London: Harrap, 197. [27] B.D.O. Anderson and S. Vongpantlerd, Network Analyss and Synthess. Englwood Clffs, NJ: Prentce Hall, [28] M.E.V. Valkenburg, Introducton to Modern Network Synthess. New York: Wley, IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 26