Longitudinal Control of a Monowheeler

Transcription

1 Proceedings of the 7 th Chinese Control Conference July 6-8, 008, Kuning,Yunnan, China Longitudinal Control of a Monoheeler Chen Pingho Det. of EE, ungnan University Eail:hc@ail.tnu.edu.t Astract: he onoheeler, a ono-heeled vehicle is coosed of a heel, a latfor and a control odule. Design, farication and siulation y LQR control aroach oint out that the results irove a lot the eigen structure of the syste in longitudinal alancing. Key Words: Monoheeler,Mono-heeled vehicle, LQR, Balance INRODUCION he innovation of autoilot onoheeled vehicles (hereafter called ooheeler ) is excited y the traded coodity of Segay yheeled scooter on sale since 004 as shon in ig.. or energy saving, the to heels of Segay yheeled scooter ight e relaced y just one heel as studied in this aer. he goal of the research is to develo a latfor-ounted onoheeler so that it ay e used in regional transortation for a single driver. he hole scoe of ork is roken into three hases: driving and itch control that also take into account coonents assigned for rolling and heading control. Preceding researches on vehicle longitudinal control technology are classified to 3 categories, i.e. ) adative control for active susensions [-9], ) vehicle dynaics [0-], 3) vehicle control [-7].Section IV siulation and results reveals the feasiility of ileentation. MALAB and its Siulink tool ox are eloyed in the siulation. Section IV gives a data digest that suarizes the associative data in the odeling and siulation. ig. Segay yheeled scooter Phase I: orard alancing control. Phase II: Heading and rolling control. Phase III: Autoilot. his aer is focused on the hase I, in other ords, forard alancing control. ig. shos the first atent of onoheeler driven y engine-oer in 94 in rance. As shon, there is a latfor stailized in level indeendent of heel tilting in rolling. Based on the design concet of ig. and ig., a concetual design to eet our goal is foralized as given in ig. 3. Section II egins ith a hardare design configuration on hich the folloing odeling is to e ased. he reliinary onoheeler hardare is asically coosed of a heel odule, a latfor odule and a control odule. he functions of each odule are descried to eet the necessity of the goals of a onoheeler erforance, i.e. heel osition 0k (ax.) and latfor itch 0.5 o (ax.). Section III gives the softare regarding the odeling and control aroaches. he odeling consists in heel 63 ig. he first onoheeled vehicle driven y engine-oer. ig. 3 Concetual engineering design HARDWARE CONUGURAION. Hardare Assignent he hardare of the ooheeler is coosed of a heel odule, a latfor odule and a control odule. Coonents of each odule are roken don elo and given as shon in ig. 4. Coonents of heel odule are heel, heel otor and heel axle earing. Coonents of latfor odule are: susension fraes, aluinu latfor, itch all scre, roll all scre (ig. 6) and attery. Coonents of control odule are )heel controller, heel otor servo alifier, heel Authorized licensed use liited to: Sungkyunkan University. Donloaded on June 9, 009 at 0:0 fro IEEE Xlore. Restrictions aly.

2 tachoeter ) Pitch controller, itch otor servo alifier, and itch all scre otentioeter 3) Roll controller, roll otor servo alifier, and roll all scre otentioeter. = 40 (kg), =. (kg), L = 6(c), L =3(c), I = (kgc ), I R =50(kgc ) It follos that I P =4679 (kgc ), I R =3004 (kgc ) ig. 4 Assignent of coonents. Mass Balancing Platfor odule has to e alanced statically. he centers of gravity of all scre set and aluinu latfor are here calculated for instances. he center of gravity of all scre set is at 6.3c right to the otor as shon in ig. 5. Since the aluinu latfor is divided into four traezoidal sections as shon in ig. 6, center of gravity of aluinu latfor is at c left to the origin of the aluinu latfor in ters of those traezoids. he center of gravity of the latfor and ounted odules as denoted in ig. 6 is (-0.9, -0.4) ith residue off-center ar.0 c. ig. 5 Center of gravity of all scre.3 Moent of inertia he latfor is susended to the heel axle ith length L underneath. herefore oents of inertia I and I R of the latfor ith resect to the heel axle are I =I + L () I R = IR + r + L () here I and I R are oents of inertia of the latfor along itch and roll direction resectively ith resect to its origin or equivalently the rojection of heel center onto the latfor as shon in ig. 6. and are ass of heel odule and ass of latfor odule resectively. As calculated e have 3 MODELING ig. 6 Weight distriution of latfor he onoheeler, coosed of heel odule and latfor odule can e regarded as a to-ody syste as if it ere a root ar ith latfor acting like a aniulator. Basically, the control inuts to the heel odule and latfor are ) heel otor torque, ) latfor alancing ar r (roortional to latfor torque ). he desired oututs are the heel osition x (say 0k) and the latfor itch attitude ϑ (say0.5 o, i.e. in horizontal level). hat is to say, the reference is R= [ x, ϑ ]. herefore the syste of the heel odule and latfor is of MIMO syste. Prior to the odeling, the free-ody analysis of to-ody dynaics is necessary. ig. 7(a) and () sho free odies of heel odule and latfor resectively. ig.7(c) shos the attitude coordinates of itch, roll and heading. In general, the attitude of the latfor is to e ket in horizontal level regardless of tilting in itch and roll siilar to an inverted endulu. ig. 7 (a)ree ody of heel () ree ody of latfor (c)attitude axis 64 Authorized licensed use liited to: Sungkyunkan University. Donloaded on June 9, 009 at 0:0 fro IEEE Xlore. Restrictions aly.

3 Assuing that there is no sliding at the heel center, e have silified equations of otion for the heel odule fro ig. 7(a): g N = 0 (3) Since I = r r (4) g = x cos = r ( + )cos r + ) θ 0 ) (5) ( r I = (6) x = r θ (7) Assuing g 0 for linear syste, Eq. (4) can e reritten y r + r ( + ) = (8) i.e. here a + + = (9) a = r, = r (9A) : Noral reaction force fro the ground, : Weight of heel odule, N: Noral utual reaction force eteen the heel and the latfor, g :Ground friction, : Bearing viscosity, : Ground contact thrust generated fro otor torque, I : Moent of inertia of heel odule. Note that the aerodynaic lift and drag on the surface of the latfor are teorary neglected and ight e estiated in further study. We also have silified equations of otion for the latfor ro ig. 7 (): g N= 0 (0) x + L θ () aking the oent along the heel center e have xl + I θ = gl θ + gr ( ) r () Coing Eq. (5), (7) and (), it follos c + + d + + e = fr (3) Coing Eq. (9) and (3) e have d + ( c ) + e + ( c ) c = + fr a a a (4) here c= Lr, d= I, e= gl, f= g (5) hen Eq. (9) and (4) are equations for the lant and can e exressed in state-sace for ith state X a a 0 a = (6) r c e c 0 ( ) ( ) c f d a d d a da d i.e. X = AX + BU (7) Its corresonding easureent equation is: x r Y = X CX θ = = (8) here is easured y heel tachoeter and y IMU gyroscoes. he scheatic of this tye of control syste is shon as ig.8. ig. 8 he scheatic of itch control in state-sace for 4 SIMULAION he data set associated ith control of heel osition and latfor itch is = 0kg, r = 0.3, =.8 kg /rad/s r = 0.05, I = kg, I = kg, = 40 kg, L = 0.6, g = 9.8/s, It follos a=.04, = 0.005, c=.048, d= 4.679, e= 6.7, f= 39 Based on Eq. (7) and (8), X =, Y = x θ, U = r (9) A = B =, C = (0) Since the syste has full-states feedack, LQR (Linear Quadratic Regulator) design is eloyed as follos. he cost is J(0) = X ( SX ) ( ) + [ ( 0 ) X QX+ U RU + λ ( AX + BU X )]dt () here Q: state eighting, R: control eighting and 65 Authorized licensed use liited to: Sungkyunkan University. Donloaded on June 9, 009 at 0:0 fro IEEE Xlore. Restrictions aly.

4 Lagrange ultilier. After several algera oerations to iniize the cost, e have the folloing Hailtonian atrix as the transition atrix in Eq. (): - X A BR B X = λ Q A λ () λ () t = S() t X () t (3) λ ( ) = S( ) X ( ) (4) - - U() t = R B λ() t = R B S() t X() t = K( t) X () t (5) he steady gain is - K= li Kt ( ) = RB li St ( ) t t - - = R Bψ ( ψ ) (6) here ψ and ψ are the first-colun lock atrix of hichis the odal for of the Hailtonian atrix in Eq. (). he siulation result shos that the oen-loo oles are o = 0, , j3.8794, j (7) he gain is K = (8) he closed-loo oles are c = j0.703, j 0.703, j7.479, j7.479 (9) It is ovious that the oles of oen loo in Eq. () sho less daing ith natural frequency at Hoever the oles of closed loo in Eq. (3) sho ore satisfied daing close to =0.7. In the aove design, the heel osition and latfor itch have equal eighting. Its state lot is given as ig. 9. Various eightings for instance e ay assign to x= and y=. Its resultant figure is given as ig. 0. Siilarly e ay also assign x=and y= and otain ig.. he aove three design otions oint out that the eighting x=and y= is the est one aong the. ig. 9 State lot of onoheeler syste ithx=,y= ig. 0 State lot of onoheeler syste ith x=, y=. ig. State lot of onoheeler syste ith x=, y=. 5 CONCLUSION his aer, focused on the goal of studying an autoilot onoheeler, descries the hardare of a heel odule, a latfor odule and a control odule together ith its softare odeling and siulation. A latfor ounted on oard the onoheeler is to e alanced dynaically as if it ere a horizontal lane regardless of the straight forard otion of the heel, either in acceleration or de-acceleration. hree susystes, i.e. a heel odule, a latfor odule and a control odule, are concetually designed, odeled and further siulated. he analysis reveals that the syste is one of MIMO systes ith heel otor torque and alancing ar as inuts and heel osition and latfor itch attitude as oututs. Siulation results also oint out that the LQR aroach iroves a lot the eigen structure of the onoheeled syste. REERENCES [] ALLEYNE A NEUHAUS P D, HEDRICK J K, Alication of Nonlinear Control heory to Electronically Controlled Susensions. Vehicle Syste Dynaics, 993():5-6, [] ALLEYNE A, HEDRICK J K, Nonlinear Adative Control of Active Susensions.IEEE ransactions on Control Systes echnology, 995(3):, Authorized licensed use liited to: Sungkyunkan University. Donloaded on June 9, 009 at 0:0 fro IEEE Xlore. Restrictions aly.

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