System Design, Modelling, and Control of a Four-Wheel-Steering Mobile Robot

Transcription

1 System Desgn, Modellng, and Control of a Four-Wheel-Steerng Moble Robot Maxm Makatchev Dept. of Manufacturng Engneerng and Engneerng Management Cty Unversty of Hong Kong Hong Kong John J. McPhee Dept. of Systems Desgn Engneerng Unversty of Waterloo Ontaro Canada NL 3G1 S. K. Tso Sherman Y. T. Lang Centre for Intellgent Desgn, Automaton and Manufacturng Cty Unversty of Hong Kong Hong Kong Integrated Manufacturng Technologes Insttute Natonal Research Councl of Canada London, Ontaro, Canada N6G 4X8 Abstract Ths paper descrbes a dynamc model and crosscouplng control of a moble robot wth four ndependently steered and drven wheels. The dynamc model ncorporates the components of the wheel mechansms, backlash effects of the wheel actuators and nonlnear forces produced by the tre-ground nteracton. The problem of computatonally effcent control of the vehcle s solved usng a low-level control loop that cross-couples steerng and drvng motors and s utlzed n conjuncton wth a knematcs-based trajectorytrackng controller. The results of smulaton show sgnfcant reducton of the vehcle slp wth moderate trajectory-trackng errors. 1 Introducton Research on modellng and control of wheeled vehcles can be dvded nto two major categores: one that s orented towards automobles and terran vehcles, the other s orented towards ndoor wheeled moble robots (WMRs). Ths dvson s based on the dfferences n the desgn of the vehcles, operatonal and envronmental condtons. To menton a few: wheels of WMRs are often ndependently actuated, as opposed to mechancally coupled pars of steerng wheels of automobles. a wheel plane of a typcal WMR s normally perpendcular to the ground plane, whle a wheel plane of an automoble may have a nonzero nclnaton angle (camber) [, 1]. The work s supported by a CERG grant 944 and a Cty Unversty of Hong Kong grant 769 n Hong Kong. ndoor WMRs usually operate wthn a lower speed range compared to automobles. Mechancal couplng of the wheels n automobles often allows the use of a smplfed, sngle-track model of two-wheel-steerng and four-wheel-steerng cars for the desgn of moton controllers [1, 4]. In modellng and control of moble robots wth ndependently steered and/or drven wheels t s requred to explctly account for each of the wheels of the vehcle. Such complex models of four-wheel vehcles has been developed n works [3, 7, 8]. However the controllers n these works are stll desgned under the assumpton that the wheels are steered n pars. The models also utlze smplfed wheel actuator dynamcs that may not be approprate for modellng of some types of WMRs, e. g. heavy vehcles wth DC motor actuated wheels, for whch the effects of backlash and wheel mechansm dynamcs may be sgnfcant. Such WMRs are often utlzed as autonomous guded vehcles (AGVs) n tght manufacturng envronments havng hgh requrements on maneuverablty. These requrements mply the necessty of actve utlzaton of the capablty of such WMRs for ndependent wheel steerng. Knematcs-based control of WMRs wth multple steerng wheels has receved consderable attenton over the last decade. Effcent trajectory-trackng controllers are developed usng dynamc feedback [9]. To the authors best knowledge, controllers that utlze complex dynamc models of the WMRs wth multple steerng wheels are not used n practce due to ther computatonal nfeasblty and dffcultes related to analytcal nvestgaton of ther propertes.

2 One of the possble ways to deal wth the tradeoff between computatonal effcency of control and a granulaton of a correspondng model s to augment a knematcs-based controller wth a controller that accounts for a partcular dynamc effect. In ths work we descrbe such a controller that ams specfcally at mnmzaton of slppage due to wheel msalgnment. The controller extends the approach presented n [6] by () cross-couplng all four wheels of the vehcle, nstead of couplng them parwse, () ncorporatng not only steerng but also drvng actuator dynamcs, () generalzng the control objectves so that the wheels are able to track a lnear approxmaton of the vehcle knematc constrants durng transent states of control. Ths couplng scheme allows the four wheels to be steered at arbtrary angles, not necessarly the same angles as n [6]. The performance of the cross-couplng controller s evaluated wth the dynamc model developed for the AGV prototype of the project. The dynamc model of the vehcle dffers from those descrbed n [3, 7, 8] n that, n addton to nonlnear forces at the contact between the tre and the ground, t ncorporates the dynamcs of the components of the wheel mechansms and a more realstc model of the wheel actuators ncludng backlash effects of DC motors. The paper s organzed as follows. Secton descrbes the knematcs of the WMR. The dynamc model s presented n secton 3. The cross-couplng controller and smulaton results are addressed n secton 4. Concludng remarks are gven n secton. Knematc model A dagram of the vehcle s presented n fgure 1. Assumng that the vehcle frame s a rgd body, the veloctes v of the wheel base ponts W are related to the velocty v c of the reference pont C of the frame as v = v c + θk CW, = 1,..., 4. (1) Pure rollng condton mples that the wheel s rotaton rate φ and steerng angle ψ wth respect to the frame coordnate system are such that φ = v r, tan ψ = v y v x, = 1,..., 4. () The knematc equaton relatng acceleraton a of the wheel base pont W and acceleraton a c of the vehcle reference pont C s a = a c + θk CW θ CW. (3) Usng the fact that a c = ( v x θv y ) + ( v y + θv x )j we have a = a x + a y j, where a x = v x θv y CW x θ CW y θ, (4) a y = v y + θv x CW y θ + CW x θ. () Y O W 3 X y A y4 W 4 T ψ4 y 1 C W 1 A x4 v 1 W Slp angle 1 Fgure 1: Notaton of knematcs of the WMR and dynamcs of the frame. 3 Dynamc model The dynamc model of the vehcle s derved by consderng dynamcs of the vehcle frame, wheel struts, wheels, and wheel motors. 3.1 Frame dynamcs The frame lateral, longtudnal and yaw dynamc equatons are: ma x = m( v x θv y ) = ma y = m( v y + θv x ) = I θ = ψ 1 θ x 1 x v c A x, (6) A y, (7) P (8) respectvely, where A x and A y are the x-component and the y-component of reacton at the steer jont W, P = T ψ +CW x A y CW y A x s the vertcal component of the total torque appled to the frame va the steer jont W. We neglect horzontal components P x, P y of the torque appled to the vehcle frame va the steer jont, assumng that all wheels of the vehcle are always n contact wth the ground and that the change of normal reacton at the wheel-ground contact pont s neglgble. 3. Wheel strut dynamcs We assume that for each wheel gravty centers of the strut and the wheel, the pont of the wheel contact wth the ground, geometrc center of the wheel, and steer jont W are all on a sngle vertcal lne. As wth the frame, we neglect horzontal components of the total moment of the wheel strut.

3 3.3 Wheel dynamcs m s a x = S x A x, (9) m s a y = S y A y, (1) w s = S z A z, (11) I s ( θ + ψ ) = T ψ D z. (1) The forces appled to the wheel are related as follows: m w a x = F x S x, (13) m w a y = F y S y, (14) w w = N S z. (1) The total moment produced by couples and forces appled to wheel s M = T φ + D z E + D x rk (F x + F y j ) = (D x + rf y ) + (T φ rf x )j + (D z E )k. (16) We assume the wheel to be cylndrcal so that the axes x, y, z are ts prncpal axes of nerta. The angular moments of the wheel are: M x = I x x ω x + (I z z I y y )ω y ω z, (17) M y = I y y ω y + (I x x I z z )ω x ω z, (18) M z = I z z ω z + (I y y I x x )ω x ω y, (19) where I x x = I z z = I t s the transverse moment of nerta and I y y = I a s the axal moment of nerta of the wheel. Assumng that the x -component of the angular velocty of the wheel s zero (no camber), the angular velocty of the wheel s ω = ( θ + ψ )k + φ j. () Substtuton of () nto (17) (19) and usng (16) and (1) gves us the expressons for the steerng and drvng torques respectvely: T ψ = (I s + I t )( θ + ψ ) + E, (1) T φ = I a φ + rf x. () The lateral and longtudnal tre frcton forces F y, F x, and the algnng torque due to frcton E are modelled as commonly done n the automotve ndustry [], [1]. 3.4 Dynamcs of wheel motors Assumng that the armature-wndng nductance of the motors s neglgble, the equatons governng the dynamcs of the () steerng and () drvng motors of the wheel are as follows: () steerng motors k 1 e ψ = k 3 ψ + R ψ, (3) k ψ = (I s + I t )( θ + ψ ) + b ψ + E, (4) where (I s +I t )( θ+ ψ)+e = T ψ s the steerng torque appled to the wheel, and b ψ s the energy loss due to the frcton n the motor and n the gear tran; () drvng motors k 1e φ = k 3 + R φ, () k φ = I a φ + b φ + rf x, (6) where I a φ +rf x = T φ s the drvng torque appled to the wheel, and b φ s the energy loss due to the frcton n the motor and n the gear tran. 3. Vehcle Dynamcs Combnng (6) (8), (9) (1), (13), (14), (4), () wth (3) (6), (1), (), the state-space model of the vehcle can be presented as: v x v ÿ θ ψ 1 4 φ 1 4 = M 1 Q(v x, v y, θ, ψ 1 4, φ 1 4, ψ 1 4, φ 1 4, e ψ1 4, e φ1 4 ), (7) where matrces M and Q are specfed n the Appendx. 4 Control 4.1 Cross-couplng controller We consder the problem of control of the moble robot equpped wth multple steerng and drvng wheels. The effcent knematcs-based trajectorytrackng control va lnearzng dynamc feedback s proposed n [9]. To preserve the computatonal feasblty of knematcs-based control and to account for tre slp due to msalgnment of the wheels, we propose to use a cross-couplng controller n conjuncton wth a knematcs-based trajectory-trackng controller. The proposed cross-couplng controller s based on the dea of a pecewse lnear approxmaton of the pure rollng condton () wth the node ponts n the space of steerng angles and drvng rates generated by a hgher level trajectory trackng controller [, 6]. For example, the lnear approxmaton of the curve n the space of steerng angles s specfed by the relatonshp ψ ψ d δ = ψ j ψ jd δ j, δ, δ j, (8)

4 Lateral Slp Angle Tme (s) Lateral Slp Angle Tme (s) 1 - Lateral Slp Angle Tme (s) Lateral Slp Angle Tme (s) Fgure : Lateral slp angles correspondng to the four wheels wth the dfferent control schemes: dash-dotted lne uncoupled control; dashed lne partally coupled control (uncoupled controller for the steerng motors, wth the drvng motors coupled wth the respectve steerng motors); sold lne cross-couplng control of the steerng and drvng motors. Concluson In ths paper we develop a dynamc model of a moble robot wth four ndependently steered and drven wheels. Detaled modellng of wheel plants allows utlzaton of the model for smulaton nvolvng such dynamc effects as slppage due to wheel msalgnment and drvng rate errors. To mnmze slppage due to msalgnment of the wheels, a cross-couplng controller s proposed to follow a pecewse lnear approxmaton of the trajectory correspondng to the pure rollng condton n the space of steerng and drvng actuators. The smulaton performed wth the dynamc model shows sgnfcant reducton of lateral slp wth moderate trajectorytrackng errors. Acknowledgments The authors would lke to thank N. Lee for hs assstance wth the AGV prototype and K. F. Man for hs advce on smulaton. Appendx A Nomenclature where ψ s and ψ d are the ntal and desred steerng angles respectvely, δ = ψ s ψ d are the ntal errors. We refer to (8) as the condton of proportonal errors between the steerng actuators and j. A smlar relatonshp can couple the drvng and steerng actuators. The LQR-based controller that couples the steerng and drvng actuators accordng to the condton of proportonal errors s descrbed n detal n []. 4. Smulaton results The performance of the cross-couplng controller s evaluated by smulaton wth the dynamc model mplemented n Matlab. The reference trajectory conssts of a straght-lne nterval and a crcular arc of the radus R = m wth the center closer to wheels 1 and 4. The vehcle undergoes the turnng maneuver so that durng the lne-nterval followng all reference steerng angles are equal to zero, and durng the crcular-arc followng the reference steerng angles are related as ψ 1d = ψ 3d = 38.8, ψ d = ψ 4d =.1. Lateral slp angles obtaned for dfferent couplng schemes are presented n fgure. The smulated and the deal trajectory n the space of steerng angles are shown n fgure 3. The trajectory trackng errors under the crosscouplng control are larger than under the uncoupled control, but consdered moderate (more detals n []). reacton force and torque at steer jont A, T ψ mass and moment of nerta of the frame m, I reacton force and torque at wheel-strut jont S, D mass and weght of the strut m s, w s moment of nerta of the strut I s lateral tre frcton force F y longtudnal tre frcton (tractve) force F x normal force at the tre-ground contact N algnng torque due to frcton E steerng torque T ψ drvng torque T φ mass and weght of the wheel m w, w w radus of the wheel r nput voltage of the motors e ψ, e φ amplfer constants k 1, k 1 motor torque constants k, k back emf constants of the motors k 3, k 3 armature current of the motors ψ, φ resstance of the motors R, R vscous-frcton coeffcents b, b B Generalzed mass and force matrces The elements of matrx of generalzed masses M are as follows: M 1,1 = M, = m + (m s + m w), M 1,3 = M 3,1 = (m s + m w)cw y, M,3 = M 3, = (m s + m w)cw x,

5 3 3.. Tme (s) 1. 1 Tme (s) Steerng angle (deg) Steerng angle 1 (deg) 1 1 Steerng angle (deg) Steerng angle 1 (deg) (a) Uncoupled control. (b) Coupled control. Fgure 3: Steerng angles of wheels 1 and under dfferent control schemes. For each subfgure the sold lne corresponds to the actual steerng angles of the wheels, the x-marked lne s the projecton of the 3D trajectory represented by the sold lne on the D space of steerng angles (wthout tme), and the o-marked lne s the trajectory n the space of steerng angles that corresponds to the no-slp steerng. M 3,3 = I + (m s + m w)cw, M 4,3 = M 4,4 = I s + I t, M, = I a, and the rest of elements are zeros. The elements of the vector of generalzed forces Q 1 are as follows: Q 1,1 = mv y θ + ( + Fx + (m s + m ( w) v y θ )) + CWx θ, Q,1 = mv x θ + ( + Fy (m s + m ( w) v x θ )) + CWy θ, Q 3,1 = (T ψ + (CW x F y CW y F x ) θ (m s + m w) (CW x v x + CW y v y) ), Q 4,1 = T ψ E, Q,1 = T φ rf x, where steerng and drvng torques are derved from (3) (6) wth the armature voltage e ψ, e φ of the steerng and drvng motors gven. References [1] J. Ackermann, Robust Decouplng, Ideal Steerng Dynamcs and Yaw Stablzaton of 4WS Cars, Automatca, 1994, vol. 3, no. 11, pp [] T. D. Gllespe, Fundamentals of vehcle dynamcs. Warrendale: Socety of Automotve Engneers, 199. [3] W. Langson, A. Alleyne, Multvarable Blnear Vehcle Control Usng Steerng and Indvdual Wheel Torques, Journal of Dynamc Systems, Measurement and Control, December 1999, vol. 11, pp [4] A. Y. Lee, A Prevew Steerng Autoplot Control Algorthm for Four-Wheel-Steerng Passenger Vehcles, Journal of Dynamc Systems, Measurement, and Control, vol. 114, September 199, pp [] M. Makatchev, S. Y. T. Lang, S. K. Tso, J. J. McPhee, Cross-Couplng Control for Slppage Mnmzaton of a Four-Wheel-Steerng Moble Robot, Proc. of the 31st Int. Symposum on Robotcs (ISR ), Montreal, Canada, May 14 17,, pp [6] N. Matsumoto, H. Kuraoka, M. Ohba, An expermental study on vehcle lateral and yaw moton control, Proc. of Int. Conf. on Industral Electroncs, Control and Instrumentaton (IECON), 1991, vol. 1, pp [7] N. Matsumoto, M. Tomzuka, Vehcle Lateral Velocty and Yaw Rate Control Wth Two Independent Control Inputs, Trans. of the ASME, Vol. 114, December 199, pp [8] S.-H. Yu, J. J. Moskwa, A Global Approach to Vehcle Control: Coordnaton of Four Wheel Steerng and Wheel Torques, Journal of Dynamc Systems, Measurement and Control, vol. 116, December 1994, pp [9] B. Thulot, B. d Andréa-Novel, A. Mcael, Modelng and Feedback Control of Moble Robots Equpped wth Several Steerng Wheels, IEEE Trans. on Robotcs and Automaton, vol. 1, no. 3, June 1996, pp [1] J. Y. Wong, Theory of Ground Vehcles. New York: Wley, 1993.