STABILITY CONTROL FOR SIX-WHEEL DRIVE ARTICULATED VEHICLE BASED ON DIRECT YAW MOMENT CONTROL METHOD

Transcription

1 STABILITY CONTROL FOR SIX-WHEEL DRIVE ARTICULATED VEHICLE BASED ON DIRECT YAW MOMENT CONTROL METHOD 1 YITING KANG, 2 WENMING ZHANG 1 Doctoral Caniate, School of Mechanical Engineering, University of Science an Technology Beijing, CHINA 2 Professor, School of Mechanical Engineering, University of Science an Technology Beijing, CHINA 1 kangyiting@sina.com, 2 wmzhang@ustb.eu.cn ABSTRACT Six-wheel rive (6WD) articulate ump truck has excellent passing ability an high prouctivity, but its stability shoul be improve. A ynamic moel of 6WD articulate vehicle with 3 egree-of-freeom (DOF) is establishe. With the control variables of yaw velocity an sie-slip angle at mass center of front boy, a feefowar-feeback control system is esigne by linear quaratic regulator (LQR) theory base on irect yaw-moment control (DYC) metho. A simulation with a step input of articulate angle is conucte, an the result shows that the value of sie-slip angle at mass center, yaw velocity an lateral acceleration all rop after control, an the control system is effective to improve the vehicle stability. The weight coefficients are regulate to analyze their influence on control effect, an the result inicates that it is necessary to regulate them to reach the optimum of both energy an error of control. Keywors: Articulate Vehicle, Vehicle Stability Control, Direct Yaw Moment Control (DYC), Linear Quaratic Regulator (LQR) 1. INTRODUCTION Articulate ump truck is a kin of off-highway vehicle with an articulate boy, which allows front boy rotate inepenently relative to rear boy. Articulate ump truck is a 6x6 rive vehicle with 3 rive axles [1, 2]. Its steering angle an pitch angle reach 45 an 15 respectively, which can ensure that all tires contact groun simultaneously, so it has goo passing ability an high prouctivity. In spite of the avantages mentione above, the lateral an longituinal stability of articulate ump truck get worse because of the increase of vehicle DOF, so its stability shoul be improve. Increasing the friction of articulate boy an regulating the amping of steering hyraulic cyliner are the conventional methos to improve the vehicle stability, but these methos waste too much energy an have low reliability. With the evelopment of control technology, DYC is applie in controlling vehicle stability. DYC is to change longituinal forces on left an right wheels to generate an aitional yaw moment to improve vehicle stability [3]. The irect yaw moment control using in wheelmotors an the active front steering (AFS) control algorithm are escribe in Ref.4, an the combination of DYC control an AFS control is researche in [5, 6]. Feeforwar-feeback control is use to enhance the stability of a 4WD vehicle, an the feeback coefficient is ecie by optimal control theory in [7]. A fuzzy feeforwarfeeback control system is esigne an teste with ABS to enhance the lateral stability of vehicle in [8], an a feeback control system base on H theory is stuie in [9]. Massto Abe an Ossama Mokhiamar [10-13] have researche an evelope a DYC control system base on slip moe control metho, an the system has been teste on real vehicle an prove to be effective. Electric rive is a new rive form of articulate ump truck, which utilizes 6 wheel-motors to rive wheels respectively. Base on the precise an fast response characters of wheel-motor, rive torques on 6 wheels can be controlle inepenently to generate an aitional yaw moment [14,15], which can improve the vehicle stability. 258

2 In orer to enhance the stability of 6WD articulate vehicle, a 3-DOF ynamic moel of 6WD articulate vehicle is firstly establishe. Moreover, a feeforwar-feeback control system base on LQR theory is esigne. In orer to verify the effectiveness of the control system, a simulation with a step input of articulate angle is conucte, an the weight coefficients are regulate to analyze their influence on control effect. 2. DYNAMIC MODEL In orer to stuy the ynamic character of articulate vehicle, the vehicle moel is simplifie to a system with 3 DOF, which inclues rotations of front boy an rear boy along vertical axis, an the lateral translation of vehicle (Fig.1). Figure 1. 3 DOF moel of a 6WD articulate vehicle The mass an rotational inertia of front an rear boy are m 1, J 1 an m 2, J 2. The sie-slip angles of front, mile an rear tires are respectively α 1, α 2, an α 3. Lateral velocity of front boy is v, an articulate angle between front an rear boy uner steering is θ. Firstly, two components of the absolute acceleration at mass center of front boy are ecie. As shown in Fig.2, XY is the absolute coorinate system, an xy is the vehicle coorinate system fixe on the front boy. The x-component an y-component of the velocity at mass center of front boy are respectively u an v at t. When the vehicle is steering, its motion inclues translation an rotation, so both value an irection of the velocity at mass center of front boy have change at t+ t. The irections of horizontal axis an vertical axis in the vehicle coorinate system also have change at the same time. Variation of velocity along axis y is Vy1 = ( v + v)cos φ v + ( u + u)sin φ (1) φ is very small, an the small quantity in the secon orer is ignore. Eq.1 is transforme to Vy1 = v + u φ (2) Figure 2. Motion analysis base on the coorinate system fixe on the vehicle Yaw velocity of front boy is ω. Eq.2 is ivie by t an then taken to the limit. The y-component of absolute acceleration at mass center of front boy in the vehicle coorinate system is a = v& + uω = u & β + uω (3) y1 Similarly, the variation of y-component of absolute acceleration at mass center of rear boy in the vehicle coorinate system is a y 2 = v& + uω ( b + c) & ω + c && θ (4) Equilibrium equations of force an moment of the vehicle are m1a y1 + m2ay 2 = Fy 1 + Fy 2 + Fy3 J & 1ω m2ay 2b = afy 1 bfy 2 bfy 3 J (&& θ & ω) + m a c = ( c + ) F + ( c + + e) F ) 2 2 y2 y2 y3 There is a linear relationship between sie-slip angle an lateral force when sie-slip angle is relatively small. The articulate ump truck always travels in a low spee, an the sie-slip angle of tire is generally no more than 3. So the lateral force an sie-slip angle are consiere to satisfy the linear relationship: Fy (5 = kα (6) where k is cornering stiffness, an α is sie-slip angle of tire. The sie-slip angles of front, mile, an rear axles can be expresse by v, ω, an θ as 259

3 aω α1 = β u ( b + c + ) ω ( c + ) & θ α 2 = θ β + u ( b + c + + e) ω ( c + + e) & θ α3 = θ β + u (7) As shown in Fig.3, two aitional riving forces, +F X1 an -F X1, are respectively loae on left an right tires in the front, which means a yaw moment M is loae on the front boy. Figure 3. Force Equilibrium Of Vehicle Uner Steay- State Steering In this case, equilibrium equation of force an moment of the vehicle is m1a y1 + m2a y2 = Fy1 + Fy 2 + Fy 3 J & 1ω m2ay 2b = afy1 bfy 2 bfy 3 + M J (&& θ & ω) + m a c = ( c + ) F + ( c + + e) F 2 2 y 2 y2 y3 (8) Eq.6 an Eq.7 are taken into Eq.8, an then the following equation can be obtaine. & β a1 a2 β a3 a4 & θ a5 = M ω b1 b ω b3 b 4 θ b (9 & 5 ) where a 1, a 2, a 3, a 4, a 5, b 1, b 2, b 3, b 4, an b 5 can be expresse by m 1, J 1, m 2, J 2, a, b, c,, e, c 1, c 2, an c 3. Eq.9 is transforme into Eq.10 an Eq.11 by Laplace transform. β a b s + a b ( b s)( a s + a ) θ ( a s)( b s) a b ( s) = ( s) ω a b a b + a s M ( s) ( a s)( b s) a b a b s + a b ( a s)( b s + b ) θ ( a s)( b s) a b ( s) = ( s) a b a b + b s M ( s) ( a s)( b s) a b (10) (11) 3. DYNAMIC MODEL Yaw velocity an sie-slip angle of vehicle at mass center are two important parameters to escribe the vehicle motion status. The yaw velocity an sie-slip angle at mass center of front boy are selecte to escribe the stability of 6WD articulate vehicle, because that of rear boy can be expresse by the front ones. The control system is esigne by LQR theory base on DYC metho. Feeforwar-feeback control is applie in control structure, an the combination control for yaw velocity an sie-slip angle of front boy are realize. The irect yaw moment as the controller input shoul make sieslip angle an yaw velocity keep up with their ieal value Fee-Forwar Controller The control objective is to make the sie-slip angle at mass center of front boy approach to zero. The relationship between the controller input an the articulate angle is M ( s) = G θ ( s) (12) ff where G ff is feeback gain, an M ff is yaw moment of feefowar control. When the vehicle is uner steay-state steering (s=0), Eq.12 can be transforme to M ff = G θ (13) ff 0 ff 0 The following equation is obtaine by taking Eq.13 into Eq.10. β a b b a + ( a b a b ) Gff = θ0 a1b 2 a2b1 Because β 0 =0, the feeforwar gain is (14) b2a4 a2b4 Gff = (15) a2b5 a5b2 By taking Eq.13 an Eq.15 into Eq.11, transfer function from articulate angle to yaw velocity can be obtaine as ω( s) [ b1 a3s + a4b1 ( a1 s)( b3 s + b4 )] = θ ( s) [( a s)( b s) a b ] ( b a a b )( b a b a + b s) [( a1 s )( b2 s ) a2b1 ]( a2b5 a5b2 ) (16) 3.2. Reference Moel The sie-slip angle at mass center of front boy shoul approach to zero (control objective). Yaw velocity is a secon-orer elaye-time system about articulate angle θ an approximate as a 260

4 first-orer elaye-time system for simplicity, which is shown in Eq.17. ( s) k = (17) θ ( s) 1 + τ s where k (u) is steay-state value, an τ (u) is time constant. The response value s=0 is taken into Eq.17, k can be expresse as k a b a b ( b a a b )( b a b a ) = + a1b2 a2b1 ( a1b2 a2b1 )( a2b5 a5b2 ) Similarly, τ (u) can be obtaine: k τ = b a + b + b a b + ( b a a b ) b The state equation of reference moel is f (18) (19) X& = A X + H θ (20) where X = β, 0 0 A =, an 0 1/ τ ω 0 H = k ( u ). τ ω 3.3. Feeback Controller Feeforwar compensation is not enough when the system is suffering external isturbance or has uncertain factors. Feeback compensation is very effective to track the ieal system characteristic, so feeforwar-feeback combination control is aopte. The state equation of real moel is X& = AX + BM fb (21) E is the error between reference value X an real value X, so E& = X& X& (22) By taking Eq.20 an Eq. 21 into Eq.22, the following equation can be obtaine: E& = AE + BM fb + ( A A ) X + ( H H ) θ f = AE + BM + W fb (23) The last term W in Eq.23 is a isturbance an can be ignore (W=0), so E& = AE + BM fb (24) Optimal control theory is applie into the esign of feeback controller, so M = G E = g ( β β ) g ( ω ω ) (25) fb fb fb1 fb2 where g fb1 an g fb2 are feeback gains of state errors between real moel an ieal moel. The performance inex function is ( T T J = E QE + M RM ) 0 fb fb t (26) where Q an R are respectively state weight matrix an control weight matrix an can be regulate. Feeback gain of controller can be obtaine by solving the Riccati Equation by means of LQR algorithm. Total aitional yaw moment is M Z = M ff + M fb (27) 3.4. Control System The control system of 6WD articulate vehicle which inclues feeforwar controller, feeback controller, reference moel an ynamic moel is establishe in SIMULINK software (Fig.5). The yaw velocity an sie-slip angle at mass center of front boy are uner combination control. Figure 4. Control System 4. DYNAMIC MODEL 4.1. Step Simulation Result A simulation with step input of θ is conucte on the 3-DOF ynamic moel to analyze the effectiveness of the control system. The stability an following characteristic are stuie by the step simulation where vehicle velocity is 15m/s, an initial articulate angle is 2. The variation of sieslip angle at mass center β, yaw velocity ω, lateral acceleration a y an aitional yaw moment M Z are the key point of simulation analysis. Fig.5 is the response curves of β, ω, a y, an M Z in step simulation respectively. 261

5 (A) Response Of Β (B) Response Of Ω 0.08ra after control (Fig.5 (a)). The ieal value of ω finally stabilizes at ra/s. ω increases from 0 to 0.05ra/s an stabilizes at 0.04ra/s before control, while the stable value of ω reuces from 0.04ra/s to 0.025ra/s after control (Fig.5 (b)). Similarly, the stable value of a y also has ecrease from 0.06g to 0.04g after control (Fig.5 (c)). M Z sharply rises from 0 to 700Nm after control (Fig.5 ()). It can be seen that β an ω get closer to their ieal values with the effect of M Z, an the vehicle stability is improve obviously Influence Of Weight Coefficient The iagonal elements of state weighte matrix Q (Eq.28) in feeback controller represent the levels of error components. Q q 0 11 = 0 q22 (28) R inicates the consuming energy of control in performance function. The objective of optimal control is to make the performance function close to its minimum, so regulating q 11, q 22 an R can lower the control energy as well as keep a relative small error, which will finally reach the optimum of both energy an error Influence of q 11 The response curves of β, ω, a y, an M Z when q 11 varies are shown in Fig.6. β gets closer to its ieal value (β =0) an a y rops with the increase of q 11. Although ω is smaller an smaller, the error between ω an ω gets bigger with the increase of q 11. The overlarge q 11 causes the fluctuations of M Z an ω. (C) Response Of A y (A) Response Of Β (D) Response Of M Z Figure 5. Simulation Results The ieal value of β is 0. β increases from 0 to 0.125ra an stabilizes at 0.12ra before control, while β raises from 0 an then tens to be stable at 262

6 (B) Response Of Ω (A) Response Of Β (C) Response Of A y (B) Response Of Ω (D) Response Of M Z Figure 6. Simulation Results With Variation Of Q 11 (C) Response Of A y Influence of q 22 The response curves of β, ω, a y, an M Z when q 22 varies are shown in Fig.7. When q 22 rises, ω gets closer to its ieal value (ω =0.0225ra/s), but the values of β an a y increase. When q 22 varies from 1000 to 10000, the control effect has little change. Compare with Fig.6, the influence of q 22 on the vehicle stability is less obvious than that of q 11. (D) Response Of M Z Figure 7. Simulation Results With Variation Of Q Influence of R The response curves of β, ω, a y, an M Z when R varies are shown in Fig.8. β is closer to its ieal value with the ecrease of R. But ω approaches to 263

7 its ieal value with the rise of R. When R varies from 10-2 to 10-5, a y ecreases from 0.035g to 0.01g, while the value of M Z rises from 900N m to 1900N m. A small R also leas to the fluctuation of M Z. The rise of R increases response time an lowers the vehicle stability. Regulating the weight coefficients have a great impact on the control effect base on the iscussion above, so it is necessary to regulate them to reach optimum effect in the simulation. (D) Response Of M Z Figure 8. Simulation Results With Variation Of R 5. CONCLUSION (A) Response Of Β (B) Response Of Ω r To enhance the stability of 6WD articulate ump truck, a feeforwar-feeback control system is esigne by DYC base on the 3-DOF ynamic moel. The feeforwar controller is esigne to keep the sie-slip angle at mass center of front boy close to its ieal value, an the feeback compensating controller is built by LQR to make the sie-slip angle at mass center an yaw velocity of front boy (β an ω) follow their ieal values (β an ω ) effectively. The simulation result shows that the control system improves stability an raise safety of the vehicle, an the control algorithmic is prove to be effective. The analysis about the influence of weight coefficients on control effect inicates that β an ω respectively get closer to β an ω with the increase of q 11 an q 22. But the influence of q 22 is smaller than that of q 11 on stability. The ecrease of R makes β close to β, but makes ω far away from ω. An R with a relatively small value will lea to the fluctuation of M Z. ACKNOWLEDGMENTS The work is supporte by the "Eleventh Five- Year Plan" Scientific an Technological Support of National Science an Technology Ministry of China through grant NO.2008BAB32B03. REFRENCES: (C) Response Of A y [1] Liu Jinxia, Zhang Wenming, an Zhang Guofen, Hanling Stability of an Articulate Dump Truck, Journal of University of Science an Technology Beijing, Vol.29, No.11, 2007, pp [2] Zhang Jing, Shi Boqiang, an Gu Jie, Topology Optimization on the Articulate 264

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