ACTIVE FOUR WHEEL STEERING SYSTEM FOR ZERO SIDESLIP ANGLE AND LATERAL ACCELERATION CONTROL

ACTIVE FOUR-WHEEL STEERING SYSTEM FOR ZERO SIDESLIP ANGLE AND LATERAL ACCELERATION CONTROL Toshihiro Hiraoka ∗ Osamu Nishihara ∗ Hiromitsu Kumamoto ∗ ∗ Graduate School of Informatics, Kyoto University, Kyoto, JAPAN Abstract: This paper proposes an active four-wheel steering system It has three additional points to an active front steering system proposed by the authors: 1) active rear steering to realize zero sideslip angle, 2) variable steering ratio to prevent abrupt involution during slowdown, and 3) model following sliding mode controller that is robust against the system uncertainty Computer simulations demonstrate good maneuverability of the proposed system Keywords: Active four wheel steering, Model following sliding mode control, Variable steering ratio INTRODUCTION Recently, active steering systems (Ackermann, 1997; Shimada et al., 1997) have been studied and developed for improvement of active safety Authors (Hiraoka et al., 2001; Hiraoka et al., 2002) also proposed an active front steering (AFS) law based on the dynamics of lateral acceleration at a center of percussion with respect to rear wheels By using the control law, lateral acceleration at the center of percussion can be proportional to steering wheel angle without influence of vehicle sideslip angle and yaw rate The AFS improves maneuverability of the vehicle in the difficult driving situation such as packed snow road However, the AFS law has problems: 1) deterioration of response from driver’s steering input to sideslip angle and yaw rate, 2) abrupt involution at slowdown, and 3) no consideration of system uncertainty such as cornering power perturbation First, active four-wheel steering law is proposed in this paper by addition of active rear steering (ARS) to AFS in order to realize zero sideslip angle, so that the dynamic characteristics of yaw rate and lateral acceleration at a center of gravity become the first order lag system Second, variable steering ratio is introduced to the relationship between driver’s steering input and front wheel angle And last, model following controller is introduced for the active four-wheel steering system to become robust against system uncertainty ACTIVE FRONT STEERING 2.1 2DOF vehicle model This paper assumes that a vehicle with constant velocity moves with only two degrees of freedom, right and left transition, and yaw rotation Figure shows 2DOF vehicle model used in this paper A state vector is defined as x = [β γ] , and an input vector as u = [uf ur ] , where β is the sideslip angle, γ is the yaw rate, and uf /ur are the front/rear steering angle A linearized equation of motion of four-wheel steering vehicle becomes where um /ua are the mannual/automatic steering component, δ is the driver’s steering wheel angle input, and ks is the steering ratio y0 y lr Target course G Lp lf ur 2.3.2 Transfer function Substitution of eq.(4) into eq.(1) yields a transfer function Gap (s) from δ to lateral acceleration at P θ β Vehicle P v εp r uf Gap (s) = x0 x Fig 2DOF vehicle model x˙ = Ax + Bu + Cw (1) ⎡ ⎤ 2(Kf lf − Kr lr ) 2(Kf + Kr ) −1 − − ⎢ ⎥ mv mv ⎢ ⎥ A=⎢ ⎥ ⎣ 2(Kf lf − Kr lr ) 2(Kf lf2 + Kr lr2 ) ⎦ − − Iz Iz v ⎡ ⎤ ⎡ 2K 2Kr ⎤ f ⎢ mv ⎥ ⎢ mv mv ⎥ ⎥,C = ⎢ ⎥ (2) B=⎢ ⎦ ⎣ l ⎦ ⎣ 2K l 2Kr lr w f f − − Iz Iz Iz where m is the vehicle mass, Iz is the moment of inertia around the axis of the center of gravity G, v is the vehicle velocity, Kf /Kr are the cornering powers of front/rear tires, lf /lr are the distances between G to front/rear wheel axles, w is the lateral disturbance, and lw is the distance between G to the disturbance load center 2Kf l · ks lr m Equation (5) shows that Gap (s) has a flat frequency response Therefore, a driver can control lateral acceleration at P without taking influence of vehicle sideslip angle and yaw rate Driving simulator experiments demonstrated the improvement of path following capability on the packed snow road (Hiraoka et al., 2001) Table shows transfer functions Gβ (s), Gγ (s) from δ to β and γ, in the both case of AFS vehicle and conventional 2WS vehicle Figure shows the step responses of β and γ of the two vehicles Velocity is 90[km/h], step input of steering wheel angle is 1[rad], and other parameters are shown in Table (see Section 4) These figures illustrate that the responses from δ to β and γ of AFS are more vibratory than that of conventional 2WS 2.3.3 Steering wheel angle for constant radius turn AFS vehicle requires the steering wheel angle δ0,AF S to perform a constant radius turn (radius: r[m]): δ0,AF S = ks 2.2 A center of percussion with respect to rear wheel In this paper, a center of percussion with respect to rear wheels is considered as a datum point P for control: no acceleration is generated at P even if an impact force acts on the rear wheels (5) lr m v 2Kf l r (6) while conventional 2WS vehicle needs the angle δ0,2W S : δ0,2W S = ks0 l r 1− m K f lf − K r lr v 2l Kf Kr (7) lf 2Kf l lr − lw (β + − uf ) + w ap = − lr m v lr m (3) 2.3 Active front steering for lateral acceleration control at a center of percussion 2.3.1 Active front steering law An active front steering law based on eq.(3) is proposed by authors (Hiraoka et al., 2001; Hiraoka et al., 2002): uf ≡ um + ua = lf δ + (β + γ) ks v (4) Amplitude To: Sideslip angle [rad] To: Yaw rate [rad/s] From: Steering wheel angle [rad] Distance Lp from G to P is calculated as Lp = Iz /(mlr ) Lateral acceleration ap at P is 0.01 Conventional 2WS AFS -0.01 -0.02 -0.03 -0.04 0.5 0.4 0.3 0.2 Velocity: 90 [km/h] Steering wheel input: 1[rad] Steering ratio: 16 (constant) 0.1 0 0.5 Time [s] 1.5 Fig Step responses of sideslip angle (upper) and yaw rate (lower) Table Transfer functions (2WS vs AFS) Gβ (s) = Gβ (0) Gβ (0) Gγ (0) a0 Tβ Tγ , Gγ (s) = Gγ (0) AFS Kf (2Kr lr l − lf mv2 ) · ks 2Kf Kr l2 − (Kf lf − Kr lr )mv2 2Kf Kr lv · ks 2Kf Kr l2 − (Kf lf − Kr lr )mv2 Kf (2Kr lr l − lf mv2 ) · ks Kr lr mv2 2Kf l · ks lr mv (Kf + Kr )Iz v + (Kf lf2 + Kr lr2 )mv − (Kf lf − Kr lr Iz mv 4Kf Kr l2 − 2(Kf lf − Kr lr )mv Iz v 2Kr lr l − lf mv2 lf mv Iz + mlr2 lr mv Iz 2Kr lr Iz v 2Kr lr l − lf mv2 lf mv 2Kr l 2Kr l l2 )mv2 ⎡ where ks0 is the constant steering gear ratio Equation (6) shows that δ0,AF S is in proportion to the square of velocity v when the steering ratio ks is constant It represents that the AFS makes vehicle stable even if the original steer characteristic is oversteer (Kf lf − Kr lr > 0) However, an abrupt involution will happen when the vehicle slows down with the constant steering wheel angle ACTIVE FOUR-WHEEL STEERING 3.1 Addition of active rear steering for zero sideslip angle 3.1.1 Active four-wheel steering law An active four-wheel steering (A-4WS) law is defined as the summation of AFS shown in eq.(4) and ARS Here, the ARS consists of linear combination of steering input δ and vehicle state x u = Dδ + Ex D= ks kh ⎡ ⎤ lf ⎣ v ⎦ ,E = kb kg Kf mv − 2Kr lr , kg = Kr 2Kr v ⎤ ⎡ lf v Substitution of eq.(11) into eq.(1) yields x˙ = A x + B δ + Cw ⎡ ⎤ 2(Kf + Kr ) − ⎢ ⎥ mv ⎥ A =⎢ ⎣ 2(K l − K l ) l mv ⎦ , f f r r r − − Iz Iz ⎤ ⎡ ⎥ ⎢ B = ⎣ 2K l ⎦ f ks Iz Kγ , + T0 s vKγ , Gap (s) = vKγ + T0 s 2Kf l Iz , T0 = · Kγ = ks lr mv lr mv (11) (13) (14) (15) 3.1.2 Transfer function Transfer functions from driver’s steering input to sideslip angle, yaw rate, lateral acceleration at G and P become as follows: (9) (10) ⎤ ⎢ ⎥ ⎢ ⎥ ⎥ ⎣ ⎦,E = ⎢ ⎣ mv − 2K l ⎦ (12) ks − Kf r r Kr 2Kr v Gβ (s) = 0, However, the value of parameter kb is arbitrary Therefore, the active four-wheel steering law without the feedback of sideslip angle is defined as the following equation again u = Dδ + Ex D= (8) Similarly to the study of ARS (Harada, 1995), the gains kh , kg are obtained to satisfy zero sideslip angle (Gβ (s) = 0) kh = − + Tγ s + a0 s + a1 s2 2WS 2Kf Kr a1 + Tβ s + a0 s + a1 s2 Gγ (s) = Gag (s) = (16) (17) (18) As shown in eq.(16) and eq.(17), Gγ (s) and Gag (s) become the first order lag It represents that A4WS has better response than conventional 2WS and AFS 3.2 Variable steering ratio Active four-wheel steering vehicle with the control law (11) requires the same steering wheel angle δ0,A4W S as eq.(6) for the constant radius turn δ0,A4W S = δ0,AF S = ks lr m v 2Kf l r Active four-wheel steering controller derived from linear vehicle model (19) From eq.(7) and eq.(19), δ0,A4W S is coincides with δ0,2W S when ks is defined as follows Feedback Controller δ + + + ur K f lf − K r lr 2Kf l − lr mv K r lr ks = ks0 (20) And also, the steering angle δ0,A4W S becomes δ0,A4W S = ks0 l r x Sliding Mode Controller Reference Vehicle Model xm + e - Fig Block diagram of proposed system e˙ = Am e + (Am − A )x 2Kf l2 lr mv ¯ − f (x, t) +(Bm − B )δ − B0 u (22) 3.3 Model following sliding mode controller 3.3.1 System uncertainty Actual vehicle dynamics has multi degrees of freedom and nonlinearity Especially, a perturbation of cornering power and a lateral disturbance such as crosswind affect on vehicle lateral motion Define the nominal values of cornering powers as Kf , Kr0 and the perturbations as ∆Kf , ∆Kr Kf ≡ Kf + ∆Kf , uf Vehicle ur Model following controller (21) when ks is set as ks = ks0 uf + + + Feedforward Controller Disturbance Kr ≡ Kr0 + ∆Kr (23) Equation of motion for vehicle with the cornering power perturbation becomes x˙ = Ax + Bu + Cw 3.3.2 Addition of model following controller This paper proposes the active four-wheel steering law by addition of model following sliding mode controller u ¯ = [¯ uf u¯r ] to the steering law (11) to guarantee the robustness against the system uncertainty Figure shows a block diagram of the proposed system (30) This paper designs the model following controller u ¯ to match the actual vehicle state vector x = [β γ] with the reference model state vector xm = [βm γm ] 3.3.3 Switching function The switching function σ is defined as follows σ= σ1 σ2 = Se = p q e1 e2 (31) u + d(x, t))} σ˙ = S e˙ = S{Am e − B0 (¯ (32) Substitution of σ˙ = to eq.(32) gives the equivalent control input u¯eq u¯eq = (SB0 )−1 S(Am e − B0 d(x, t)) (33) Let the model following controller u¯ be the summation of nominal equivalent control input u¯eq,0 and nonlinear control input u ¯nl u ¯=u ¯eq,0 + u ¯nl = (SB0 )−1 SAm e + (26) f (x, t) = {∆Ax + ∆B(Dδ + Ex + u ¯) + Cw}.(27) Let the reference vehicle model be x˙ m = Am xm + Bm δ ¯ − f (x, t) e˙ = Am e − B0 u (25) Substitution of eq.(25) into eq.(24) yields x˙ = A x + B δ + B0 u ¯ + f (x, t) Let Am , Bm of the reference model be the same as A , B of eq.(13) In other words, this paper defines the transfer functions Gβ (s), Gγ (s) shown in eq.(16) as the reference model Then, eq.(29) becomes Time differentiation of σ appears as = (A0 + ∆A)x + (B0 + ∆B)u + Cw (24) u = Dδ + Ex + u ¯ (29) (28) and define the error e = xm − x The following error equation is obtained from eq.(26) and eq.(28) ρ1 sgn(σ1 ) ρ2 sgn(σ2 ) (34) where ρ1 , ρ2 are positive constant values Lyapunov function is defined as V = σ σ/2 and the time differentiation of V becomes V˙ = −σ T SB0 {¯ unl + d(x, t))} (35) Let the parameters p = Iz /(lr mv), q = −Iz /(lf mv) in the switching function, eq.(35) becomes 2Kf l V˙ = −σ1 (ρ1 sgn(σ1 ) + d ) lr mv 2Kr0 l (ρ2 sgn(σ2 ) + d2 ) −σ2 lf mv 80 (36) Finally, we have the control law: ⎡ ⎡ ⎤ lf 1 ⎣ uf ⎢ v = K ⎦δ + ⎣ lr mv ur ks − f − Kr0 2Kr0 v ⎡ ⎤ lr mv ⎢ −1 − 2Kf l ⎥ ρ1 sgn(σ1 ) ⎥e + +⎢ ⎣ ⎦ lr mv ρ2 sgn(σ2 ) −1 2Kr0 l Initial velocity: Final velocity: Running time: Steering angle: =Target radius: 40 60[km/h] 20[km/h] 10[s] 0.43[rad] 100[m] 20 A-4WS with VSR w/o SMC ⎤ ⎥ ⎦x A-4WS with VSR,SMC AFS w/o VSR 60 y[m] Therefore, the convergence of the system to the switching plane is guaranteed by the Lyapunov stable theorem because V˙ < when ρ1 > |d1 | and ρ2 > |d2 | A-4WS w/o VSR, SMC Conventional2WS 0 20 40 60 x[m] 80 100 Fig Constant radius turn test with deceleration (37) Vehicle’s steering angle is 0.43[rad] The angle is necessary for the constant radius turn r=100[m] that is calculated by substitution of r = 100 into eq.(21) A sign function sgn(σi ) of eq.(37) is approximated by the following equation to prevent systems chattering by smoothing control input Figure demonstrates that VSR prevents the abrupt involution, and also shows that A-4WS with SMC can run along the constant radius path r=100[m] by the model following controller sgn(σi ) σi , µi > (i = 1, 2) |σi | + µi (38) 4.2 Double lane change test SIMULATION This paper performed two simulations using CarSim Ver.5.12 by Mechanical Simulation Corporation: 1) constant radius turn test with deceleration and 2) double lane change test CarSim is a software package for simulating real vehicle dynamics by using 19 degrees of freedom vehicle model Table shows the simulation parameters 4.1 Constant radius turn test with deceleration In order to verify the effectiveness of VSR (Variable Steering Ratio, eq.(22)), simulations were performed with five vehicles: (1) Conventional 2WS (2) AFS without VSR (3) A-4WS without VSR, SMC(Sliding Mode Controller) (4) A-4WS with VSR, without SMC (5) A-4WS with VSR, SMC The vehicle starts with an initial velocity 60[km/h], and decelerates for 10[s] by -4[km/h] per second Table Simulation parameters m 1707 kg ρ1 , ρ2 0.5 - Iz 2741.9 kgm2 µ1 , µ2 0.02 - lf 1.014 m τ 1.0 s lr 1.676 m τl 0.25 s Kf 68909 N/rad ks0 16 - Double lane change tests were performed to verify the performance of obstacle avoidance and drivability Here, three vehicles ran along the path (see Figure 5) under two conditions Vehicles: (1) A-4WS with SMC (2) A-4WS without SMC (3) Conventional 2WS Conditions: (1) Dry asphalt road (µ=0.9), v = 90[km/h] (2) Packed snow road (µ=0.2), v = 60[km/h] In the simulations, a look-ahead driver model was used The reference point is L = τ v[m] ahead from the control datum point P The model outputs a steering wheel angle δ based on the course error ε at the reference point δ(s) = he−τl s ε(s) (39) where h is the proportional gain, and τl is the dead time Figure shows that A-4WS with SMC obtains desired responses of sideslip angle and yaw rate y[m] P2 (130,3.5) Kr0 51406 N/rad O (0,0) P1 (100,0) P3 (155,3.5) P4 (180,0) Fig Path of double lane change test x[m] A-4WS with SMC A-4WS w/o SMC Conventional 2WS y [m] y [m] A-4WS with SMC A-4WS w/o SMC Conventional 2WS 2 0 -2 -2 100 x [m] 200 300 100 x [m] 200 300 0.1 A-4WS with SMC A-4WS w/o SMC Conventional 2WS 0.05 β [rad] β [rad] 0.02 0 A-4WS with SMC A-4WS w/o SMC Conventional 2WS -0.05 -0.02 -0.1 Time [s] 10 10 Time [s] 15 20 0.4 0.5 γ [rad/s] γ [rad/s] 0.2 A-4WS with SMC A-4WS w/o SMC Conventional 2WS -0.5 0 Time [s] 10 -0.4 5 10 Time [s] 15 20 2 ag [m/s ] 10 ag [m/s ] 10 0 A-4WS with SMC A-4WS w/o SMC Conventional 2WS -5 -10 A-4WS with SMC A-4WS w/o SMC Conventional 2WS -0.2 Time [s] 10 (1) µ = 0.9 (Dry asphalt road, v=90[km/h]) A-4WS with SMC A-4WS w/o SMC Conventional 2WS -5 -10 10 Time [s] 15 20 (2) µ = 0.2 (Packed snow road, v=60[km/h]) Fig Double lane change test even on the packed snow road by the model following sliding mode controller that works effectively to compensate a state error Therefore, it has an adequate path following capability Moreover, the lateral acceleration response of A-4WS with SMC becomes faster than others It implies the improvement of obstacle avoidance capability Next, A-4WS w/o SMC is compared with Conventional 2WS On the dry asphalt road, a sideslip angle of A-4WS w/o SMC, that should be zero in the linear region because of eq.(16), is equivalent to that of Conventional 2WS, but yaw rate response and path following capability are better than Conventional 2WS Furthermore, on the packed snow road where tire characteristic shows strong nonlinearity, A-4WS w/o SMC gets worse in the control results than Conventional 2WS CONCLUSION This paper proposed the active four-wheel steering system for zero sideslip angle and lateral acceleration control at a center of percussion by using model following sliding mode controller Theoretical analysis and computer simulations showed that the proposed system had a good maneuverability and the robustness against system uncer- tainty such as cornering power perturbation and lateral disturbance REFERENCES Ackermann, J (1997) Robust control prevents car skidding IEEE Control Systems 17(3), 23–31 Harada, H (1995) Control strategy of active rear wheel steering in consideration of system delay and dead times Transaction of JSAE (in Japanese) 26(1), 74–78 Hiraoka, T., H Kumamoto and O Nishihara (2002) Side slip angle estimation and active front steering system based on lateral acceleration data at centers of percussion with respect to front/rear wheels Proceedings of 2002 JSAE Annual Congress (in Japanese) Hiraoka, T., H Kumamoto, O Nishihara and K Tenmoku (2001) Cooperative steering system based on vehicle sideslip angle estimation from side acceleration data at percussion centers Proceedings of IEEE International Vehicle Electronics Conference 2001 pp 79–84 Shimada, Y., S Nohtomi, S Horiuchi and N Yuhara (1997) An adaptive LQ control system design for front and rear wheel steering vehicle Transaction of JSAE (in Japanese) 28(4), 111–116  ... the constant steering wheel angle ACTIVE FOUR- WHEEL STEERING 3.1 Addition of active rear steering for zero sideslip angle 3.1.1 Active four- wheel steering law An active four- wheel steering (A-4WS)... gets worse in the control results than Conventional 2WS CONCLUSION This paper proposed the active four- wheel steering system for zero sideslip angle and lateral acceleration control at a center... than conventional 2WS and AFS 3.2 Variable steering ratio Active four- wheel steering vehicle with the control law (11) requires the same steering wheel angle δ0,A4W S as eq.(6) for the constant radius