7 Special Issue Estimation and Control of for Active Safety Estimation of Circle and Vehicle Dynamics Research Integrated Control for Four- Distributed Steering and Report Four-wheel Distributed /Braking Systems Eiichi Ono, Yoshikazu Hattori, Yuji Muragishi

Abstract To improve the performance of vehicle forces acting on each tire. Then, we propose a dynamics control systems, it is important to be vehicle dynamics control system for four-wheel able to estimate the friction force characteristics distributed traction/braking and four-wheel between the and the road. In this paper, we distributed steering that is based on an on-line estimate the radius of the tire friction circle by nonlinear optimization algorithm that minimizes using the relationship between the Self-Aligning the maximum µ rate of the four tires by using the Torque (SAT), and the lateral and longitudinal estimation of the radius of the tire friction circle.

Vehicle dynamics control, , Vehicle dynamics integrated management, Keywords Tire friction circle

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1 3 F 1. Introduction =++εεε()1 13 23+⋅x ⋅+()12 ε 13 + 3 ε 23 + 4 ε. 2 5 Kβ

In turns with low lateral accelerations, the • • • • • • • • • • • • • • • • • • • • • • • • • (4) characteristics of Self-Aligning Torque (SAT) Figure 1 shows the tire grip margin ε as a function γ become more saturated than those of the lateral of the SAT model rate SAT and normalized force, such that an algorithm for estimating the tire longitudinal force Fx/Kβ, as calculated from Eq. (4). characteristics based on the relationship between This figure shows that the tire grip margin can be 1, 2) SAT and the lateral force has been developed. described by a monotonous function of the SAT While such methods can be applied to pure lateral model rate and the normalized longitudinal force. γ motion, we clarified the relationship between Then, the tire grip margin can be estimated from SAT the tire grip margin and SAT with combined lateral and Fx/Kβ by using a three-dimensional map as 3, 4) and longitudinal slip by using a brush model and shown in Fig. 1. are proposing a method for estimating the radius of Furthermore, the radius of the tire friction circle F0 the tire friction circle with combined lateral and can be estimated as: longitudinal slip.5) Furthermore, we are proposing a 22+ vehicle dynamics control system for four-wheel FFxy F = • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (5) distributed steering and four-wheel distributed 1−ε traction/braking systems, that aims to optimize the tire friction circle of each wheel. Then, the effects 3. Vehicle dynamics control for four-wheel of the vehicle dynamics control system on four- distributed steering and traction/braking wheel distributed steering and four-wheel distributed system traction/braking systems are shown by simulations. 3. 1 Hierarchical control structure The hierarchical control structure shown in Fig. 2 2. Estimation of tire friction circle is adopted for vehicle dynamics control.6) Each tire grip margin is estimated from the SAT, [Vehicle Dynamics Control] longitudinal and lateral force of each individual This layer calculates the target force and the 5) ε tire. The tire grip margin , normal SAT TSAT0 and moment of the vehicle in order to achieve the γ SAT model rate SAT are defined as desired vehicle motion corresponding to the driver's pedal input and the steering wheel angle. The 22+ FFxy ε ≡−1 , • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (1) designed target force and moment also satisfy the F robust stability condition.7) [Force & Moment Distribution] ⎪⎧ ll2 F ⎪⎫ T ≡+⎨ x ⎬ F , SAT 0 y • • • • • • • • • • • • • • • • • • • • • • • • • (2) The target force and moment of the vehicle motion ⎩⎪6 3 Kβ ⎭⎪ are distributed to the target forces of each wheel,

TSAT γ ≡ ,• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (3) based on the tire grip margin in this layer. TSAT 0 where, l : ground contact length, Kβ : cornering stiffness, F : radius of tire friction circle, Fx, Fy : longitudinal and lateral force, and TSAT : self-aligning 1 torque (SAT). By using the brush model, the relationship between the SAT model rate, 0.5 longitudinal force, and tire grip margin can be 0 Tire grip margin derived as -0.2 1 0.5 ⎛ ⎞ 0 0 1 2 F 2 ⎜ + x ⎟γεε()++13 23 Fx/K [rad] ⎜ ⎟ SAT 1 0.2 -0.5 SAT model rate ⎝ 6 3 Kβ ⎠ γ Fig. 1 Relationship between , Fx/Kβ and e.

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[Wheel Control] controlled so as to have the same value as γ γ γ This layer controls the motion of each wheel so as i = i = i, (i, j = 1, 2, 3, 4) • • • • • • • • • • • • • • • • • • (7) to achieve the target force. Furthermore, the angle between the direction of the

This paper describes a force & moment friction force and the X-axis is expressed as qi (i = 1, distribution algorithm in detail. 2, 3, 4), as shown in Fig. 4. In the second layer of γ 3. 2 Nonlinear optimal control hierarchical control, the value of qi that minimizes A vehicle model is described using the coordinates by satisfying the following constraints is calculated shown in Fig. 3, with the X-axis being the direction by using Sequential Quadratic Programming (SQP). 4 of the target resultant force and the Y-axis being the γ = ∑ FqFiicos x0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (8) µ γ = direction perpendicular to the X-axis. The rate i i 1 4 (i = 1, 2, 3, 4) of each wheel is defined as γ = ∑ FqFiisin y0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (9) i=1 FF22+ 4 γε≡− = xi yi γ ()−+ = ii1 , • • • • • • • • • • • • • • • • • • • • • • • • (6) ∑ Fdiicos ql iii sin q Mz0 .• • • • • • • • • • • • • • • (10) Fi i=1 where, Fxi, Fyi : longitudinal and lateral force of the Eq. (8) represents a constraint indicating that the i-th wheel, Fi : the radius of the friction circle of the resultant force in the X-axis direction is the target µ i-th wheel. In this study, the rate of each wheel is value Fx0. Eq. (9) represents a constraint indicating that the resultant force in the Y-axis direction is

target value Fy0. Eq. (10) represents a constraint indicating that the moment around the center of Information Driver input (Target) gravity of the vehicle is the desired yaw moment M0. (Result) Vehicle Dynamicsdynamics control Control By eliminating γ from Eqs. (8)-(10), the constraints

Force & moment of vehicle of qi can be obtained. 4 ()−− + = Force & momentMoment distributionDistribution ∑ FFdMixiz{}00cos qFlq ixii 0 sin 0 • • • • • • (11) i=1 Lateral & longitudinal forces of tires 4 −+−()= Wheel Controlcontrol ∑ FFdqiyii{}000cos FlM yiz sin q i0 • • • • • (12) i=1 Driving torque, Brake pressure, Steer angle The performance function J that minimizes γ is Actuator defined as

()2 + 2 + 2 dF00xyz() lF00 M0 J = • • • • • • • • • • • • • • • • • • • (13) Fig. 2 Hierarchical vehicle dynamics management γ algorithm. where, d0, l0 : constants of a typical moment arm. In this paper, we use the following constants. TT+ X-axis d = fr• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (14) 0 4

d1 -d2 F1 F F 2 x0 l2 l1 F Y-axis y0 Target resultant

Force & moment qi -l M X-axis 3 0 -l 4 Tire force γ Fi

d3 d4 F4 F3 Friction circle Fi Friction circle Y-axis

Fig. 3 Coordinate system corresponding to resultant Fig. 4 Relationship between direction of target resultant force. force and direction of friction force.

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LL+ 2 fr ++()()+ } l = • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (15) lF0 yizi0000 lM qsin q icos c qqii0 0 2 ⎪⎧⎛ q 2 ⎞ ⎪⎫ Since the numerator on the right side of Eq. (13) is a +−()dF2 dM ⎨⎜1− i0 ⎟cosqq+ sin q⎬ 0 xiz00⎪ 2 ii00 i 0⎪ constant, maximizing J implies minimizing γ. By ⎩⎝ ⎠ ⎭ substituting Eqs. (8)-(10) into Eq. (13), J can be ⎪⎧⎛ q 2 ⎞ ⎪⎫⎤ rewritten as, ++()lF2 lM ⎨⎜1− i0 ⎟sinqq− cos q⎬⎥ 0 yiz00⎪ 2 ii00 i 0⎪ 4 4 ⎩⎝ ⎠ ⎭⎦⎥ JdF=+2 ∑∑ Fcos qlF2 F sin q 0 xii0 0 yii0 4 ⎧ ⎫ i==1 i 1 1 2 =−∑ FXqXY⎨ () −+ ⎬ , • • • • • • • • • • • • • (23) 4 iDiiii⎩ ⎭ +−+() i=1 2 izMFdql0 ∑ iiiicos sinn qi i=1 4 where, =−∑ FdF{}()2 dMcos q++() lFlM2 sin q X i0 x00 iz i 0 y00 iz i. = Ni = X i ,• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (24) i 1 X • • • • • • • • • • • • • • • • • • • • • (16) Di 2 Then, the optimization problem can be formulated as XNi = (d0 Fx0 - diMz0)(qi0cosqi0 - sinqi0) 2 follows. + (l0 Fy0 + liMz0)(qi0sinqi0 + cosqi0), • • • • • • (25)

Problem 1: =−2 ++2 XdFdMqlFlMqDi()0 x0000 i zcos i() y000 i z sin i , We need to find qi (i = 1, 2, 3, 4), that maximizes the performance function (16) with constraint • • • • • • • • • • • • • • • • • • • • • • (26) equations (11) and (12). ⎪⎧⎛ q 2 ⎞ ⎪⎫ =−2 − i0 + Problem 1 can be solved by using SQP. First, sinqi YdFdMixiz()0 00⎨⎜1 ⎟cosqqii00 sin q i 0⎬ ⎩⎪⎝ 2 ⎠ ⎭⎪ and cosqi can be linearized around qi0 as:

sin qi = sin qi0 + cos qi0 (qi - qi0) • • • • • • • • • • • • • • • • • (17) ⎪⎧⎛ q 2 ⎞ ⎪⎫ X 2 ++2 − i0 − + Ni cos qi = cos qi0 + sin qi0 (qi - qi0)• • • • • • • • • • • • • • • • • (18) ()lF0 yiz00 lM ⎨⎜1 ⎟sinqqii00 cos q i 0⎬ . ⎩⎪⎝ 2 ⎠ ⎭⎪ 2X Then, constraint equations (11) and (12) can be Di expressed as follows. • • • • • • • • • • • • • • • • • • • • • • (27) 4 ()+ + Assuming that XDi >0, the problem of maximizing J ∑ FFdMixiz{}00000sin q i Fl xiii cos qq i=1 can be rewritten as the problem minimizing 4 4 =+{()()+ = 2 ∑ FFdMixiz00000 q isin q i cos q i Kp∑ i , • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (28) i=1 i~1 +−()} Flxi00 q icos q i 0 sin q i 0• • • • • (19) by the variable transformation 4 {}+−() =−()• • • • • • • • • • • • • • • • • • • • • • • • • • (29) ∑ FFdiyi00000sin q i Fl yi M z cos q i q i pFXqXiiDiii. i=1 4 =+() Further, linearized constraint equations (19) and (20) ∑ FFdqiyii{ 00sin q i 0 cos q i 0 i=1 can be expressed as +−()− ()Flyi00000 M z q icos q i sin q i } • • • • • (20) ⎡ AAAA⎤ ⎡B ⎤ 11 12 13 14 = 1 Next, sinqi and cosqi can be approximated around qi0 ⎢ ⎥ p ⎢ ⎥, • • • • • • • • • • • • • • • • • (30) ⎣AAAA⎦ ⎣B ⎦ by a 2nd order Taylor expansion as: 21 22 23 24 2

sin q 2 where, sinqq=+ sin cos qqq() −−−i0 ()qq ii000 iii ii0 F 2 =⋅i () + + A100000i {}Fdxi M zsin q i Fl xi cos q i , • • (31) X • • • • • • • • • • • • • • • • • • • • • • (21) Di

cos q 2 cosqqqqq=− cos sin () −−−i0 ()qq iiiii000 ii0 F 2 =⋅i +−() A200000i {Fdyisin q i Fl yi M z cos q i}, • • (32) X • • • • • • • • • • • • • • • • • • • • (22) Di 4 Then, performance function (16) can be expressed as =+⎡(){}()− + BFFdMqXq100000∑ ixiz⎣ i isin i cos q i 4 ⎡ 1 i=1 JF=−∑ {}() dFdMqlFlMqq2 − cos++()2 sin 2 i⎢ 0 x0000 iz i y000 iz i i +−{}()− ⎤ i=1 ⎣ 2 Flxi00 q i X icos q i 0 sin q i 0⎦ , • • • • • • • • (33)

+−2 ()− {()dF0 xizi00000 dM qcos q i sin q i

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4 4 =−⎡ {}()+ JFFdqFlMq()q =−∑ {}cos +−() sin BFFdqXqq20000∑ iyii⎣ isin i cos i Fy i y000 i i y i z i . i=1 i=1

• • • • • • • • • • • • • • • • • • • • • • • (41) +−(){}()− − ⎤ Flyi000 M z q i X icos q i 0 sin q i 0⎦ . • • (34) JFx(q) corresponds to the left side of Eq. (11), and

Since minimizing K implies minimizing the JFy(q) corresponds to the left side of Eq. (12). If T Euclidian norm of p = [p1 p2 p3 p4] , the optimal P(q) < P(q0), then solution can be derived by using pseudo-inverse q0 = q, • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (42) matrix calculation as and we can go to STEP 2. Otherwise, q^ which ^ < () satisfies PP()qq0 is a straight line searched in [q, + ⎡ AAAA⎤ ⎡B ⎤ p = 11 12 13 14 ⋅ 1 , q0]. Then, ⎢ ⎥ ⎢ ⎥ • • • • • • • • • • • • • • • (35) = ^ ⎣AAAA21 22 23 24 ⎦ ⎣B2 ⎦ qq0 , • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (43) where, A+ represents the pseudo-inverse of matrix A. and we can go to STEP 2. Then, We know that SQP is one of the most effective methods of nonlinear optimization.8) Further, ⎡ 1111⎤ q = diag ⎢ ⎥ optimal γ can be obtained as ⎣⎢ FX11DDDD FX 22 FX 33 FX 44⎦⎥

⎡ ⎤ ()2 + 2 + 2 X1 dF00xyz() lF00 M0 + ⎢ ⎥ γ = ⎡ AAAA⎤ ⎡B ⎤ X 4 . ⋅ 11 12 13 14 ⋅ 1 + ⎢ 2 ⎥ ∑ FdF{}()2 − dMcos q++(() lFlM2 sin q ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ i0 x00 iz i 0 y00 iz i ⎣AAAA⎦ ⎣B ⎦ X , • • • • • • • • • • • (36) i=1 21 22 23 24 2 ⎢ 3 ⎥ ⎢ ⎥ ⎣X 4 ⎦ • • • • • • • • • • • • • • • • • • • • • (44) where, The magnitude and direction of each tire force are T q = [q1 q2 q3 q4] . translated to the steer angle and longitudinal force of

Then, recursion formula solving of optimal qi based each wheel, and these are controlled in the last layer on SQP is shown as follows. of the hierarchical control. The lateral and

STEP 1 To satisfy the condition XDi >0, the longitudinal forces can be obtained as γ initial value of qi0 is calculated as Fxi = Fi cos qi , • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (45) γ Fyi = Fi sin qi . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (46) lF2 + lM q = tan−1 0 yiz00 Further, by using a brush model, the of i0 2 − . • • • • • • • • • • • • • • • • • • • • • • (37) dF0 xiz00 dM each wheel can be calculated as

By substituting Eq. (37), XDi can be rewritten as − ⎛ K −κ sin q ⎞ α = tan 1 ⎜ sii⎟ XdFdMqlFlMq=−()2 cos++()2 sin i −κ , • • • • • • • • • • • • • • • • • • • • (47) Di0 x0000 i z i y000 i z i ⎝ Kα 1 iicos q ⎠

2 2 where, =−()dF2 dM++() lF2 lM 0 xiz000 yiz00 3F ⎧ 1 ⎫ κγ=−−i ()3 i ⎨11 ⎬ ,• • • • • • • • • • • • • • • • • • • • • • • • • • (48) 2 K ⎩ ⎭ ⎛ lF+ lM ⎞ s ⋅−cos⎜ q tan −1 0 yiz00⎟ ⎜ i0 2 − ⎟ ⎝ dF0 xiz00 dM ⎠ Ks : braking stiffness, Kα : cornering stiffness.

2 2 =−2 ++2 > • ()dF0 xiz00 dM() lF0 yiz00 lM 0 . (38) 4. Simulation results

STEP 2 qi is calculated from Eq. (36). Figure 5 shows the simulation results for four- STEP 3 We define a penalty function as wheel distributed steering and traction/braking control in the case of a single-shot sine-wave ()= 1 + ρ ()+ () steering maneuver at ν = 22 m/s. A linear model of P q ()JJFxqq Fy , • • • • • • • • • • • • (39) J ()q the vehicle is used for the reference of control. The where, µ rate is controlled such that it is the same for each ρ > 0, wheel, and is restricted to a maximum value of 0.95, 4 ()=−−()+ so that the grip margin is uniform, while the limit on JFFdMqFlqFxq ∑ i{} x00 i zcos i x 0 i sin i , i=1 the friction circles is satisfied. This implies that the

• • • • • • • • • • • • • • • • • • • • • • • (40)

R&D Review of Toyota CRDL Vol. 40 No. 4 12 tire forces are used efficiently in the friction circles. steering maneuver at ν = 22 m/s. This figure shows Figure 6 compares the maximum resultant force that the limited performance (maximum resultant of the simulation result for a single-shot sine-wave force) is improved through the use of integrated control. The maximum resultant force is improved by 7.8 % as a result of adding active front-rear 0.05 steering control to four-wheel distributed traction/braking control, and by a further 2.5 % as a

[rad] 0

Slip angle Reference (linear model) result of adding an independent steering mechanism -0.05 0 0.5 1 1.5 2 2.5 3 (four-wheel distributed steering). 0.5 5. Conclusion 0 [rad/s] In this paper, the radius of each tire friction circle

Yaw velocity velocity Yaw -0.5 0 0.5 1 1.5 2 2.5 3 is estimated from the relationship between the Self- 2000 Left Aligning Torque (SAT) and the lateral and 0 [N] Front : solid longitudinal forces acting on each tire, and a method Right Rear : dashed - 2000 of vehicle dynamics control that uses the tire friction

Longitudinal force 0 0.5 1 1.5 2 2.5 3 0.2 circle of each wheel for the optimum is proposed. A distribution algorithm using Sequential Quadratic 0 [rad] Programming (SQP) is used to calculate the Steer angle angle Steer -0.2 magnitude and direction of the tire forces, such that 0 0.5 1 1.5 2 2.5 3 1 they satisfy the constraints corresponding to the

rate) target force and moment of the vehicle motion and µ 0.5 (

γ µ ( rate) minimize each tire rate. The proposed algorithm, 0 0 0.5 1 1.5 2 2.5 3 being characterized by pseudo-inverse matrix Time [s] calculation, features high-speed calculation and accurate calculation that satisfy the constraints, so that it can be applied to four-wheel distributed Fig. 5 Simulation result for four-wheel distributed steering and traction/braking control (single-shot steering and four-wheel distributed traction/braking sine-wave steering at ν = 22 m/s). systems for which the optimization of the eight parameters is necessary. Then, the effectiveness of the vehicle dynamics control for four-wheel distributed steering and four-wheel distributed Without control traction/braking systems is shown by simulations.

7.7% References Four-wheel4-wheel distributed distributed traction/braking traction/braking control control 1) Pasterkamp, W. R. and Pacejka, H. B. : "On Line 7.8% Estimation of Tyre Characteristics for Vehicle Control", AVEC '94, (1994), 521-526 Active front-rear steering and four-wheel Active front-rear steering and 4-wheel 2) Fukada, Y. : Japanese Unexamined Patent 1995- distributed traction/braking control 137647 (in Japanese), (1995) 2.5% 3) Bernard, J. E., et al. : "Tire Shear Force Generation Four-wheel distributed steering and four-wheel 4-wheel distributed steering and 4-wheel during Combined Steering and Braking Maneuvers", distributed traction/braking control SAE Tech. Pap. Ser., No.770852 (1977) 0 5000 10000 15000 4) Burkard, H. and Calame, C. : "Rotating Wheel Maximum resultant force [N] Dynamometer with High Frequency Response", Tire Technol. Int. 1998, (1998), 154-158 5) Ono, E., et al. : "Estimation of Tire Grip Margin Fig. 6 Comparison of maximum resultant force of Using Electric Power Steering System", Proc. 18th simulation result for a single-shot sine-wave IAVSD Symp. Dyn. of Veh. on Roads and Tracks, steering maneuver at ν = 22 m/s. (2003)

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6) Hattori, Y., et al. : "Force and Moment Control with Nonlinear Optimum Distribution for Vehicle Eiichi Ono Dynamics", AVEC '02, (2002), 595-600 Research fields : Vehicle Dynamics 7) Ono, E., et al. : "Bifurcation in Vehicle Dynamics and Control Robust Front Wheel Steering Control", IEEE Trans. Academic degree : Dr. Eng. on Control Syst. Technol., 6-3(1998), 412-420 Academic society : Soc. Instrum. Control 8) Han, S. P. : "A Globally Convergent Method for Eng., Soc. Automot. Eng. Jpn Nonlinear Programming", J. Optim. Theory Appl., Awards : SICE Award for Outstanding 22(1978), 297-309 Paper, 1995 (Report received on Sep. 22, 2005) SICE Chubu Chapter Award for Outstanding Research, 1999 SICE Chubu Chapter Award for Outstanding. Technology, 2002 IFAC Congress Applications Paper Prize, 2002 Paper Award of AVEC, 2002 Paper Award of AVEC, 2004

Yoshikazu Hattori Research fields : Vehicle Dynamics Control, Vehicle Dynamics Analysis & Modeling Academic society : Soc. Instrum. Control Eng., Inst. Syst., Control Inform. Eng., Soc. Automot. Eng. Jpn.

Yuji Muragishi Research fields : Vehicle Dynamics Control Academic society : Jpn. Soc. Mech. Eng., Jpn. Fluid Power Syst. Soc.

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