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Estimation of Tire Friction Circle and Vehicle Dynamics Integrated 7 Special Issue Estimation and Control of Vehicle Dynamics for Active Safety Estimation of Tire Friction Circle and Vehicle Dynamics Research Integrated Control for Four-wheel Distributed Steering and Report Four-wheel Distributed Traction/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 tires 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, Self aligning torque, Vehicle dynamics integrated management, Keywords Tire friction circle R&D Review of Toyota CRDL Vol. 40 No. 4 8 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 slip 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. R&D Review of Toyota CRDL Vol. 40 No. 4 9 [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. R&D Review of Toyota CRDL Vol. 40 No. 4 10 LL+ 2 fr ++()()+ } l = • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (15) lF0 yizi0000 lM qsin q i ccos 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 , • • • • • • •
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