Optimum Vehicle Dynamics Control Based on Tire Driving and Braking

23 Special Issue Modeling, Analysis and Control Methods for Improving Vehicle Dynamic Behavior Optimum Vehicle Dynamics Control Based on Tire Driving and Research Braking Forces Report Yoshikazu Hattori

Abstract

This paper discusses a method of controlling This report proposes a method of distributing the tire force exerted by each wheel of a vehicle the target force and moment of a vehicle to each as a means of ensuring steerability and stability. tire by considering variations in the tire force Conventional vehicle stability control systems caused by changes in the tire's vertical load, the generally rely on feedback of vehicle states such longitudinal slip, and so on. As a result, the as the side slip angle and yaw rate to enable proposed strategy achieves seamless behavior stabilization under a range of vehicle/road surface between the normal and the critical limit regions. combinations. On a low-friction road surface that And, we have confirmed that excellent levels of is incapable of applying sufficient force, steerability and stability can be achieved, relative however, it is unreasonable to expect a system to to conventional stability control systems, by provide sufficient control upon the occurrence of simulating a slalom maneuver. Also, this undesirable vehicle behavior. The tire force has a strategy enables effective vehicle control in the non-linear saturating characteristic, the value of face of unstable phenomena involving which varies with the vehicle state and road unbalanced lateral forces arising from changes in surface, making it difficult to determine the force the vehicle behavior, such as closing the throttle to be generated by each tire to ensure the desired during a turn maneuver. vehicle behavior.

Keywords Vehicle dynamics control, Nonlinear optimum control, Tire model, Steerability, Stability

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1. Introduction Figure 1 illustrates the control provided by VDM. We refer to this illustration as the "Ball in a bowl." Since the 1980s, various chassis active control The ball corresponds to the vehicle state that is systems have been developed to ensure vehicle maintained within a conceptual "bowl" that stability. Direct-yaw moment control systems with corresponds to the control systems. Inside the bowl active braking, as typified by VSC (Vehicle Stability is the stable region, while outside it is the unstable Control) have realized good vehicle stability in the region. In conventional systems, the wall is 1-4) critical limit region. Those papers have mainly configured by several systems such as ABS, VSC, discussed control methods that feedback various and TCS (Traction Control System), all of which are aspects of the vehicle state, such as the side slip sheer just before facing an emergency situations. As angle and yaw velocity. a result, the vehicle motion can be stabilized, Some subsequent proposals have introduced although non-smooth motion may sometimes occur. vehicle and tire models to estimate the vehicle state VDM, on the other hand, enables smoother behavior, and calculate the yaw moment required to stabilize because it involves the conventional control systems 5-6) the vehicle. In earlier methods that handed over being restructured to form a continuous and smooth control of the vehicle immediately after detecting a "wall." To realize this level of control, the following vehicle skid, however, it was difficult to attain elements are important: smooth control. Therefore, a method of directly • Hierarchical algorithm involving the restructuring controlling the force and moment of the vehicle was of the conventional control strategy 7) proposed. Meanwhile, the tire force exhibits non- • Feedforward force and moment control for vehicle linear saturating characteristics according to the dynamics vehicle state and road surface conditions. • Nonlinear optimum distribution method that Additionally, these values change frequently. coordinates the operations of each actuator Therefore, it is not easy to determine the force that Each of the above is discussed in the following should be generated at each tire in order to obtain chapters. the target force and moment of the vehicle under any given driving situation. Hence, earlier systems used 3. Hierarchical algorithms for VDM rule-based methods to calculate the longitudinal Since vehicle control systems have continued to force at each wheel from the direct yaw moment. become more diversified, their algorithms need to be Against this background, next-generation vehicle able to easily manage the cooperative control of dynamics control should be able to provide seamless multiple systems such as the drive train, braking and vehicle steerability and stability at any time while steering. Accordingly, it is important to be able to driving, integrated control systems to use most of the ensure the compatibility of the algorithms with tire performance. different system configurations. To ensure the high First, we propose the new VDM (Vehicle level of performance, we require a model-based Dynamics Management) concept. Next, we suggest algorithm that uses feedforward control in addition three key technologies that are central to realizing to feedback control of the vehicle state. this concept. Finally, the performance of the proposed strategy is investigated by simulation. TCS VDM 2. Concept and key technologies of VDM VSC VSC The ultimate goal of vehicle dynamics control is to produce vehicles that anyone can drive "safely,"

"pleasantly," and "as one wishes." In realizing this, ABS we must embrace the key words of "seamlessly" and "anytime." To achieve this, we are proposing the VDM concept. Fig. 1 The evolution of control by VDM.

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An HVDM (Hierarchical Vehicle Dynamics individual tire forces, with the total becoming the Management) algorithm has been proposed to satisfy desired force and moment of the vehicle. the above requirements (Fig. 2). The HVDM [Wheel control] algorithm consists of small systems, each connected This layer calculates the target values for each hierarchically. actuator, such as the engine, braking, steering and so [Vehicle dynamics control] on. The target values are determined to generate the This layer calculates the desired longitudinal and desired tire forces. lateral forces, as well as the yaw moment of the [Actuator control] vehicle. The forces and moment are determined as Each actuator system is controlled. The braking the vehicle assumes the desired motion, taking the system, for example, uses actuated pressure valves driver's operations into account. to control the braking torque. [Force and moment distribution] The upper layer outputs the target values to the This layer determines the distribution of the lower layer, while the lower layer feeds back the results. The upper layer then recalculates the target value depending on the results. Information DriverInputs (Steer, Throttle,Brake) This both-way communication enables each layer Vehicle Dynamics Control to cooperate with the other and maintain greater Force &Moment of Vehicl e robustness corresponding to the characteristic Force & Moment Distribution change of the controlled system and environment. Tire Forces(Slip Ratios) 4. Adoption of force and moment control Wheel Control Figire 3 shows the proposed control flow. This Driving Torque Brake Pressure flow consists of two parts: One is feedforward Drive Train Control Brake Control control of the force and moment of the vehicle and (Target) the other is feedback control of the vehicle states. (Result) This type of system is generally called "Two- Degrees-of-Freedom Control." The main advantage Fig. 2 Hierarchical vehicle dynamics management of this system is that it enables the independent algorithm. design of controllability and stability.8) The feedforward control estimates the tire forces. FForce orce and MMoment oment VVehicle ehicle States Therefore, the part controlled as FeFeedforwarded forward FFeedback eedba ck the force and moment of the + - vehicle, as calculated by γ * β * , estimating the tire force, is RReference eferenc e Model * * * Feedback equal to the value obtained by Fx , Fy , Mz γ Controller , β the reference model. Feedback control stabilizes Driver + δ + Distribution the vehicle behavior as it δ - + ControllerC ontroller approaches the limit region Observe ˆ ˆ ˆ where the modeling error is ModelModel F , Fy , M VehiVehicle x z large. Fxi , Fyi , M zi , By combining these two parts, δ vehicle behavior can be made smooth and the probability of approaching the critical limit Fig. 3 Vehicle dynamics control algorithm. can be reduced.

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proposed. It is based on the following tire model. 5. Nonlinear optimum distribution method 5. 1 Tire model The other core technology employed by VDM is a The characteristics that the tire force distribution force and moment distribution method that uses algorithm requires of the tire model are as follows. nonlinear optimization. As shown in Fig. 4, this • Saturation of tire force method distributes the target force and moment of • Dependency of driving and cornering stiffness the vehicle to the longitudinal and lateral forces of on vertical load each wheel. • Relationship between longitudinal and lateral It is well known that, when braking in a turn, the force (friction circle) distribution of the braking force in proportion to the • Fewer parameters for describing the model vertical load of each wheel causes the yaw moment The simple brush model described by Eqs. (1)-(10) to remain the same, if any reduction in the lateral satisfies these requirements.9) The characteristics of force corresponding to an increase in the the longitudinal and lateral forces for the applied tire longitudinal force is ignored. The tire force, model are shown in Fig. 5. κ α however, features non-linear saturating Where i , i are the slip ratio and slip angle of characteristics where the value varies with the each tire, Kα , Kκ are the cornering and driving vehicle state and road surface. Therefore, it is not stiffness, Kα 0 , Kκ 0 , are the coefficients of stiffness easy to determine the force that should be generated corresponding to the vertical load, Fzi is the vertical µ at each tire to obtain the desired vehicle behavior in load on each wheel, and i is the maximum tire force any driving situation. normalized by Fzi . κ The proposed nonlinear optimizing method is i cos θ = • • • • • • • • • • • • • • (1) efficient in the following two points. λ • When a target value that exceeds the physical α Kα tan i sin θ = • • • • • • • • • • • • • • (2) limit is input, the algorithm automatically executes a Kκ λ trade-off based on a pre-designed performance Kκ = Kκ 0 F • • • • • • • • • • • • • • (3) function. zi = • When the target value is sufficiently small, the Kα Kα 0 Fzi • • • • • • • • • • • • • • (4) algorithm is able to distribute a reasonable force to 2 2 α λ = κ 2 + Kα tan i each wheel, making it a redundant system regulated i • • • • • • • • • • • • • • 2 (5) Kκ by the performance function. For these reasons, a nonlinear optimum method is ξ = − Kκ λ 1 µ −κ • • • • • • • • • • • • • • (6) 3 i Fzi (1 i )

1. Where ξ > 0, Target

Result 2 Force & Moment ξ Kκ

Target = −κ of the Vehicle Fxi −κ i (1 i ) −µ θ − ξ 2+ 2ξ 3 ) Nonlinear i Fzi cos (1 3 • • • • • • • • (7) OptimumOptimum ξ 2 α Method = − Kα tan i Fyi 1 −κ

Target i Result 2 3 Tire Forces −µι F sin θ (1− 3ξ + 2ξ ) • • • • • • • • • (8)

Target zi Result Result 2. Where ξ < 0, = −µ θ Fxi i Fzi cos • • • • • • • • • • • • • • (9) = −µ θ Fyi i Fzi sin • • • • • • • • • • • • • • (10) Fig. 4 Force & moment distribution control.

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5. 2 Nonlinear optimizing algorithm Also, WE , Wδκ , and Wκ are the weights The object is to solve for the optimum balance of corresponding to E, δκ and κ. The force and the tire force. When a balance is achieved, the error moment of the vehicle calculated by summing the

between the target force and moment of the vehicle distributed tire forces of each wheel (Fx, Fy, Fz) are and those generated by each tire force are almost described by Eq. (15) using each tire force and the

equal and, simultaneously, each ratio of generated Jacobian (J) value given by Eq. (16). [ ^ [ tire force to the maximum tire force is almost equal. Fx 4 Fxi ^ δκ • • • • • • • • • • • The optimizing algorithm minimizes the following F = Fyi + J (15)

y i−1 ^ performance function. Sequential quadratic [ [Mzi Mz 11) programming is used to solve this problem. [ Fx Fx Fx Fx T δκ T δκ κ δκ T κ δκ κ κ κ κ L = E WE E + Wδκ + ( + ) Wκ ( + ) fl fr rl rr Fy Fy F F • • • • • • • • • • • • • • (11) J= κ κ κ y κ y • • • • • (16) fl fr rl rr Where κ is the slip ratio of each wheel. Mz Mz Mz Mz [ κ κ κ κ This controller is discretized. And, δκ is the fl fr rl rr operating value of this system. Next, κ is modified κ by δκ to κ + δκ. Here, J is described as the function of the slip ratio ( ) This method involves calculating δκ to minimize L and the slip angle based on the tire model. δκ for the linearized system around the operating point After the above preparation, satisfies Eq. (17), at each control sampling time. minimizing L. L/ δκ = 0 • • • • • • • • • • • • • • • (17) κ κ κ κ κ T =[ fl fr rl rr ] • • • • • • • • • • • • • (12) δκ δκ δκ δκ δκ T We solve Eq. (17) for δκ by substituting Eqs. (11)- =[ fl fr rl rr ] • • • • • • • • • (13) (16). Finally, we obtain Eq. (18). E is the error between the target force and the δκ T −1 T ∆ − κ * * * = (Wδκ +Wκ+J WE J) (J WE E Wκ ) moment of the vehicle (Fx , Fy , Mz ), and that calculated from the distributed tire forces • • • • • • • • • • • • • • • (18) 4

^ ^ ^ ∆ * * * T − T (Fx ,F y ,M z ), and is given by Eq. (14). E = [ Fx , Fy , Mz ] [ Fxi Fyi Mzi ] i−1 ^ [ • • • • • • • • • • • • • • • (19) * − Fx Fx = * − ^ E • • Fy Fy • • • • • • • • • • • • • • (14) 6. Simulation [M * − ^ z Mz This section explains the simulations that we The purpose of the term "κ + δκ " is to equalize performed to show the effect of the proposed control each ratio of tire force to the maximum force, while method. The conventional system calculates a target that of "δκ " is to prevent the actuator from yaw moment by regulating the weighted-sum of the responding too quickly. ) [N] x F ) [N] y F Lateral Force ( Longitudinal Force (

Slip Ratio (κ) Slip Angle (α) [rad]

Fig. 5 Characteristics of tire force.

R&D Review of Toyota CRDL Vol. 38 No. 4 28 slip angle and the angular velocity of the vehicle. Then, if the target yaw moment is in the outward 0.2 direction during cornering, it is applied by the Proposed System 0.15 Conventional System outside front wheel. On the other hand, if the target Without Control is in the inward direction, it is applied by the three 0.1

y [rad/s] it wheels other than the inside rear wheel. 0.05 6. 1 Slalom maneuver 0 The control inputs and vehicle states during a slalom maneuver on a low-friction road surface are -0.05 shown in Fig. 6. -0.1 Slip Angular Veloc The proposed method compensates for any -0.15 inbalance in the yaw moment by means of -0.2 -0.05 0 0.05 0.1 feedforward prior to the vehicle skidding. As a Slip Angle [rad] result, unstable vehicle behavior is highly unlikely to occur. (a) Stability performance Figure 6(a) shows the phase plane trajectory of the slip angle and angular velocity of the vehicle. The proposed control draws the smallest trajectory, indicating the highest stability. Furthermore, Fig. 6(b) 0.2 2 shows differences in the steering angle, yaw velocity PrProposed oposed System 0.15 Conventional System 1.5 and control inputs between the conventional and WithoutCWithou t Control ontr ol SteeringASteering A ngle proposed controls. The proposed control starts 0.1 1 applying control earlier than the conventional 0.05 0.5 control. The proposed control also indicates good 0 0 tracking performance between the steering angle and -0.05 -0.5 yaw velocity in order to start the application of RateYaw [rad/s] -0.1 -1 -1.5 control to the balance of the four wheel forces before -0.15 AngleSteering Steerin [rad] ngle [rad] the vehicle skids. -0.2 -2 6. 2 Closing the throttle during a turn maneuver -0.25 -2.5 0 2 4 6 8 If we close the throttle during limit cornering in a Time [s] rear-wheel drive vehicle, the vehicle may deviate 3 from the desired trace line. Because deceleration Pr oposed System 2 RR RL slip of the rear wheels increases, the weight is FR FL transferred to the front wheels, causing the front 1 FR RR lateral force to increase and the rear lateral force to 0 0 2 4 6 8 decrease. Brake Pressure [MPa] Time [s] 3 As shown in Fig. 7, variations in the slip angle and Conventional System the yaw velocity increase very slowly. As a 2 FL FR consequence, conventional control that mainly uses 1 feedback of the vehicle states cannot produce the 0 desired effect. This is caused by the error in the 0 2 4 6 8 vehicle behavior being so small that it is masked by Brake Pressure [MPa] Time [s] the dead zone to avoid signal noise. On the other hand, the proposed method calculates (b) Steerability performance & control inputs the variance in the yaw moment caused by closing the throttle based on the vehicle model and driver Fig. 6 Slalom maneuver. inputs. This method is able to cancel the desired yaw

R&D Review of Toyota CRDL Vol. 38 No. 4 29 moment before the vehicle skids. Consequently, excellent steerability and stability at any time while undesirable vehicle behavior almost never occurs driving. The key technologies of VDM, namely, the (Fig. 8). methods for feedforward control of the force and moment and the nonlinear optimum distribution 7. Conclusion algorithm are described. The performance of the We have proposed a VDM concept that realizes proposed control system was confirmed by simulation.

0 References -0.02 1) Shibahata, Y., et al. : "Improvement of Vehicle -0.04 Throttle Off Maneubarability by Direct Yaw Moment Control'', -0.06 Vehicle System Dynamics, 22(1993), 465-481 0 0.5 1 1.5 2 2.5 3 Time [s] 2) Inagaki, S., et al. : "Analysis on Vehicle Stability in

s]0.14 Slip Angle[rad] Critical Cornering Using Phase-Plane Method",

d/ Throttle Off 0.13 5 AVEC'94, No.50(1994), 287-292 3) Koibuchi, K., et al. : "Vehicle Stability Control in 0.13 Limit Cornering by Active Brake'', SAE Tech. Pap. 0.12 5 0 0.5 1 1.5 2 2.5 3 Ser., No.960487(1996) Yaw RateYaw [ra Time [s] ] 4) van Zanten, A. T. : "Control Aspects of the BOSCH- 2 0.06 VDC'', AVEC'96(1996), 573-608 0.04 Throttle Off 0.02 5) Fukada, Y. : "Estimation of Vehicle Slip-angle with 0 Combination Method of Model Observer and Direct oment [1/s -0.02 0 0.5 1 1.5 2 2.5 3 Integration'', AVEC'98(1998), 201-206 w M Time [s] 6) Furukawa, Y., Abe, M., et al. : "On-board-tire-model Ya Reference Control for Cooperation of 4WS and Direct Yaw Moment Control for Improving Active Fig. 7 Closing the throttle durig a turn maneuver Safety of Vehicle Handling'', AVEC'96(1996), 573-608 7) Katsuyama, E., Fukushima, N. : "Improvement of Turning Behavior Using Yaw Moment Feedback 1.5 Control'', JSAE20005171, (2000) RR PrProposed op osed System 8) Ono, E., Hayashi,Y., Doi, S. and Takanami, K. : 1 "Coordination of Vehicle Steering Suspension RL Systems by Integrated Control Strategy'', 0.5 FR AVEC'92(1992), 384-389 FL 9) Abe, M. : "Vehicle Dynamics and Control", (1992), 0 00.511.522.53 Sankaido (in Japanese) Time [s] 2.5 10) Hattori, Y., Koibuchi, K. and Yokoyama, T. : "Force 2 CC onven onven tion al System and Moment Control with Nonlinear Optimum

Brake PressureBrak [MPa] 1.5 ressure [MPa] Distribution for Vehicle Dynamics'', AVEC2002 FR 1 (2002), 595-600 0.5 11) Seraji, H., Colbaugh, R. : "Improved Configuration Control for Redundant Robots'', J. Robotic Systems 0 00.511.522.53 7-6(1990), 897-928 Time [s][s] (Report received on Oct. 2, 2003) 0.5 Pr oposed System 0.3 Conventional System Yoshikazu Hattori 0.1 Year of birth : 1965 00.511.522.53

Yaw RateYa [rad/s] at d/s] Time [s][s] Division : Vehicle Control Lab. 0 Research fields : Vehicle control, Vehicle -0.2 dynamics analysis, Driver behavior -0.4 analysis 00.511.522.53 Academic society : The Soc. Instrum.

Slip Angle[rad] Time [s][s] Control Eng., Inst. Syst., Control Inform. Eng., Soc. Autom. Eng. Jpn. Fig. 8 Control results.

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