Integrated Yaw and Roll Stability Control for Sport Utility
Vehicles in Highway Turning Scenario
GE Jin, GAO Feng, XU Guoyan, SONG Xiaolin
5 (School of Transportation Science and Engineering, Beihang University, Beijing 100191) Abstract: This paper designs an integrated control strategy of vehicle yaw and rollover stability to enhance the active safety of highway vehicles in turning scenario. A rollover predictor is built based on Time-To-Rollover index. Modified Linear Quadratic Regulation is used in roll stability control, and Partial-Integral design is used in yaw stability control. With Carsim and MATLAB, simulations of
10 fishhook maneuver and double lane-change test are conducted to evaluate the effectiveness of the control configuration and to validate the control method. The result shows that this controller performs well in maintaining vehicle roll and yaw stability. This study provides a theoretical basis towards a unified chassis control of coupled roll and yaw stability.
Keywords: vehicle dynamics; yaw stability control; roll stability control; simulation; Carsim
Rollover has been one of the important causal factors in highway casualty despite of its small
percentage in the number of accidents. Of the nearly 11,000,000 passenger car, sport utility
vehicle, pickup, and van crashes in 2002 in the USA, only 3 percent is produced directly by
20 rollovers; however, these rollovers accounted for nearly 33 per cent of all deaths. Therefore, rollover prevention is meaningful and very useful in terms of enhancing transportation safety and
Vehicle rollover can be categorized into two types: un-tripped and tripped rollover. Tripped rollover happens after the vehicle loses stability and slides on the road. Once there is disturbance
25 from the road, such as encountering a small bump or hitting road curb, the vehicle is highly possible to roll over. Therefore tripped rollover should be prevented by decreasing the possibility of losing control at the first place. Different from tripped rollover, un-tripped rollover may happen before a driver realizes that the vehicle has lost control, and thus requires more attention. It may be induced by sharp cornering under high velocity, emergent obstacle avoidance, or severe
30 lane-change maneuvers.
Vehicles with relatively high center of gravity are more subject to the risk of rollover, especially Sport Utility Vehicles. While SUVs are very popular in consumers, their static rollover index (ratio of C.G. height to half of the track width) is higher than those of sedans. The rollover may happen when the vehicle is avoiding unexpected obstacles on highway, which is not an
35 uncommon incident. Through reducing the lateral acceleration and the vehicle roll angle, the performance of vehicle roll stability can be enhanced. Various control techniques including active steering, differential braking, and active suspension have been used in vehicle rollover avoidance
[3-5]during the past decades, but the configurations and algorithms mainly focus on the roll stability control. Thus significant improvements can be expected for the integrated control of yaw and roll
This paper presents an integrated controller of vehicle yaw and roll stability, which design methodology is based on differential braking and active steering. Predicted load transfer ratio is utilized as measurement of rollover risk. Modified LQR design is used in roll stability control. The
control performance of proposed algorithm is investigated by combined simulation on Carsim and
Foundations: National Natural Science Foundation (No. 51105021)
Brief author introduction:GE Jin, (1990-),female,graduate student, intelligent vehicle.
Correspondance author: Gao Feng, (1955-),Male, professor, intelligent vehicle. E-mail: email@example.com
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45 Matlab. The organization of this paper is as follows: In section 1, 2-dof and 3-dof vehicle models
are built. In section 2, the general control strategy is designed. Yaw and roll stability controller
design is presented respectively. In section 3, the proposed control algorithm is testified in
simulated fishhook maneuver and double lane change test. Finally, section 4 provides some
50 1 Vehicle models
In this section, both 2-degree-of-freedom (2-dof) bicycle and 3-dof yaw-roll models are built
and verified. The models ignore vehicle suspension characteristics and vertical motion. Also the
longitudinal velocity is assumed to be constant. Standard coordinate systems and units are used
here. The meaning of variables are in accordance with those in.
55 1.1 Bicycle Model
The 2-dof bicycle model is widely used in vehicle yaw stability control. Combined with
steering angle input, this bicycle model will be used as reference model in determining vehicle
yaw stability status.
60 Fig. 1 Two-dof bicycle model
The bicycle model is expressed in state-space form. (1)
+ C l C?l C C ? ? C ? ?f r f f r r f ? 1? ? ? ? ?2 mv mv mvβ ?? ? ?β x ? ?? ? δ=? ? ? ?
+ 22 ? ? x x ? ? r(1) rl C ?l c l C + l c l C
f f ? ? ? ? f f r r f f r ? = ? ?? ?
r I v I ? I ? z xzz??
In which β is yaw rate, r is slip angle, C is front axle cornering stiffness, Cis rear axle f r
65 cornering stiffness, vis vehicle longitudinal velocity, l is the distance between front axle and x f
center of gravity, lis the distance between rear axle and center of gravity,I is the moment of r z
inertia about yaw axis, m is the vehicle mass, δ is the steering angle of steering wheel. The cornering stiffness of front and rear axle are estimated using the static response of a
given SUV Carsim model. Given a reasonable initial value, the Nelder-Mead unstrained nonlinear 70 optimization method is employed to minimize the error of response (2) between the bicycle model
and given Carsim model.
β? βr? r 2dofCARSIM CARSIM 2dof J = α dt + α (2) dt 1 2 ?? max βmax rCARSIM CARSIM r, r,α αCARSIM 2 dof J 1 2 In (2), is the cost function, are the weighing factor, are respectively the yaw β, β CARSIM 2 dof rate of Carsim and bicycle model, are respectively the side slip angle of Carsim and 75 bicycle model.
1.2 Yaw-Roll model
The 3-dof model considering roll movement is a simple yet most commonly used vehicle
model in rollover prevention as in Fig. 2. This model is developed from the Euler-Lagrange
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80Fig. 2 Three-dof model This yaw-roll 3-dof model is presented as follows:
X= AX + B u(3)1 1 0 0 ?mhmv?? xs e
T ? ? ?
I ?1?1xs ?= β X 85 ? ?= δ u , = E V , ,B r φ φ A = E U , E = In which: ?I 0mhv ,xzs s e x
1 1 ? ? ?0I 0 I ?
1 z xz ? ? ?(l C ? l??0 0 1 0f f r ?(C + + mv0 0 ? ?f x ?C ? ? f C)C)r r ? v? x
? ? ??m h v C ? m gh K ? 0 0 φ2φ s e ?? ?(l C U = ? ? ? .V = , s e x ? ? 2 1 + l C )???l C
f f ?(l C ?l C )0 0 ? ? f f?
? ? 0 f f r r v ? ??? ? ?1 ?x ? 0 0 0 1.3 Model Verification
The 3-dof yaw-roll model is verified in a procedure of step steering input. The vehicle
longitudinal velocity is set to be 80 km/h, and the steering wheel steer angle is 50 deg.
90 The simulation results is shown in Fig. 3-6. The lateral acceleration, roll angle, yaw rate, and
side slip angle of 3-dof model agree well with those of the tested Carsim model.
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Fig. 3 Comparison of vehicle lateral acceleration between the two vehicle models under step steer maneuver
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95 Fig. 4 Comparison of vehicle yaw rate Fig. 5 Comparison of vehicle roll angle
2 Integrated Controller Design
The control strategy includes four main subsystems: yaw stability controller (YSC), roll stability controller (RSC), tire force allocation, and ESC-RSC enabling block. RSC is activated by
100 rollover index, and YSC is activated by slip angle and yaw rate error. Tire force allocation block
assigns tire forces to four wheels after the desired yaw moment is computed. The desired yaw
moment is computed using weighed sum method as presented in (4):
M = M + αM α1YSC 2 RSC (4)
In which the weighing factors αandα show the comparative importance between yaw 1 2
105stability control and roll stability control. They are computed as follows:
λYSCα= 1 λ+ λYSC RSC (5)
λ RSCα = 2 λ + λYSC RSC (6)
λandλare the individual importance of yaw stability control and roll stability control. YSC RSC
They can be determined as in (7) and (8):
* βr ?r λ= max ? ?YSC ?rβ??maxmax ? ?110 (7) TTR λ= 1 ?RSC TTR threshold (8) In YSC, when there is over steer tendency, the outer front wheel will increase brake pressure;
when there is under steer tendency, the inner rear wheel will increase brake pressure. In RSC, the brake forces are distributed among four wheels according to the sign of steering wheel angle and
desired yaw moment. Also the sudden jump or dive of brake pressure and potential lock braking of 115
wheels are carefully avoided in this allocation process. The general control scheme is presented in
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Fig. 6 RSC/YAC based integrated control strategy 120 2.1 Yaw Stability Controller Design When rollover index is below the set threshold, maintaining desired yaw stability is given the
higher priority. It can be achieved through monitoring the yaw rate error and the slip angle and thus generating direct yaw moment.
Since actuators will be activated before tire slip ratio approaches non-linear area, linearized 125
tire model is sufficient to be used in the controller design. The stability of linear vehicle model has
been validated in many existing publications.
The dynamic equation of yaw motion can be expressed as following: 2 2 C ?l Cl C + lCl l C 1 f f r rf r rff f r ?δ + Mr=β + (9) z I I vI I z z x z z
130 Thus the steady-state yaw rate rcan be written as follows: ide v/ l x r= δ(10) ide 2 1 + kv x m(l C ? lC) f f r r k = 22C C l f r in which The yaw rate error is defined as: e= r ?r (11) r ide
135 One PI controller is designed to decrease the yaw rate error. Similarly, another PI controller is
designed to reduce the side slip angle. And the overall direct yaw moment is presented as follow:
M = K e+ K edt + K e+ Ke(12) pr r ir r p β β iβ β ? ?
2.2 Rollover Predictor Design In this section, a rollover prediction algorithm is proposed to predict the rollover risk and
140 activate the rollover controller, and subsequently a roll stability controller is built using modified
Based on the 3-dof yaw-roll model built in section 2, the state-space model used in rollover
control is written as follows: X= AX + Bu(13)
?C 0? ?f ? ?0 1 ? ?V = ???l C 0 f f T ? ? ??1?1X = βr φ φ [ ] A = E UB = E V ?145 in which ， , , , ,? ? ,other
u = δ M 0 0 z
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parameters are the same as in section 1.
Load Transfer Ratio is used as the rollover index here. It is defined as follow:
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F?F zr z l LTR = ............. ...................................................(14)F+ Fzr zl Vehicle roll motion equations can be expressed as follows:
Cφ ?mgh φ = mah?φ ey e?150 (15) ? (F?F )t / 2 ?mgh φ = mah?zr zl ey ?
Thus the rollover index can be written as:
2 ?? mh 2 h e = + LTR a?? (16)y ? ?t g C?mgh φ e ? ?
To control vehicle rollover more effectively, it is desired to avoid rollover before the accident
happened. While low rollover threshold can activate corresponding controller early, and thus
lessens the times of accident, this method brings too much intervention to vehicle dynamics, and 155
may affect vehicle maneuverability. Therefore a rollover prediction algorithm is necessary. This algorithm is also based on the 3-dof yaw-roll model. The rollover risk is evaluated by
TTR(Time To Rollover). When TTR is no less than the threshold of 1 second, the vehicle is considered as roll stable. The rollover index threshold is set to be LTR = 0.8. Given the current
steering wheel angle and direct yaw moment, future LTR can be calculated until the predicted 160
time or the LTR reaches given threshold. The algorithm flow chart is presented in Fig. 7:
Fig. 7 Rollover Prediction Flow Chart
165 2.3 Rollover Stability Controller Design When TTR is smaller than the set threshold of one second, the rollover controller is engaged. Optimal control can produce satisfying control results by minimizing or maximizing cost  function. LQR method is widely used in vehicle dynamics control. To avoid the difficulty of assigning state weighing matrix and to enhance the controller performance, the rollover index is
expressed as quadratic performance index. Thus LTR is further written as: 170
LTR = y= CX + Du(17) RI RI RI
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T vA?h A??x 11 e 41 ? ?v( A+ 1) ? hAx 12 e 42 ?? ? ? C2h2hφR R D = ?h B v B ? h B Av B ] A+ ?h A[ vIn which:C= ?? . and , e 41x 12e 42 ij RI x 11x 13 e 43 RI t ? g m h t ? g
? ?s R
? ? K v A + ?h A φ ? ?
x 14 e 44 mh s R
Bare elements of matrix A and B in the 3-dof state-space vehicle model.ij
The cost function can be written as:
? ? T T T T T = [ yQy+ uRu]dτ = [ xQx+ 2 xNu+ uRuJ ]dτ175(18) R RIRI τ τ τ τ τ τ τ τ ? () () ? () () () () () () t 0t 0 T T T Q = CQC; N = CQD; R = DQD + R In which: . Q and are state weighing matrices set inR
Q the beginning of calculation. and R are symmetric matrixes. Because the physical meaning of
these weighing matrices are very clear, it is more convenient to assign values to Q and . R
By solving Algebraic Ricatti Equation (19), the optimal control input can be acquired as in 180 (20). ?1 T T ?1 T ?1 T ?1 T P( A ?BR N ) + ( A? NRB)P ? PBRBP + Q ? NRN = 0(19)
?1T Tu* = ? R( BP + N ) x= ? Kx(τ ) (τ ) (20) With the parameters used in this paper, the controlled system is observable and controllable. Thus the closed-loop control system is asymptotically stable.
185 3 Simulation Simulated tests are carried out to verify the proposed integrated control algorithm. 3.1 Simulation of open-loop fishhook maneuver The fishhook steer input is used in this simulation. In the standard NHTSA fishhook maneuver, tested vehicle is driven at a target speed with given steering wheel commands.
190 As Fig. 9 and Fig. 10 show, the spikes of yaw rate and roll angle are reduced, and the vehicle side slip angle is significantly decreased in Fig. 12. This demonstrates that with proposed control
method, the lateral stability is prominently improved while the roll stability is maintained. The trajectories of controlled and uncontrolled vehicle in Fig. 8 further illustrate that this controller
design helps to stabilize vehicle when there is a sharp steering input. 195 Fig. 8 Trajectories of controlled and uncontrolled vehicle. (The yellow vehicles are uncontrolled, and the red ones are controlled.
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200 Fig. 9 Comparison of vehicle yaw rate Fig. 10 Comparison of Vehicle roll angle Fig. 11 Comparison of vehicle lateral acceleration Fig. 12 Comparison of vehicle slip angle Fig. 13 Velocity of controlled and uncontrolled vehicle 205 3.2 Simulation of closed-loop double lane change With the proposed stability control algorithm, simulated vehicle can pass the Double
Lane-Change test without spinning out or rolling over, while simulated vehicle without control exhibits lower maneuverability and higher propensity to lose lateral and roll stability. (See Fig.
In Fig. 15, the controlled lateral acceleration is clearly decreased, and thus reduces the risk of
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