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Study on a vehicle dynamics model for improving roll stability
JSAE Review 24 (2003) 149–156
Study on a vehicle dynamics model for improving roll stability Shuichi Takanoa, Masao Nagaib, Tetsuo Taniguchic, Tadashi Hatanoc a
Graduate School, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei-shi, Tokyo 184-8588, Japan b Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei-shi, Tokyo 184-8588, Japan c National Traffic Safety and Environment Laboratory, 7-42-27 Jindaijihigashi, Choufu, Tokyo 182-0012, Japan Received 25 November 2002; received in revised form 24 December 2002
Abstract In this paper, a three-degree-of-freedom model is employed for computer simulation to determine the relationship between the planar and roll motions of a large-size vehicle, so that the roll motion could be eventually predicted to prevent the vehicle from going dynamically unstable. Factors such as the varying center of gravity height and roll steer are also taken into consideration. An experiment is conducted using an actual truck with and without load to investigate its effect on the dynamics of the vehicle. r 2003 Society of Automotive Engineers of Japan, Inc. and Elsevier Science B.V. All rights reserved.
1. Introduction
2. Vehicle model
In recent years, improvement in highway systems has enabled trucks and buses to travel at higher speeds. As a result, there is now a strong need for developing safer versions of these vehicles that can be driven even by inexperienced drivers. For passenger cars, much research is being conducted to control their motions to provide more safety [1,2]. However, there are not as many studies for large-size vehicles. Accidents involving large vehicles are often hazardous to motorists. In 2001, 86% of the people killed in accidents involving large-size trucks in the United States were those who were not the driver or passengers of the trucks [3]. Therefore, methods for predicting the motions of these vehicles and preventing them from entering unstable states are being strongly called for. One factor that distinguishes trucks and buses from passenger cars is their high and varying centers of gravity, which could cause them to roll over in extreme conditions. To predict this behavior, understanding of the vehicle’s roll motion and what causes it to be unstable is important. In this report, a vehicle model with three degrees of freedom is employed to determine how the planar and roll motions of a large vehicle affect each other. Also, the dynamic effects of the center of gravity height, roll stiffness and roll steer are clarified. Finally, an experiment is conducted to confirm the effects of changes in the e.g. height.
A truck with total mass of 7600 kg is represented by a vehicle model with three degrees of freedom including the side slip angle b; yaw rate r and roll angle f: This model is shown in Fig. 1 and used for computer simulation. Taking the roll angle into consideration enables examination of its dynamic characteristics, which is not possible with the conventional ‘‘bicycle’’ model. Dynamics of this model can be represented by Eqs. (1)–(3). Lateral acceleration: . ¼ 2Ff þ 2Fr : mVðb’ þ rÞ ms hs f ð1Þ Yaw motion: I r’ Ixz f. ¼ 2lf Ff þ 2lr Fr :
ð2Þ
Roll motion: . ms hs Vðb’ þ rÞ Ixz r’ ¼ SMx : If f
ð3Þ
In the above equations, m stands for the total vehicle mass, ms the sprung mass, and I and If the yaw and roll moments of inertia, respectively. Ixz is the product of inertia with respect to the x and z directions in Fig. 1 and is assumed to be zero in this report. lf and lr are the distances from the center of gravity to the front and rear axle, respectively. Ff and Fr are the lateral forces produced by each of the front and rear tires, respectively. hs is the length of the roll moment arm, which is the distance from the vehicle roll center to the center of
0389-4304/03/$30.00 r 2003 Society of Automotive Engineers of Japan, Inc. and Elsevier Science B.V. All rights reserved. doi:10.1016/S0389-4304(03)00012-2
JSAE20034111
S. Takano et al. / JSAE Review 24 (2003) 149–156
150
lfr
βf
y r
V β
lr r
2 Ff
lf
βr
x’ ¼ Ax þ Bu;
φ
φ << 1 sin φ ≈ φ
− Kφ φ . −Cφφ
hs
ms g
y
ms g / 2
2
abb
6 6 arb A¼6 6a 4 fb
abr
abf’
arr
arf’
afr
aff’
0
1
0
abf
3
7 arf 7 7; aff 7 5 0
0
RC
Fr
ð9Þ
where 2 3 b 6 7 6r7 7; u ¼ d; x¼6 7 6f 4’5 f 2 3 bb 6b 7 6 r7 B¼6 7 4 bf 5
lr
2 Fr
z
the roll angle of the vehicle. Fig. 2 represents the block diagram of this 3-DOF vehicle model. Table 1 contains the parameters used for this model. Since Eqs. (1)–(6) are too complicated for analyzing the characteristics of the vehicle dynamics, they are expressed using the state equations below:
x
δ
In the matrix A, aij where iaj is a coupling term between the terms i and j. When the coupling terms involving the roll and planar motions, i.e., abf’ ; abf ; arf’ ;
Fr ms g / 2
Fig. 1. Three-DOF vehicle model. “ Bicycle” model
gravity. Mx stands for the roll moment acting on the sprung mass. Finally, V represents the vehicle speed and g the gravitational acceleration. The tire model and roll moment are described as follows: Tire model: Ff ¼ Kf ðb þ lf r=V d af Þ;
ð4Þ
Fr ¼ Kr ðb lr r=V ar Þ:
ð5Þ
δ
2 Kf
+ + + +
@ar f; ar ¼ @f
where @af =@f and @ar =@f are the roll steer angle of the front and rear wheels for a unit roll angle. These roll steer coefficients are assumed to be constant in this study, so that the amount of roll steer is proportional to
2( l f K f − l r Kr ) − mV + V
Yaw moment
+
− 2(l f2 K f + l r2 Kr )
+ + +
2lfKf
. r
1 I
+
+
+
Possible yaw moment control
+
+ + + +
Roll steer moment
Roll steer force
+
+
+ Possible roll moment control
V 1 S
r
I xz
+
+
ð8Þ
β
S
− 2(l f K f − lr Kr )
ð6Þ
Note that Kf and Kr are the cornering stiffness of the front and rear tire, respectively. They are assumed to be linear with respect to the vertical loads on the wheels. The steer angle of the front wheels is denoted by d: Kf is the roll stiffness and Cf is the roll damping coefficient. af and ar are the roll steer of the front and rear wheels, respectively, and can be described by the following equations: @af f; ð7Þ af ¼ @f
mV
+ +
Roll moment: ’ SMx ¼ ðKf þ ms ghs Þf Cf f:
Lateral force
− 2 ( K f + Kr ) . β 1 1
+
+ m hV s s
+
+
− ( K φ − ms ghs )
− Cφ 1 Iφ
1 . S φ
.. φ
1 S
Roll moment
Ixz m s hs
∂ α f ∂α 2 l f K f − r lr K r ∂φ ∂φ
∂α ∂α r 2 f Kf + K r ∂φ ∂φ
Fig. 2. Block diagram for 3-DOF vehicle model.
φ
S. Takano et al. / JSAE Review 24 (2003) 149–156
151 “Bicycle” model
Table 1 Vehicle parameter Definition
Symbol Unit
Vehicle mass Yaw moment of inertia Roll moment of inertia Roll moment arm Distance from front axle to c.g. Distance from rear axle to c.g. Front cornering stiffness (without load) Rear cornering stiffness (without load) Roll stiffness Roll damping coefficient Gravitational acceleration
Value (Loaded)
δ
m I If hs lf lr Kf
kg kg m2 kg m2 m m m N/rad
4665 (7660) 13500 (24000) 5000 (15000) 0.4813 (0.9499) 1.716 (2.056) 2.014 (1.674) 22750 (32750)
Kr
N/rad
34500 (46000)
Kf Cf g
a βr
+
2
arf
ð12Þ
m2s h2s Þ
ð13Þ ms hs IKf þ ms hs IKr þ Ixz mlf Kf Ixz mlr Kr ; 2 m þ I mI m2s h2s I Ixz f ð14Þ
afr ¼ 2
a rφ a
. rφ
+
+
+ + −
1 s + a φ φ.
. φ
1 s
φ
aφφ
Fig. 3. Alternative block diagram for 3-DOF vehicle model.
3. Theoretical analysis and computer simulation
3 @af @ar lf Kf lr Kr Ixz Kf m 7 6 2ðIf m @f @f 6 7 6 7 4 5 @af @ar Kf þ Kr ms hs þ Ixz ms ghs m þ2Ixz @f @f ; ¼ 2 m þ I mI m2s h2s I Ixz f
afb ¼ 2
+
ð10Þ
3 @af @ar 2 K K 2ðI I I Þ þ h K I m f r s s f 7 6 f xz @f @f 6 7 6 7 4 5 @af @ar 2 2 lf K f lr Kr Ixz þ ms hs gI þ2ms hs @f @f ; ð11Þ ¼ 2 2 2 m þ I mIÞV ðms hs I Ixz f Ixz Cf m ; 2 m þ I mI m2s h2s I Ixz f
r
aφ r
2
arf’ ¼
a βφ
1 s + a rr
+ + +
br
aφβ
arf ; afb and afr ; are all zeroes, the roll motion is independent of the planar motions. By examining these terms, the relationship between the roll and planar motions can be analyzed. These coupling terms are expressed as the following:
abf
a βφ.
a rβ
N m/rad 300000 N m s/rad 22800 9.81 m/s2
m s hs C f I ; 2 m þ I mIÞ Vðm2s h2s I Ixz f
+ +
+ +
bφ
abf’ ¼
β
1 s + a ββ
+ +
bβ
ms hs Ilf Kf ms hs Ilr Kr þ Ixz mlf2 Kf þ Ixz mlr2 Kr : 2 m þ I mIÞ Vðm2s h2s I Ixz f ð15Þ
Fig. 3 shows a block diagram representing Eq. (9).
In this section, the effects of the changes in the coupling terms of matrix A are examined. 3.1. Interactions between side slip angle and roll angle Equation (10) indicates that abf’ is dependent on the roll damping coefficient Cf ; roll moment arm hs and sprung mass ms. Eq. (11) shows that abf becomes zero if the effect of roll steer is neglected and Kf ¼ ms hs g; provided Ixz is assumed to be zero as in this study. However, Kf is typically several times the value of mshsg, so that abf is never zero without a means of controlling vehicle dynamics. Additionally, Eq. (14) shows that afb is usually not zero if Ixz is zero. Large-size vehicles, especially trucks, have varying center of gravity heights depending on the amounts of load they carry. As the e.g. height changes, the length of the roll moment arm hs also varies. Fig. 4 shows how the values of the coupling terms are affected by changes in hs; those not affected are excluded. The absolute values of abf ; abf’ ; and afb increase in accordance with hs. Also, the figure shows the values of these terms when Kf and Cf are varied. Kf and Cf influence the values of abf and abf’ ; respectively. Their absolute values increase with Kf and Cf : Thus, the relationship between the side slip angle and roll angle depends on the c.g. height and suspension parameters such as Kf and Cf : Figs. 5 and 6 show the time response simulations of the vehicle during a lane change maneuver for various values of hs and Kf : Graphs with various values of Cf are excluded, as the results did not show sufficient changes. In these graphs, the roll angle obviously shows greater peak values when
S. Takano et al. / JSAE Review 24 (2003) 149–156
152
2
aφ r
δ [deg]
0.05 0
aβ φ.
Cφ
- 0.05
1 0 -1 -2
- 0.1
0
1
2
3
4
5
6
7
aφβ
β [deg]
aβφ , aφ r [s-1 ]
- 0.15
aφβ [s-2]
aβ φ [10*unit]
2
- 0.2
Kφ
- 0.25
1 0
-2
- 0.3
r [deg/s]
aβφ
- 0.4 - 0.45 0.2
0.4
0.6
0.8
δ [deg]
2 1
β [deg]
5
6
7
0
1
2
3
4
5
6
7
4
5
6
7
φ [deg]
Kφ
0
Kφ
-2 0
1
2
3
Fig. 6. Lane-change simulation with various values of roll stiffness Kf (increments for Kf : 710% of the std. value; @af =@f ¼ 0:1; @ar =@f ¼ 0:1; V ¼ 120 km/h).
0 -1 0
1
2
3
4
5
6
7
hs is higher and Kf is lower. The side slip angle response is affected as well but to a lesser degree. Also, the side slip angle and roll angle have approximately the same amounts of phase delay.
1
hs
0
hs
-1 0
1
2
3
4
5
6
7
5
r [deg/s]
4
Time [sec.]
2
0
0
1
2
3
4
5
6
7
4
5
6
7
4
φ [deg]
3
2
-4
2
hs
0
hs
-2 -4
2
4
Fig. 4. Values of elements in matrix A when vehicle parameters are changed (increments: 720% of normal value for Cf and Kf ; V ¼ 120 km/h).
-5
1
0
-5
1
Roll moment arm h s [m]
-2
0
5
- 0.35
-2
Kφ
-1
0
1
2
3
Time [sec.] Fig. 5. Lane-change simulation with various values of roll moment arm hs (increments for hs: 70.3 m; @af =@f ¼ 0:1; @ar =@f ¼ 0:1; V ¼ 120 km/h).
3.2. Interactions between yaw rate and roll motion Since Ixz is assumed to be zero in this study, arf’ and arf are zero by Eqs. (12), (13) and (15) if roll steer is ignored. Therefore, afr is the only coupling term related to the yaw rate and roll angle in this case. Furthermore, Fig. 4 shows that afr depends somewhat on hs but not on Kf or Cf : It is also notable that afr becomes zero when lf Kf ¼ lr Kr or when the vehicle is in neutral steer. Figs. 5 and 6 confirm that neither hs nor Kf significantly affect the yaw rate of the vehicle during a lane change. The figures also indicate that the phase delay of the yaw rate with respect to the front steer angle is small compared to that of the side slip angle and roll angle. 3.3. Effects of roll steer coefficients In the previous sections, the roll steer coefficients were set at constant values. Among the coupling terms in the
2
1
1
0 -1 -2 0
2
4
5
6
7
0
0
∂ αf
-2
∂φ 2
3
5
6
1
2
3
7
4
5
6
7
5
6
7
6
7
6
7
∂α r ∂φ 0
1
2
3
4
5
∂ αf ∂φ ∂α f
-5
∂αf ∂φ
∂φ 1
2
3
4
4
6
7
-2 1
2
4
0
1
∂α r ∂φ 2
3
4
5
4
0
5
6
7
Time [sec.] Fig. 7. Lane-change simulation with various values of front rollsteer coefficient (value for @af =@f: 0.2, 0.1, 0, 0.1, and 0.2; @ar =@f ¼ 0; V ¼ 120 km/h).
matrix A, however, abf and afr are influenced by changes in the roll steer coefficients. The effect of front roll steer on dynamics of the vehicle performing a lane change maneuver is shown in Fig. 7, and that of rear roll steer is depicted in Fig. 8. As roll understeer increases with @af =@f becoming more negative or @ar =@f becoming more positive, the phase delay of each variable and the height of the first peak decreases, while the height of the second peak increases.
4. Experimental results 4.1. Static analysis of a vehicle on a tilting platform Experiment was conducted to confirm the effects of varying the center of mass, and therefore hs, of a truck. A loaded truck with mass of 7600 kg is put on a tilting platform. Both the tilt angle of the platform and the roll angle of the vehicle can then be recorded to give a graph of the vehicle roll angle for a given lateral acceleration acting on the vehicle. The load was then removed from the truck, and the procedure was repeated. Fig. 9
∂α r
∂ αr ∂φ
∂φ
-2 -4
3
∂α r ∂φ
-5
2
∂ αf ∂φ
0
0
-10
5
∂αf ∂φ
2
-4 0
4
0
-2 -4
1
0
-10 0
β [deg]
2
5
φ [deg]
-1
2
-4 0
r [deg/s]
3
153
0
-2
1
r [deg/s]
β [deg]
δ [deg]
2
φ [deg]
δ [deg]
S. Takano et al. / JSAE Review 24 (2003) 149–156
0
1
2
3
4
5
Time [sec.] Fig. 8. Lane-change simulation with various values of rear rollsteer coefficient (value for @ar =@f: 0.2, 0.1, 0, 0.1, and 0.2; @af =@f ¼ 0; V ¼ 120 km/h).
indicates the results. The figure clearly shows that the roll angle for any given lateral acceleration is larger when the truck has the some load. The figure also shows the results obtained from simulation with the 3-DOF model. They largely agree with the experimental values. 4.2. Static analysis of a traveling vehicle Fig. 9 also shows similar data taken from the vehicle used in the previous section traveling with a constant turning radius of 70.8 m. Note that results from two different test methods largely agree. 4.3. Dynamic analysis of a vehicle performing a lanechange maneuver The previous two experiments were static. To examine the behavior of the roll and other motions in dynamic situations, the truck in the previous sections was employed to perform a lane change experiment. The truck was driven at 80 km/h and steered to move 3.6 m in the lateral direction while traveling 50 m. Five to six trials per condition were carried out. Fig. 10 shows
S. Takano et al. / JSAE Review 24 (2003) 149–156
154
Roll angle [deg]
8 7
Without load (Tilting platform)
6
Without load (Traveling vehicle; Right turn) Without load (Traveling vehicle; Left turn)
5
Without load (Simulation)
4
With load (Tilting platform)
3
With load (Traveling vehicle; Right turn)
2
With load (Traveling vehicle; Left turn) With load (Simulation)
1 0
0
2
4
6
Lateral acceleration [m/s2] Fig. 9. Roll angle of a truck for given lateral acceleration. 4
Steer angle δ [deg]
Steer angle δ [deg]
5
0 Experiment Simulation
-5 0
1
2
3
4
5
6
8
Yaw rate r [deg/s]
10 0
2
3
4
5
6
7
8
Roll angle φ [deg]
4 2 0
Lateral Acceleration ay [m/s 2]
3
4
5
6
7
8
2 0
-2
-4 0
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
2
2
4
-5
-4
1
3
5
-10
0
2
0 -5
-2
4
1
10
Roll angle φ [deg]
1
0
10 5
-10 -15
-10
Lateral Acceleration ay [m/s2]
Yaw rate r [deg/s]
20
-20 0
0 -2 -4
7
Experiment Simulation
2
1 0 -1 -2
Time [sec] 1
2
3
4
5
6
7
8
Time [sec] Fig. 10. Response of a truck without load during lane-change maneuver (V ¼ 80 km/h)
typical results for the truck without load. Fig. 11 is a similar set of results for the truck with load. These indicate that the roll angle is considerably higher for a given lateral acceleration when the vehicle has some load, and therefore higher hs, which agrees with the previous theoretical analysis. Also, the yaw rate for a given steer angle largely stays the same regardless of the load, which also agrees with the analysis. Figs. 10 and 11 also show the results obtained by simulation using the 3DOF model developed in this study. The simulation agrees well with the experiment.
Fig. 11. Response of a truck with load during lane-change maneuver (V ¼ 80 km/h).
To better clarify the relationship between the steer angle input and output variables during a lane change, the first peak is defined as peak A, and the second peak as peak B. Refer to Fig. 12 for details and nomenclatures. The peak values of the input and output and the output’s delay with respect to the input are then examined. Figs. 13–15 are the peak values of various output variables with respect to the peak value of the front steer angle input during a lane change. In Fig. 15, it is clear that the roll angle peak values are severely affected by the load and therefore, a change in hs : Figs. 13 and 15 respectively show that the peak values of the yaw rate and lateral acceleration are not significantly influenced by the load.
S. Takano et al. / JSAE Review 24 (2003) 149–156
155
Peak A Output
Input
Peak B Peak B delay
Peak value of lateral acceleration [m/s 2 ]
4
Peak A delay
3.5 3
Without load (PeakA)
2.5
Without load (PeakB)
2
With load (PeakA)
1.5
With load (PeakB)
1 0.5
Fig. 12. Definitions of the peaks.
0
0
2
4
6
Peak value of steer angle [deg] Fig. 15. Peak value of the lateral acceleration with respect to the peak value of the front steer angle.
16 14
Without load (PeakA)
12
Without load (PeakB)
0.9
10
0.8
8 With load (PeakB)
6 4
0.6 0.4 0.3 0.1
0
2
4
6
0
Peak value of steer angle [deg] Fig. 13. Peak value of the yaw rate with respect to the peak value of the front steer angle.
Yaw rate
Roll angle
Lateral acceleration
Fig. 16. Delay of peak A of various variables with respect to the front steer angle peak A.
8
0.9 0.8
7 Without load (PeakA)
6 Without load (PeakB)
5 With load (PeakA)
4
With load (PeakB)
3
Without load
0.7
Delay time [s]
Peak value of roll angle [deg]
With load
0.5
0.2
2 0
Without load
0.7
With load (PeakA)
Delay time [s]
Peak value of yaw rate [deg/s]
18
0.6
With load
0.5 0.4 0.3 0.2 0.1
2
0
Yaw rate
1 0
Roll angle
Lateral acceleration
Fig. 17. Delay of peak B of various variables with respect to the front steer angle peak B. 0
2
4
6
Peak value of steer angle [deg]
Fig. 14. Peak value of the roll angle with respect to the peak value of the front steer angle.
Figs. 16 and 17 are the delays of peaks of various variables with respect to the front steer angle peaks during a lane change. With peak A, increasing the e.g. height mainly seems to delay the roll angle. However, with peak B, increasing the e.g. height affects all of the yaw rate roll angle and lateral acceleration.
5. Conclusions 1. Simulation results show that changes in the height of the center of gravity and suspension parameters such as C/ and K/ affect the interaction between the planar motions and roll motion of a large-size vehicle. Also, the roll steer strengthens the relationship between the planar motions and roll angle of the vehicle.
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S. Takano et al. / JSAE Review 24 (2003) 149–156
2. Static analysis and experiment indicate that a change in the height of the c.g. influences the roll angle the most. However, dynamic experiment shows that it affects other variables such as the yaw rate and lateral acceleration as well in terms of delay with respect to the input steer angle. In the future, use of a vehicle model that has tires with nonlinear characteristics and a control system for suppressing the roll motion will be considered.
References [1] E.M. Elbeheiry, et al., Advanced ground vehicle suspension systems—a classified bibliography, Veh. System Dyn. 24 (3) (1995) 231–258. [2] N. Rosam, J. Darling, Development and simulation of a novel roll control system for the interconnected hydragas suspension, Veh. System Dyn. 27 (1) (1997) 1–18. [3] National Highway Traffic Safety Administration, 2001 Annual Assessment: Motor Vehicle Traffic Crash Fatality and Injury Estimates for 2001, 2001.
Study on a vehicle dynamics model for improving roll stability Shuichi Takanoa, Masao Nagaib, Tetsuo Taniguchic, Tadashi Hatanoc a
Graduate School, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei-shi, Tokyo 184-8588, Japan b Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei-shi, Tokyo 184-8588, Japan c National Traffic Safety and Environment Laboratory, 7-42-27 Jindaijihigashi, Choufu, Tokyo 182-0012, Japan Received 25 November 2002; received in revised form 24 December 2002
Abstract In this paper, a three-degree-of-freedom model is employed for computer simulation to determine the relationship between the planar and roll motions of a large-size vehicle, so that the roll motion could be eventually predicted to prevent the vehicle from going dynamically unstable. Factors such as the varying center of gravity height and roll steer are also taken into consideration. An experiment is conducted using an actual truck with and without load to investigate its effect on the dynamics of the vehicle. r 2003 Society of Automotive Engineers of Japan, Inc. and Elsevier Science B.V. All rights reserved.
1. Introduction
2. Vehicle model
In recent years, improvement in highway systems has enabled trucks and buses to travel at higher speeds. As a result, there is now a strong need for developing safer versions of these vehicles that can be driven even by inexperienced drivers. For passenger cars, much research is being conducted to control their motions to provide more safety [1,2]. However, there are not as many studies for large-size vehicles. Accidents involving large vehicles are often hazardous to motorists. In 2001, 86% of the people killed in accidents involving large-size trucks in the United States were those who were not the driver or passengers of the trucks [3]. Therefore, methods for predicting the motions of these vehicles and preventing them from entering unstable states are being strongly called for. One factor that distinguishes trucks and buses from passenger cars is their high and varying centers of gravity, which could cause them to roll over in extreme conditions. To predict this behavior, understanding of the vehicle’s roll motion and what causes it to be unstable is important. In this report, a vehicle model with three degrees of freedom is employed to determine how the planar and roll motions of a large vehicle affect each other. Also, the dynamic effects of the center of gravity height, roll stiffness and roll steer are clarified. Finally, an experiment is conducted to confirm the effects of changes in the e.g. height.
A truck with total mass of 7600 kg is represented by a vehicle model with three degrees of freedom including the side slip angle b; yaw rate r and roll angle f: This model is shown in Fig. 1 and used for computer simulation. Taking the roll angle into consideration enables examination of its dynamic characteristics, which is not possible with the conventional ‘‘bicycle’’ model. Dynamics of this model can be represented by Eqs. (1)–(3). Lateral acceleration: . ¼ 2Ff þ 2Fr : mVðb’ þ rÞ ms hs f ð1Þ Yaw motion: I r’ Ixz f. ¼ 2lf Ff þ 2lr Fr :
ð2Þ
Roll motion: . ms hs Vðb’ þ rÞ Ixz r’ ¼ SMx : If f
ð3Þ
In the above equations, m stands for the total vehicle mass, ms the sprung mass, and I and If the yaw and roll moments of inertia, respectively. Ixz is the product of inertia with respect to the x and z directions in Fig. 1 and is assumed to be zero in this report. lf and lr are the distances from the center of gravity to the front and rear axle, respectively. Ff and Fr are the lateral forces produced by each of the front and rear tires, respectively. hs is the length of the roll moment arm, which is the distance from the vehicle roll center to the center of
0389-4304/03/$30.00 r 2003 Society of Automotive Engineers of Japan, Inc. and Elsevier Science B.V. All rights reserved. doi:10.1016/S0389-4304(03)00012-2
JSAE20034111
S. Takano et al. / JSAE Review 24 (2003) 149–156
150
lfr
βf
y r
V β
lr r
2 Ff
lf
βr
x’ ¼ Ax þ Bu;
φ
φ << 1 sin φ ≈ φ
− Kφ φ . −Cφφ
hs
ms g
y
ms g / 2
2
abb
6 6 arb A¼6 6a 4 fb
abr
abf’
arr
arf’
afr
aff’
0
1
0
abf
3
7 arf 7 7; aff 7 5 0
0
RC
Fr
ð9Þ
where 2 3 b 6 7 6r7 7; u ¼ d; x¼6 7 6f 4’5 f 2 3 bb 6b 7 6 r7 B¼6 7 4 bf 5
lr
2 Fr
z
the roll angle of the vehicle. Fig. 2 represents the block diagram of this 3-DOF vehicle model. Table 1 contains the parameters used for this model. Since Eqs. (1)–(6) are too complicated for analyzing the characteristics of the vehicle dynamics, they are expressed using the state equations below:
x
δ
In the matrix A, aij where iaj is a coupling term between the terms i and j. When the coupling terms involving the roll and planar motions, i.e., abf’ ; abf ; arf’ ;
Fr ms g / 2
Fig. 1. Three-DOF vehicle model. “ Bicycle” model
gravity. Mx stands for the roll moment acting on the sprung mass. Finally, V represents the vehicle speed and g the gravitational acceleration. The tire model and roll moment are described as follows: Tire model: Ff ¼ Kf ðb þ lf r=V d af Þ;
ð4Þ
Fr ¼ Kr ðb lr r=V ar Þ:
ð5Þ
δ
2 Kf
+ + + +
@ar f; ar ¼ @f
where @af =@f and @ar =@f are the roll steer angle of the front and rear wheels for a unit roll angle. These roll steer coefficients are assumed to be constant in this study, so that the amount of roll steer is proportional to
2( l f K f − l r Kr ) − mV + V
Yaw moment
+
− 2(l f2 K f + l r2 Kr )
+ + +
2lfKf
. r
1 I
+
+
+
Possible yaw moment control
+
+ + + +
Roll steer moment
Roll steer force
+
+
+ Possible roll moment control
V 1 S
r
I xz
+
+
ð8Þ
β
S
− 2(l f K f − lr Kr )
ð6Þ
Note that Kf and Kr are the cornering stiffness of the front and rear tire, respectively. They are assumed to be linear with respect to the vertical loads on the wheels. The steer angle of the front wheels is denoted by d: Kf is the roll stiffness and Cf is the roll damping coefficient. af and ar are the roll steer of the front and rear wheels, respectively, and can be described by the following equations: @af f; ð7Þ af ¼ @f
mV
+ +
Roll moment: ’ SMx ¼ ðKf þ ms ghs Þf Cf f:
Lateral force
− 2 ( K f + Kr ) . β 1 1
+
+ m hV s s
+
+
− ( K φ − ms ghs )
− Cφ 1 Iφ
1 . S φ
.. φ
1 S
Roll moment
Ixz m s hs
∂ α f ∂α 2 l f K f − r lr K r ∂φ ∂φ
∂α ∂α r 2 f Kf + K r ∂φ ∂φ
Fig. 2. Block diagram for 3-DOF vehicle model.
φ
S. Takano et al. / JSAE Review 24 (2003) 149–156
151 “Bicycle” model
Table 1 Vehicle parameter Definition
Symbol Unit
Vehicle mass Yaw moment of inertia Roll moment of inertia Roll moment arm Distance from front axle to c.g. Distance from rear axle to c.g. Front cornering stiffness (without load) Rear cornering stiffness (without load) Roll stiffness Roll damping coefficient Gravitational acceleration
Value (Loaded)
δ
m I If hs lf lr Kf
kg kg m2 kg m2 m m m N/rad
4665 (7660) 13500 (24000) 5000 (15000) 0.4813 (0.9499) 1.716 (2.056) 2.014 (1.674) 22750 (32750)
Kr
N/rad
34500 (46000)
Kf Cf g
a βr
+
2
arf
ð12Þ
m2s h2s Þ
ð13Þ ms hs IKf þ ms hs IKr þ Ixz mlf Kf Ixz mlr Kr ; 2 m þ I mI m2s h2s I Ixz f ð14Þ
afr ¼ 2
a rφ a
. rφ
+
+
+ + −
1 s + a φ φ.
. φ
1 s
φ
aφφ
Fig. 3. Alternative block diagram for 3-DOF vehicle model.
3. Theoretical analysis and computer simulation
3 @af @ar lf Kf lr Kr Ixz Kf m 7 6 2ðIf m @f @f 6 7 6 7 4 5 @af @ar Kf þ Kr ms hs þ Ixz ms ghs m þ2Ixz @f @f ; ¼ 2 m þ I mI m2s h2s I Ixz f
afb ¼ 2
+
ð10Þ
3 @af @ar 2 K K 2ðI I I Þ þ h K I m f r s s f 7 6 f xz @f @f 6 7 6 7 4 5 @af @ar 2 2 lf K f lr Kr Ixz þ ms hs gI þ2ms hs @f @f ; ð11Þ ¼ 2 2 2 m þ I mIÞV ðms hs I Ixz f Ixz Cf m ; 2 m þ I mI m2s h2s I Ixz f
r
aφ r
2
arf’ ¼
a βφ
1 s + a rr
+ + +
br
aφβ
arf ; afb and afr ; are all zeroes, the roll motion is independent of the planar motions. By examining these terms, the relationship between the roll and planar motions can be analyzed. These coupling terms are expressed as the following:
abf
a βφ.
a rβ
N m/rad 300000 N m s/rad 22800 9.81 m/s2
m s hs C f I ; 2 m þ I mIÞ Vðm2s h2s I Ixz f
+ +
+ +
bφ
abf’ ¼
β
1 s + a ββ
+ +
bβ
ms hs Ilf Kf ms hs Ilr Kr þ Ixz mlf2 Kf þ Ixz mlr2 Kr : 2 m þ I mIÞ Vðm2s h2s I Ixz f ð15Þ
Fig. 3 shows a block diagram representing Eq. (9).
In this section, the effects of the changes in the coupling terms of matrix A are examined. 3.1. Interactions between side slip angle and roll angle Equation (10) indicates that abf’ is dependent on the roll damping coefficient Cf ; roll moment arm hs and sprung mass ms. Eq. (11) shows that abf becomes zero if the effect of roll steer is neglected and Kf ¼ ms hs g; provided Ixz is assumed to be zero as in this study. However, Kf is typically several times the value of mshsg, so that abf is never zero without a means of controlling vehicle dynamics. Additionally, Eq. (14) shows that afb is usually not zero if Ixz is zero. Large-size vehicles, especially trucks, have varying center of gravity heights depending on the amounts of load they carry. As the e.g. height changes, the length of the roll moment arm hs also varies. Fig. 4 shows how the values of the coupling terms are affected by changes in hs; those not affected are excluded. The absolute values of abf ; abf’ ; and afb increase in accordance with hs. Also, the figure shows the values of these terms when Kf and Cf are varied. Kf and Cf influence the values of abf and abf’ ; respectively. Their absolute values increase with Kf and Cf : Thus, the relationship between the side slip angle and roll angle depends on the c.g. height and suspension parameters such as Kf and Cf : Figs. 5 and 6 show the time response simulations of the vehicle during a lane change maneuver for various values of hs and Kf : Graphs with various values of Cf are excluded, as the results did not show sufficient changes. In these graphs, the roll angle obviously shows greater peak values when
S. Takano et al. / JSAE Review 24 (2003) 149–156
152
2
aφ r
δ [deg]
0.05 0
aβ φ.
Cφ
- 0.05
1 0 -1 -2
- 0.1
0
1
2
3
4
5
6
7
aφβ
β [deg]
aβφ , aφ r [s-1 ]
- 0.15
aφβ [s-2]
aβ φ [10*unit]
2
- 0.2
Kφ
- 0.25
1 0
-2
- 0.3
r [deg/s]
aβφ
- 0.4 - 0.45 0.2
0.4
0.6
0.8
δ [deg]
2 1
β [deg]
5
6
7
0
1
2
3
4
5
6
7
4
5
6
7
φ [deg]
Kφ
0
Kφ
-2 0
1
2
3
Fig. 6. Lane-change simulation with various values of roll stiffness Kf (increments for Kf : 710% of the std. value; @af =@f ¼ 0:1; @ar =@f ¼ 0:1; V ¼ 120 km/h).
0 -1 0
1
2
3
4
5
6
7
hs is higher and Kf is lower. The side slip angle response is affected as well but to a lesser degree. Also, the side slip angle and roll angle have approximately the same amounts of phase delay.
1
hs
0
hs
-1 0
1
2
3
4
5
6
7
5
r [deg/s]
4
Time [sec.]
2
0
0
1
2
3
4
5
6
7
4
5
6
7
4
φ [deg]
3
2
-4
2
hs
0
hs
-2 -4
2
4
Fig. 4. Values of elements in matrix A when vehicle parameters are changed (increments: 720% of normal value for Cf and Kf ; V ¼ 120 km/h).
-5
1
0
-5
1
Roll moment arm h s [m]
-2
0
5
- 0.35
-2
Kφ
-1
0
1
2
3
Time [sec.] Fig. 5. Lane-change simulation with various values of roll moment arm hs (increments for hs: 70.3 m; @af =@f ¼ 0:1; @ar =@f ¼ 0:1; V ¼ 120 km/h).
3.2. Interactions between yaw rate and roll motion Since Ixz is assumed to be zero in this study, arf’ and arf are zero by Eqs. (12), (13) and (15) if roll steer is ignored. Therefore, afr is the only coupling term related to the yaw rate and roll angle in this case. Furthermore, Fig. 4 shows that afr depends somewhat on hs but not on Kf or Cf : It is also notable that afr becomes zero when lf Kf ¼ lr Kr or when the vehicle is in neutral steer. Figs. 5 and 6 confirm that neither hs nor Kf significantly affect the yaw rate of the vehicle during a lane change. The figures also indicate that the phase delay of the yaw rate with respect to the front steer angle is small compared to that of the side slip angle and roll angle. 3.3. Effects of roll steer coefficients In the previous sections, the roll steer coefficients were set at constant values. Among the coupling terms in the
2
1
1
0 -1 -2 0
2
4
5
6
7
0
0
∂ αf
-2
∂φ 2
3
5
6
1
2
3
7
4
5
6
7
5
6
7
6
7
6
7
∂α r ∂φ 0
1
2
3
4
5
∂ αf ∂φ ∂α f
-5
∂αf ∂φ
∂φ 1
2
3
4
4
6
7
-2 1
2
4
0
1
∂α r ∂φ 2
3
4
5
4
0
5
6
7
Time [sec.] Fig. 7. Lane-change simulation with various values of front rollsteer coefficient (value for @af =@f: 0.2, 0.1, 0, 0.1, and 0.2; @ar =@f ¼ 0; V ¼ 120 km/h).
matrix A, however, abf and afr are influenced by changes in the roll steer coefficients. The effect of front roll steer on dynamics of the vehicle performing a lane change maneuver is shown in Fig. 7, and that of rear roll steer is depicted in Fig. 8. As roll understeer increases with @af =@f becoming more negative or @ar =@f becoming more positive, the phase delay of each variable and the height of the first peak decreases, while the height of the second peak increases.
4. Experimental results 4.1. Static analysis of a vehicle on a tilting platform Experiment was conducted to confirm the effects of varying the center of mass, and therefore hs, of a truck. A loaded truck with mass of 7600 kg is put on a tilting platform. Both the tilt angle of the platform and the roll angle of the vehicle can then be recorded to give a graph of the vehicle roll angle for a given lateral acceleration acting on the vehicle. The load was then removed from the truck, and the procedure was repeated. Fig. 9
∂α r
∂ αr ∂φ
∂φ
-2 -4
3
∂α r ∂φ
-5
2
∂ αf ∂φ
0
0
-10
5
∂αf ∂φ
2
-4 0
4
0
-2 -4
1
0
-10 0
β [deg]
2
5
φ [deg]
-1
2
-4 0
r [deg/s]
3
153
0
-2
1
r [deg/s]
β [deg]
δ [deg]
2
φ [deg]
δ [deg]
S. Takano et al. / JSAE Review 24 (2003) 149–156
0
1
2
3
4
5
Time [sec.] Fig. 8. Lane-change simulation with various values of rear rollsteer coefficient (value for @ar =@f: 0.2, 0.1, 0, 0.1, and 0.2; @af =@f ¼ 0; V ¼ 120 km/h).
indicates the results. The figure clearly shows that the roll angle for any given lateral acceleration is larger when the truck has the some load. The figure also shows the results obtained from simulation with the 3-DOF model. They largely agree with the experimental values. 4.2. Static analysis of a traveling vehicle Fig. 9 also shows similar data taken from the vehicle used in the previous section traveling with a constant turning radius of 70.8 m. Note that results from two different test methods largely agree. 4.3. Dynamic analysis of a vehicle performing a lanechange maneuver The previous two experiments were static. To examine the behavior of the roll and other motions in dynamic situations, the truck in the previous sections was employed to perform a lane change experiment. The truck was driven at 80 km/h and steered to move 3.6 m in the lateral direction while traveling 50 m. Five to six trials per condition were carried out. Fig. 10 shows
S. Takano et al. / JSAE Review 24 (2003) 149–156
154
Roll angle [deg]
8 7
Without load (Tilting platform)
6
Without load (Traveling vehicle; Right turn) Without load (Traveling vehicle; Left turn)
5
Without load (Simulation)
4
With load (Tilting platform)
3
With load (Traveling vehicle; Right turn)
2
With load (Traveling vehicle; Left turn) With load (Simulation)
1 0
0
2
4
6
Lateral acceleration [m/s2] Fig. 9. Roll angle of a truck for given lateral acceleration. 4
Steer angle δ [deg]
Steer angle δ [deg]
5
0 Experiment Simulation
-5 0
1
2
3
4
5
6
8
Yaw rate r [deg/s]
10 0
2
3
4
5
6
7
8
Roll angle φ [deg]
4 2 0
Lateral Acceleration ay [m/s 2]
3
4
5
6
7
8
2 0
-2
-4 0
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
2
2
4
-5
-4
1
3
5
-10
0
2
0 -5
-2
4
1
10
Roll angle φ [deg]
1
0
10 5
-10 -15
-10
Lateral Acceleration ay [m/s2]
Yaw rate r [deg/s]
20
-20 0
0 -2 -4
7
Experiment Simulation
2
1 0 -1 -2
Time [sec] 1
2
3
4
5
6
7
8
Time [sec] Fig. 10. Response of a truck without load during lane-change maneuver (V ¼ 80 km/h)
typical results for the truck without load. Fig. 11 is a similar set of results for the truck with load. These indicate that the roll angle is considerably higher for a given lateral acceleration when the vehicle has some load, and therefore higher hs, which agrees with the previous theoretical analysis. Also, the yaw rate for a given steer angle largely stays the same regardless of the load, which also agrees with the analysis. Figs. 10 and 11 also show the results obtained by simulation using the 3DOF model developed in this study. The simulation agrees well with the experiment.
Fig. 11. Response of a truck with load during lane-change maneuver (V ¼ 80 km/h).
To better clarify the relationship between the steer angle input and output variables during a lane change, the first peak is defined as peak A, and the second peak as peak B. Refer to Fig. 12 for details and nomenclatures. The peak values of the input and output and the output’s delay with respect to the input are then examined. Figs. 13–15 are the peak values of various output variables with respect to the peak value of the front steer angle input during a lane change. In Fig. 15, it is clear that the roll angle peak values are severely affected by the load and therefore, a change in hs : Figs. 13 and 15 respectively show that the peak values of the yaw rate and lateral acceleration are not significantly influenced by the load.
S. Takano et al. / JSAE Review 24 (2003) 149–156
155
Peak A Output
Input
Peak B Peak B delay
Peak value of lateral acceleration [m/s 2 ]
4
Peak A delay
3.5 3
Without load (PeakA)
2.5
Without load (PeakB)
2
With load (PeakA)
1.5
With load (PeakB)
1 0.5
Fig. 12. Definitions of the peaks.
0
0
2
4
6
Peak value of steer angle [deg] Fig. 15. Peak value of the lateral acceleration with respect to the peak value of the front steer angle.
16 14
Without load (PeakA)
12
Without load (PeakB)
0.9
10
0.8
8 With load (PeakB)
6 4
0.6 0.4 0.3 0.1
0
2
4
6
0
Peak value of steer angle [deg] Fig. 13. Peak value of the yaw rate with respect to the peak value of the front steer angle.
Yaw rate
Roll angle
Lateral acceleration
Fig. 16. Delay of peak A of various variables with respect to the front steer angle peak A.
8
0.9 0.8
7 Without load (PeakA)
6 Without load (PeakB)
5 With load (PeakA)
4
With load (PeakB)
3
Without load
0.7
Delay time [s]
Peak value of roll angle [deg]
With load
0.5
0.2
2 0
Without load
0.7
With load (PeakA)
Delay time [s]
Peak value of yaw rate [deg/s]
18
0.6
With load
0.5 0.4 0.3 0.2 0.1
2
0
Yaw rate
1 0
Roll angle
Lateral acceleration
Fig. 17. Delay of peak B of various variables with respect to the front steer angle peak B. 0
2
4
6
Peak value of steer angle [deg]
Fig. 14. Peak value of the roll angle with respect to the peak value of the front steer angle.
Figs. 16 and 17 are the delays of peaks of various variables with respect to the front steer angle peaks during a lane change. With peak A, increasing the e.g. height mainly seems to delay the roll angle. However, with peak B, increasing the e.g. height affects all of the yaw rate roll angle and lateral acceleration.
5. Conclusions 1. Simulation results show that changes in the height of the center of gravity and suspension parameters such as C/ and K/ affect the interaction between the planar motions and roll motion of a large-size vehicle. Also, the roll steer strengthens the relationship between the planar motions and roll angle of the vehicle.
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S. Takano et al. / JSAE Review 24 (2003) 149–156
2. Static analysis and experiment indicate that a change in the height of the c.g. influences the roll angle the most. However, dynamic experiment shows that it affects other variables such as the yaw rate and lateral acceleration as well in terms of delay with respect to the input steer angle. In the future, use of a vehicle model that has tires with nonlinear characteristics and a control system for suppressing the roll motion will be considered.
References [1] E.M. Elbeheiry, et al., Advanced ground vehicle suspension systems—a classified bibliography, Veh. System Dyn. 24 (3) (1995) 231–258. [2] N. Rosam, J. Darling, Development and simulation of a novel roll control system for the interconnected hydragas suspension, Veh. System Dyn. 27 (1) (1997) 1–18. [3] National Highway Traffic Safety Administration, 2001 Annual Assessment: Motor Vehicle Traffic Crash Fatality and Injury Estimates for 2001, 2001.