# Analysis of Vehicle Steering and Driving Bifurcation Characteristics

Authors: Xianbin Wang; Shuming Shi
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

The typical method of vehicle steering bifurcation analysis is based on the nonlinear autonomous vehicle model deriving from the classic two degrees of freedom (2DOF) linear vehicle model. This method usually neglects the driving effect on steering bifurcation characteristics. However, in the steering and driving combined conditions, the tyre under different driving conditions can provide different lateral force. The steering bifurcation mechanism without the driving effect is not able to fully reveal the vehicle steering and driving bifurcation characteristics. Aiming at the aforementioned problem, this paper analyzed the vehicle steering and driving bifurcation characteristics with the consideration of driving effect. Based on the 5DOF vehicle system dynamics model with the consideration of driving effect, the 7DOF autonomous system model was established. The vehicle steering and driving bifurcation dynamic characteristics were analyzed with different driving mode and driving torque. Taking the front-wheel-drive system as an example, the dynamic evolution process of steering and driving bifurcation was analyzed by phase space, system state variables, power spectral density, and Lyapunov index. The numerical recognition results of chaos were also provided. The research results show that the driving mode and driving torque have the obvious effect on steering and driving bifurcation characteristics.


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## Body

## 1. Introduction

In recent years, with the development of tyre mechanics and nonlinear dynamics theory, the fact that the vehicle dynamics bifurcation mechanism due to nonlinear characteristics of the tyre lateral force has been basically confirmed [1–3]. The 2DOF vehicle model assumes that the longitudinal velocity is constant and does not consider the effect of tyre longitudinal force on the system stability [4]. But, under the steering and driving conditions, with the increase of the longitudinal velocity, the tyres have to provide more longitudinal driving force. From the essence of dynamic mechanics, this phenomenon is caused by the combined slip characteristics of tyre longitudinal and lateral forces and the coupling effect of multidegree of freedom.Tyres under different driving conditions can provide various lateral forces. Therefore, the steering bifurcation mechanism without driving torque involved cannot fully reveal the vehicle steering and driving bifurcation characteristics and also cannot explain driving torque effects on vehicle handling stability. The bifurcation analysis is usually based on the nonlinear vehicle model deriving from the classic 2DOF linear vehicle model. In the steering bifurcation analysis, the most representative research result is the phase plane of vehicle body sideslip angle and body sideslip angle rate proposed by Inagaki [5] in 1994. Using the method of Inagaki, Ono [6, 7] also got the same conclusion that vehicle system had three equilibrium points. One was the stable focus; the other two were unstable saddle points. Ono also found that when the body sideslip angle increased to a certain extent, vehicle instability was mainly caused by the saddle node bifurcation. Shen et al. [8] used the geometry trajectory intersection method to study the stability of vehicle steering dynamics. Their results also were close to Inagaki’s conclusion; namely, the vehicle steering dynamics system had three equilibrium points. One was a stable focus and the other two were instable saddle points. On the basis of 2DOF vehicle model, longitudinal velocity was added into the vehicle body model by Liu et al. [9]. As a result, the constant longitudinal velocity assumption was removed, and the stability criterion based on the 2DOF model was revised. Furthermore, the chaos phenomenon of the vehicle steering movement was found. And it reflected that the dynamics essential of vehicle steering instability was the chaotic motion [10]. Horiuchi et al. [11] proposed a method of analyzing the vehicle stability during acceleration and braking adopting constrained bifurcation and continuation methods. Based on the vehicle model including the longitudinal velocity, lateral velocity, yaw rate, and roll velocity, the bifurcation caused by the braking deceleration was analyzed. According to the wheel rotation dynamics analysis, the classical tire longitudinal slip rate equation was verified under instable driving situation by Wang et al. [12]. And on the basis of 5DOF nonlinear vehicle model including the driving effect the driving mode effects on vehicle stability with different steering angle input were analyzed.Above all, by employing the 2DOF vehicle model, the vehicle steering stability study has achieved fruitful research results. The 3DOF vehicle model including the longitudinal velocity reveals the chaos motion nature of the vehicle instability. The 5DOF vehicle model containing the driving force indicates that different driving modes affect vehicle steering stability. However, the dynamic evolution process of vehicle steering and driving bifurcation still lacks the comprehensive research, especially the fully accurate numerical recognition results of chaos.


## 2. Vehicle System Model

### 2.1.
5DOF Vehicle System Model

Based on the 3DOF vehicle steering model [13], the 5DOF [12] planar motion nonlinear dynamics vehicle system model with driving torque involved is established. The equations of 5DOF vehicle model are shown in (1). Figure 1 is the sketch of the bicycle vehicle model:(1)v˙x=vyω+Flfcos⁡δf-Fsfsin⁡δf+Flr-sgn⁡vx·Cair_xAL_xρ/2vx2m,v˙y=-vxω+Flfsin⁡δf+Fsfcos⁡δf+Fsr-sgn⁡vy·Cair_yAL_yρ/2vy2m,ω˙=Flfsin⁡δf+Fsfcos⁡δflf-FsrlrIz,ω˙f=Td-sgn⁡ωf·Tbf-Re·FlfJ,ω˙r=Td-sgn⁡ωr·Tbr-Re·FlrJ,where m is vehicle mass; I
z is vehicle yaw moment of inertia; v
x is longitudinal velocity; v
y is lateral velocity; ω is yaw rate; ω
f is angular velocity of front wheel; ω
r is angular velocity of rear wheel; l
f is distance from front axle to the mass center; l
r is distance from rear axle to the mass center; J is wheel moment of inertia; δ
f is front wheel steering angle; C
a
i
r
_
x, C
a
i
r
_
y are longitudinal and lateral air resistance coefficient; A
L
_
x, A
L
_
y are vehicle longitudinal and lateral area; ρ is air density; T
d is driving torque; T
b is braking torque; R
e is wheel rolling radius; F
l
f, F
l
r, F
s
f, and F
s
r are longitudinal and lateral tyre force of front and rear wheel.Figure 1
Vehicle bicycle model.The vehicle parameters are listed in Table1.Table 1
Vehicle parameters.
Parameters
Values
m/kg
1500
I
z/kg⋅m2
3000
l
f/m
1.2
l
r/m
1.3
J/kg⋅m2
1
C
air_
x/
0.3
C
air_
y/
0.4
A
L
_
x/m2
1.7
A
L
_
y/m2
3.5
ρ/kg/m3
1.29
R
e/m
0.224The tyre force is modeled by the Magic Formula [14]:(2)F=Dsin⁡Carctan⁡Bx-EBx-arctan⁡Bx,where B, C, D, and E are coefficients. F is the longitudinal or lateral tyre force, and x is the longitudinal slip or sideslip angle.The typical longitudinal slip [14] is defined as follows:(3)k=ωw·Re-vwxvwx,where ω
w is angular velocity of wheel; v
w
x is linear velocity at the wheel center in the longitudinal direction.The longitudinal tyre parameters are listed in Table2.Table 2
Longitudinal tyre parameters.
Tyre
Coefficient
B
C
D
E
Front tyre
11.275
1.56
2574.8
0.4109
Rear tyre
18.631
1.56
1749.6
0.4108The sideslip angles of front and rear tyre are given in the following:(4)αf=δf-atan⁡vy+ω·lfvx,αr=δr-atan⁡vy-ω·lrvx,where δ
r
=
0.Table3 shows the lateral tyre parameters [15].Table 3
Lateral tyre parameters.
Tyre
Coefficient
B
C
D
E
Front tyre
11.275
1.56
2574.7
−1.999
Rear tyre
18.631
1.56
1749.7
−1.7908Figures2 and 3 show the steady state tyre characteristics of the longitudinal force and lateral tire force.Figure 2
Steady state tyre characteristics.Figure 3
Steady state tyre characteristics.By using models proposed by Pacejka [14], the combined slip characteristics, the tyre forces can be calculated as follows: (5)Flf=Flf0·Gx,Flr=Flr0·Gx,Gx=cos⁡arctan⁡Bg,xα·α,Bg,xα=rx,1cos⁡arctan⁡rx,2·k,Fsf=Gy·Fsf0,Fsr=Gy·Fsr0,Gy=cos⁡arctan⁡Bg,yk·k,Bg,yk=ry,1cos⁡arctan⁡ry,2·α,where F
l
f
0, F
l
r
0, F
s
f
0, and F
s
r
0 are steady state tyre force, calculated by Magic Formula, (2); G
x, G
y are tyre force weighting function; r
x
,
1, r
x
,
2, r
y
,
1, and r
y
,
2 are combined slip coefficient.The best interactive curves are produced with parameters mentioned in Table4.Table 4
Combined slip coefficient.
Longitudinal slip coefficient
Lateral slip coefficient
r
x
,
1
r
x
,
2
r
y
,
1
r
y
,
2
35
40
40
35


### 2.2.
7DOF Autonomous System Model

In order to focus attention on the dynamics characteristics of the vehicle itself, an effective way of introducing periodic front wheel steering angle input into the model without the driver model is establishing the autonomous model.The typical vehicle handling stability test condition is selected. The front wheel steering angle input is the sinusoidal form as follows:(6)δf=Asin⁡ωsft,where A is amplitude of front wheel angle input; ω
s
f is frequency of front wheel angle input.The 5DOF vehicle system model is transformed into 7DOF autonomous system model with the front wheel steering angle as an intermediate variable:(7)X˙=fX,where X
=
v
y
,
ω
,
v
x
,
ω
f
,
ω
r
,
δ
f
,
z and z is intermediate variable:(8)z=ωsfcos⁡ωsft,(9)v˙x=vyω+Flfcos⁡δf-Fsfsin⁡δf+Flr-sgn⁡vx·Cair_xAL_xρ/2vx2m,v˙y=-vxω+Flfsin⁡δf+Fsfcos⁡δf+Fsr-sgn⁡vy·Cair_yAL_yρ/2vy2m,ω˙=Flfsin⁡δf+Fsfcos⁡δflf-FsrlrIz,ω˙f=Td-sgn⁡ωf·Tbf-Re·FlfJ,ω˙r=Td-sgn⁡ωr·Tbr-Re·FlrJ,δ˙f=Az,z˙=-ωsf2Aδf.


## 2.1.
5DOF Vehicle System Model

Based on the 3DOF vehicle steering model [13], the 5DOF [12] planar motion nonlinear dynamics vehicle system model with driving torque involved is established. The equations of 5DOF vehicle model are shown in (1). Figure 1 is the sketch of the bicycle vehicle model:(1)v˙x=vyω+Flfcos⁡δf-Fsfsin⁡δf+Flr-sgn⁡vx·Cair_xAL_xρ/2vx2m,v˙y=-vxω+Flfsin⁡δf+Fsfcos⁡δf+Fsr-sgn⁡vy·Cair_yAL_yρ/2vy2m,ω˙=Flfsin⁡δf+Fsfcos⁡δflf-FsrlrIz,ω˙f=Td-sgn⁡ωf·Tbf-Re·FlfJ,ω˙r=Td-sgn⁡ωr·Tbr-Re·FlrJ,where m is vehicle mass; I
z is vehicle yaw moment of inertia; v
x is longitudinal velocity; v
y is lateral velocity; ω is yaw rate; ω
f is angular velocity of front wheel; ω
r is angular velocity of rear wheel; l
f is distance from front axle to the mass center; l
r is distance from rear axle to the mass center; J is wheel moment of inertia; δ
f is front wheel steering angle; C
a
i
r
_
x, C
a
i
r
_
y are longitudinal and lateral air resistance coefficient; A
L
_
x, A
L
_
y are vehicle longitudinal and lateral area; ρ is air density; T
d is driving torque; T
b is braking torque; R
e is wheel rolling radius; F
l
f, F
l
r, F
s
f, and F
s
r are longitudinal and lateral tyre force of front and rear wheel.Figure 1
Vehicle bicycle model.The vehicle parameters are listed in Table1.Table 1
Vehicle parameters.
Parameters
Values
m/kg
1500
I
z/kg⋅m2
3000
l
f/m
1.2
l
r/m
1.3
J/kg⋅m2
1
C
air_
x/
0.3
C
air_
y/
0.4
A
L
_
x/m2
1.7
A
L
_
y/m2
3.5
ρ/kg/m3
1.29
R
e/m
0.224The tyre force is modeled by the Magic Formula [14]:(2)F=Dsin⁡Carctan⁡Bx-EBx-arctan⁡Bx,where B, C, D, and E are coefficients. F is the longitudinal or lateral tyre force, and x is the longitudinal slip or sideslip angle.The typical longitudinal slip [14] is defined as follows:(3)k=ωw·Re-vwxvwx,where ω
w is angular velocity of wheel; v
w
x is linear velocity at the wheel center in the longitudinal direction.The longitudinal tyre parameters are listed in Table2.Table 2
Longitudinal tyre parameters.
Tyre
Coefficient
B
C
D
E
Front tyre
11.275
1.56
2574.8
0.4109
Rear tyre
18.631
1.56
1749.6
0.4108The sideslip angles of front and rear tyre are given in the following:(4)αf=δf-atan⁡vy+ω·lfvx,αr=δr-atan⁡vy-ω·lrvx,where δ
r
=
0.Table3 shows the lateral tyre parameters [15].Table 3
Lateral tyre parameters.
Tyre
Coefficient
B
C
D
E
Front tyre
11.275
1.56
2574.7
−1.999
Rear tyre
18.631
1.56
1749.7
−1.7908Figures2 and 3 show the steady state tyre characteristics of the longitudinal force and lateral tire force.Figure 2
Steady state tyre characteristics.Figure 3
Steady state tyre characteristics.By using models proposed by Pacejka [14], the combined slip characteristics, the tyre forces can be calculated as follows: (5)Flf=Flf0·Gx,Flr=Flr0·Gx,Gx=cos⁡arctan⁡Bg,xα·α,Bg,xα=rx,1cos⁡arctan⁡rx,2·k,Fsf=Gy·Fsf0,Fsr=Gy·Fsr0,Gy=cos⁡arctan⁡Bg,yk·k,Bg,yk=ry,1cos⁡arctan⁡ry,2·α,where F
l
f
0, F
l
r
0, F
s
f
0, and F
s
r
0 are steady state tyre force, calculated by Magic Formula, (2); G
x, G
y are tyre force weighting function; r
x
,
1, r
x
,
2, r
y
,
1, and r
y
,
2 are combined slip coefficient.The best interactive curves are produced with parameters mentioned in Table4.Table 4
Combined slip coefficient.
Longitudinal slip coefficient
Lateral slip coefficient
r
x
,
1
r
x
,
2
r
y
,
1
r
y
,
2
35
40
40
35


## 2.2.
7DOF Autonomous System Model

In order to focus attention on the dynamics characteristics of the vehicle itself, an effective way of introducing periodic front wheel steering angle input into the model without the driver model is establishing the autonomous model.The typical vehicle handling stability test condition is selected. The front wheel steering angle input is the sinusoidal form as follows:(6)δf=Asin⁡ωsft,where A is amplitude of front wheel angle input; ω
s
f is frequency of front wheel angle input.The 5DOF vehicle system model is transformed into 7DOF autonomous system model with the front wheel steering angle as an intermediate variable:(7)X˙=fX,where X
=
v
y
,
ω
,
v
x
,
ω
f
,
ω
r
,
δ
f
,
z and z is intermediate variable:(8)z=ωsfcos⁡ωsft,(9)v˙x=vyω+Flfcos⁡δf-Fsfsin⁡δf+Flr-sgn⁡vx·Cair_xAL_xρ/2vx2m,v˙y=-vxω+Flfsin⁡δf+Fsfcos⁡δf+Fsr-sgn⁡vy·Cair_yAL_yρ/2vy2m,ω˙=Flfsin⁡δf+Fsfcos⁡δflf-FsrlrIz,ω˙f=Td-sgn⁡ωf·Tbf-Re·FlfJ,ω˙r=Td-sgn⁡ωr·Tbr-Re·FlrJ,δ˙f=Az,z˙=-ωsf2Aδf.


## 3. Amplitude of Front Wheel Angle Effect on Bifurcation Characteristics

In order to be close to the practice vehicle operation condition, the steering and driving simulations under constant driving torque are carried out based on 7DOF autonomous system model.The selection of driving torque is based on the related technical parameters of passenger car equipped with 1.6 L engine. In general, the maximum power of naturally aspirated engine is about 77/5600 of kw/rmp and the maximum engine torque is about 155/3500 Nm/rpm. Therefore, the driving torqueT
d = 140 Nm is selected as the upper limit of the simulation.For the specific simulation parameters, we get the following.Initial longitudinal velocityv
x
=
20 m/s. The constant driving torque on driving wheel T
d = 140 Nm. Note that, for all-wheel-drive mode, driving torque applying on the front and rear wheels is half of the total driving torque, respectively; namely, T
d = 70 Nm. The varying range of amplitude value of front wheel steering angle A is (
0.0005,0.1
) rad, with the interval value being 0.0005 rad. The frequency of front wheel steering angle f
=
0.4 Hz. The expression of the angular frequency of front wheel steering angle is as follows:(10)ωsf=2π·0.4=0.8π.Figures4–6 show the bifurcation diagrams of three state variables (v
x, v
y, and ω) under front-wheel-drive mode.Figure 4
Bifurcation diagrams ofv
y (T
d = 140 Nm, v
x = 20 m/s, front-wheel-drive mode).Figure 5
Bifurcation diagrams ofω (T
d = 140 Nm, v
x = 20 m/s, front-wheel-drive mode).Figure 6
Bifurcation diagrams ofv
x (T
d = 140 Nm, v
x = 20 m/s, front-wheel-drive mode).These figures are obtained as follows. For a given value ofT
d, differential system (9) is solved numerically using MATLAB, starting from a constant initial point, v
x = 20 m/s, v
y = 0 m/s, ω
=
0 rad/s, ω
f = 0 rad/s, ω
r = 0 rad/s, δ
f
=
0, and ω
s
f = 0.8π, for t
=
0. After the transient behavior vanishes, the variables (v
x
,
v
y
,
ω
,
ω
f,  and ω
r) are sampled in the range t
<
500 s at the frequencyf = 0.4 Hz. The bifurcation diagram is composed of all the sampling points of state variables and the corresponding amplitude of front wheel steering angleA.By the results of bifurcation diagram, we get the following.(1) When the amplitude of front wheel steering angleA
<
0.02 rad, the lateral velocity v
y is gradually increasing as A increases. When A
=
0.0195 rad, v
y varying range eventually reaches [
1.381,1.879
] m/s. But the yaw angular velocity is gradually decreasing with the increasing A. WhenA = 0.0195 rad, ω varying range is [
-
0.1087
,
-
0.09283
] rad/s. When driving torque T
d = 140 Nm, since the driving torque is greater than the air resistance, the longitudinal velocity v
x increases. But, with the increase ofA, whenA = 0.0195 rad, the varying range of v
x gradually reduces from [
39.47,44.71
] m/s to [
37.54,42.09
] m/s.(2) When the amplitude of front wheel steering angleA = 0.02 rad, the varying range of v
y sharply increases to [
-
35.21,37.53
] m/s. Then the varying range of v
y decreases as A increases. WhenA = 0.1 rad, varying range of v
y is [
-
16,14.68
] m/s. WhenA = 0.02 rad, the varying range of ω also sharply increases to [
-
0.7432,0.7834
] m/s. Then the varying range of ω increases as A increases. WhenA = 0.1 rad, the varying range of ω is [
-
0.9546,1.176
] rad/s. WhenA = 0.02 rad, the varying range of v
x also sharply increases to [
-
36.12,41.36
] m/s. Then the varying range of v
x decreases as A increases. WhenA = 0.1 rad, the varying range of v
x is [
-
13.54,13.71
] m/s.Figure7 is the bifurcation diagrams of three state variables (v
x, v
y, and ω) under all-wheel-drive mode. Figure 8 corresponds to rear-wheel-drive mode.Figure 7
Bifurcation diagrams ofv
y, ω, and v
x (T
d = 140 Nm, v
x = 20 m/s, all-wheel-drive mode).Figure 8
Bifurcation diagrams ofv
y, ω, and v
x (T
d = 140 Nm, v
x = 20 m/s, rear-wheel-drive mode).It is obvious that the overall trend of the bifurcation diagrams of all-wheel-drive mode and rear-wheel-drive mode is in accordance with the bifurcation diagram of front-wheel-drive mode. The varying range of yaw rate expands to(
-
4,4
) rad/s. But the bifurcation parameterA (amplitude of front wheel steering angle) has the obvious differences under different driving mode. A of front-wheel-drive mode (A
=
0.02 rad) > A of all-wheel-drive mode (A
=
0.0175 rad) > A of rear-wheel-drive mode (A
=
0.014 rad). So, it can be conclude that the different driving mode has obvious effects on the bifurcation characteristics.Figures9–11 are the system state variable bifurcation diagrams of different driving mode with driving torque T
d = 140 Nm and initial longitudinal velocity v
x = 30 m/s.Figure 9
Bifurcation diagrams ofv
y, ω, and v
x (T
d = 140 Nm, v
x = 30 m/s, front-wheel-drive mode).Figure 10
Bifurcation diagrams ofv
y, ω, and v
x (T
d = 140 Nm, v
x= 30 m/s, all-wheel-drive mode).Figure 11
Bifurcation diagrams ofv
y, ω, and v
x (T
d = 140 Nm, v
x = 30 m/s, rear-wheel-drive mode).It is clear that the overall trend of the bifurcation diagrams is in accordance with the bifurcation diagram of initial longitudinal velocityv
x = 20 m/s. The bifurcation parameter A varies a little when the initial longitudinal velocity increases to v
x = 30 m/s. The bifurcation parameter of front-wheel-drive mode is still A
=
0.02 rad. The bifurcation parameter of all-wheel-drive mode is A
=
0.0175 rad as before. But the bifurcation parameter of rear-wheel-drive mode decreases to A
=
0.0135 rad. There is no sort order change of A under different driving mode. A of front-wheel-drive mode (A
=
0.02 rad) > A of all-wheel-drive mode (A
=
0.0175 rad) > A of rear-wheel-drive mode (A
=
0.0135 rad).Figures12–14 are the system state variable bifurcation diagrams of different driving mode with driving torque T
d = 70 Nm and initial longitudinal velocity v
x = 30 m/s.Figure 12
Bifurcation diagrams ofv
y, ω, and v
x (T
d = 70 Nm, v
x = 30 m/s, front-wheel-drive mode).Figure 13
Bifurcation diagrams ofv
y, ω, and v
x (T
d = 70 Nm, v
x = 30 m/s, all-wheel-drive mode).Figure 14
Bifurcation diagrams ofv
y, ω, and v
x (T
d = 70 Nm, v
x = 30 m/s, rear-wheel-drive mode).It is clear that the overall trend of the bifurcation diagrams is in accordance with the previous bifurcation diagrams. The bifurcation parameterA varies a lot when the driving torque reduces to T
d = 70 Nm. The bifurcation parameter of front-wheel-drive mode increases to A
=
0.028 rad. The bifurcation parameter of all-wheel-drive mode is A
=
0.0265 rad. The bifurcation parameter of rear-wheel-drive mode is A
=
0.02 rad. There is no sort order change of A under different driving mode. A of front-wheel-drive mode (A
=
0.028 rad) > A of all-wheel-drive mode (A
=
0.0265 rad) > A of rear-wheel-drive mode (A
=
0.02 rad).


## 4. Dynamic Evolution Process of Steering and Driving Bifurcation

In fact, the system state variables bifurcation diagrams in previous section indicate that the 5DOF vehicle system dynamic characteristics are the typical chaotic motion. Namely, the 5DOF vehicle system turns into the chaotic motion from the quasiperiodic motion with the constant driving torque and periodic front wheel steering angle input. In this section, the details of the dynamic evolution process of steering and driving bifurcation are confirmed.Taking the front-wheel-drive mode as an example, the simulations are carried out in this section. The simulation condition is identified with previous section, namely, the steering and driving simulations under constant driving torque.For the specific simulation parameters, we get the following.The initial simulation values includev
x = 20 m/s, v
y = 0 m/s, ω
=
0 rad/s, ω
f = 0 rad/s, and ω
r = 0 rad/s. The constant driving torque T
d = 140 Nm. The driving mode is front-wheel-drive one. The amplitudes of front wheel steering angle are A
=
0.015 rad, A
=
0.02 rad, andA = 0.04 rad. The frequency of front wheel steering angle f
=
0.4 Hz. The expression of the angular frequency of front wheel steering angle is still (10).


### 4.1. Amplitude of Front Wheel Steering AngleA
=
0.015 rad

Figures15–22 are the numerical results with A
=
0.015 rad.Figure 15
v
x-v
y-ω phase space trajectories (A = 0.015 rad, front-wheel-drive mode).Figure 16
v
x-v
y-ω phase space portrait onto v
y-ω plane (A
=
0.015 rad, front-wheel-drive mode).Figure 17
State variables (A
=
0.015 rad, front-wheel-drive mode).Figure 18
Part of state variables sequence (A
=
0.015 rad, front-wheel-drive mode).Figure 19
Poincare section of phase trajectory (A
=
0.015 rad, front-wheel-drive mode).Figure 20
Power spectral density of state variables (A
=
0.015 rad, front-wheel-drive mode).Figure 21
Lyapunov exponents (A
=
0.015 rad, front-wheel-drive mode).Figure 22
Lyapunov exponents (A
=
0.015 rad, front-wheel-drive mode).Figure15 is v
x-v
y-ω phase space trajectories.Figure16 is phase space trajectories in Figure 15 onto v
y
-
ω plane. It is shown that when A
=
0.015 rad, the phase space trajectories are quasiperiodic torus. In the whole simulation period, the initial longitudinal velocity v
x increases gradually and stabilizes in a certain range, while the lateral velocity v
y and yaw rate ω gradually transformed into stable periodic oscillation.By analyzing the state variablesv
x, v
y, and ω in Figures 17 and 18, the same conclusion can be drawn too. Figure 18 is the calculation result in first 100 s.Figure19 is the Poincare section of phase trajectory captured in period of front wheel steering angle. It can be seen that each state variable is in a straight line. This is the typical characteristics of the periodic motion.Figure20 is the power spectral density diagram of each state variable. In this figure, all state variables are in typical periodic motion; namely, the power spectral density shows single peak distribution characteristics. Note that the varying frequency of v
y is consistent with the frequency of front wheel steering angle. They are both 0.4 Hz.Figures21 and 22 show the Lyapunov exponents of state variables. Figure 22 is the local enlargement of Figure 21. The algorithm employed in this work for determining Lyapunov exponents was proposed by Wolf et al. [16]. The Lyapunov spectra are (
0,0
,
0
,
-
,
-
,
-
,
-
), which prove that the system is not chaotic.


### 4.2. Amplitude of Front Wheel Steering AngleA
=
0.02 rad

Figures23–30 are the numerical results with A
=
0.02 rad. From the previous analysis, it is known that the system bifurcation occurs.Figure 23
v
x-v
y-ω phase space trajectories (A
=
0.02 rad, front-wheel-drive mode).Figure 24
v
x-v
y-ω phase space portrait onto v
y-ω plane (A
=
0.02 rad, front-wheel-drive mode).Figure 25
State variables (A
=
0.02 rad, front-wheel-drive mode).Figure 26
Part of state variables sequence (A
=
0.02 rad, front-wheel-drive mode).Figure 27
Poincare section of phase trajectory (A
=
0.02 rad, front-wheel-drive mode).Figure 28
Power spectral density of state variables (A
=
0.02 rad, front-wheel-drive mode).Figure 29
Lyapunov exponents (A
=
0.02 rad, front-wheel-drive mode).Figure 30
Lyapunov exponents (A
=
0.02 rad, front-wheel-drive mode).Figures23 and 24 are v
x-v
y-ω phase space trajectories. Compared with phase space trajectories, while A
=
0.015 rad, the phase space trajectories transform into bounded chaos.By the state variablesv
x, v
y, and ω in Figures 25 and 26, it can be concluded that, at the beginning of the simulation, the absolute value of longitudinal velocity v
x increases gradually and stabilizes in a certain range. The absolute value of lateral velocity v
y and yaw rate ω increases gradually and then begins to swing within a certain range. But when the simulation time at aboutt = 251 s (at this moment, the v
x = 38.16 m/s, ω
=
0.1383 rad/s, v
y
=
-
1.083 m/s), the first dramatic changes of system dynamic state suddenly take place.(1) The lateral velocity decreases quickly at the beginning and then begins to increase rapidly. Att = 257 s, lateral velocity reaches the minimum v
y = −38.72 m/s. Whent = 261.8 s, lateral velocity increased to v
y = 38.54 m/s. Whent = 264.5 s lateral speed turns back to the post scope.(2) At the beginning, yaw rate begins to rapidly increase and then decreases rapidly, after that increases rapidly again, and finally returns to the initial state range. Att = 257 s, yaw rate reaches the peak ω = 0.742 rad/s. Att = 259.6 s yaw rate reduces to ω = 0.4464 rad/s. When t
=
261.8 s, yaw rate reaches the peak again ω = 1.026 rad/s. Att = 265 s yaw rate turns back to the initial state range.(3) The longitudinal velocity begins to decrease rapidly firstly, then increases rapidly, and finally stabilizes within the original scope. Whent = 259.5 s, longitudinal velocity reaches the minimum v
x = −37.22 m/s. Whent = 264 s, longitudinal velocity increases to v
x = 34.51 m/s. After that, the longitudinal velocity increases slowly.When the simulation time at aboutt = 446 s (at this moment, the v
x = 41.5 m/s,   ω
=
-
0.05763 rad/s, v
y = 2.843 m/s), system dynamic state suddenly shows second dramatic change.(1) The lateral velocity begins to rapidly increase and then decreases quickly. Whent
=
449.8 s, the lateral velocity increases to v
y = 38.71 m/s. When t
=
456 s, lateral velocity quickly reduces to v
y = −38.82 m/s. Then after vibrating many times within negative scope, lateral velocity eventually stabilizes within the original domain at t
=
490 s.(2) The yaw rate shows repeated oscillation at first and eventually returns to the initial state scope. Whent = 449.7 s, yaw rate reaches ω
=
-
0.7104 rad/s. Whent = 453.4 s, yaw rate increases to ω
=
-
0.2618 rad/s. Yaw rate reaches its peak att = 456 s, ω
=
-
0.9067 rad/s. Then the continuous repeated vibrations occur. The maximum yaw rate reaches ω
=
0.7415 rad/s at t
=
477.3 s. And the minimum value achieves ω
=
-
0.7982 rad/s att = 470.1 s. Finally, yaw rate returns to the post state range att = 490 s.(3) The longitudinal velocity increases quickly again after decreases for three times consecutively. The minimum longitudinal velocityv
x = −37.39 m/s appears att = 452.9 s to 37.39 m/s. Whent = 459.4 s, longitudinal velocity increases to the maximum v
x = 34.04 m/s. When t
=
488 s, the longitudinal velocity goes back to the previous steady state. And then the longitudinal velocity increases slowly.Figure27 is the phase trajectory of the Poincare section captured in period of front wheel steering angle. It can be seen from the picture that each state variable shows a limited point set, which is the typical characteristics of chaotic motion.Figure28 is the power spectral density diagram of each state variable. It is obvious that the lateral velocity and yaw rate show the broadband multipeak distribution characteristics of typical chaos. Compared with the power spectral density, whileA = 0.015 rad, frequency range of longitudinal velocity and front and rear wheel rotational velocity is increased.Figures29 and 30 are the Lyapunov exponents of state variables.Figure29 is the local enlargement of Figure 30. When the simulation timet = 210 s, system of Lyapunov exponents spectrum turns from 0,0
,
-
,
-
,
-
,
- to (
+
,
0,0
,
-
,
-
,
-
,
-
), which is kept to the end of the whole simulation. This is the proof that the system enters the state of chaos.


### 4.3. Amplitude of Front Wheel Steering AngleA
=
0.04 rad

Figures31–38 are the simulation results with A
=
0.04 rad. It is clear that the system bifurcation has occurred, and the system has completely entered into chaos state.Figure 31
v
x-v
y-ω phase space trajectories (A
=
0.04 rad, front-wheel-drive mode).Figure 32
v
x-v
y-ω phase space portrait onto v
y-ω plane (A
=
0.04 rad, front-wheel-drive mode).Figure 33
State variables (A
=
0.04 rad, front-wheel-drive mode).Figure 34
Part of state variables sequence (A
=
0.04 rad, front-wheel-drive mode).Figure 35
Poincare section of phase trajectory (A
=
0.04 rad, front-wheel-drive mode).Figure 36
Power spectral density of state variables (A
=
0.04 rad, front-wheel-drive mode).Figure 37
Lyapunov exponents (A
=
0.04 rad, front-wheel-drive mode).Figure 38
Lyapunov exponents (A
=
0.04 rad, front-wheel-drive mode).The phase space trajectories in Figures31 and 32 show that the vehicle state has lost stability. In other words, the vehicle has entered into chaos. The state variables of time series in Figures 33 and 34 also show the severe oscillation. But its overall range is bounded. The Poincare section diagram in Figure 35 can also get the same conclusion that the Poincare section is the limited point set in bounded scope.In Figure36, the power spectral density of the state variables all presents the typical chaotic characteristics of broadband multimodal. Figures 37 and 38 are the Lyapunov exponents of state variables. It is obvious that, at the beginning of the simulation, the maximum Lyapunov exponent quickly stabilizes to the state in which maximum Lyapunov exponent is positive. It also shows that the system has completely entered into chaos.


## 4.1. Amplitude of Front Wheel Steering AngleA
=
0.015 rad

Figures15–22 are the numerical results with A
=
0.015 rad.Figure 15
v
x-v
y-ω phase space trajectories (A = 0.015 rad, front-wheel-drive mode).Figure 16
v
x-v
y-ω phase space portrait onto v
y-ω plane (A
=
0.015 rad, front-wheel-drive mode).Figure 17
State variables (A
=
0.015 rad, front-wheel-drive mode).Figure 18
Part of state variables sequence (A
=
0.015 rad, front-wheel-drive mode).Figure 19
Poincare section of phase trajectory (A
=
0.015 rad, front-wheel-drive mode).Figure 20
Power spectral density of state variables (A
=
0.015 rad, front-wheel-drive mode).Figure 21
Lyapunov exponents (A
=
0.015 rad, front-wheel-drive mode).Figure 22
Lyapunov exponents (A
=
0.015 rad, front-wheel-drive mode).Figure15 is v
x-v
y-ω phase space trajectories.Figure16 is phase space trajectories in Figure 15 onto v
y
-
ω plane. It is shown that when A
=
0.015 rad, the phase space trajectories are quasiperiodic torus. In the whole simulation period, the initial longitudinal velocity v
x increases gradually and stabilizes in a certain range, while the lateral velocity v
y and yaw rate ω gradually transformed into stable periodic oscillation.By analyzing the state variablesv
x, v
y, and ω in Figures 17 and 18, the same conclusion can be drawn too. Figure 18 is the calculation result in first 100 s.Figure19 is the Poincare section of phase trajectory captured in period of front wheel steering angle. It can be seen that each state variable is in a straight line. This is the typical characteristics of the periodic motion.Figure20 is the power spectral density diagram of each state variable. In this figure, all state variables are in typical periodic motion; namely, the power spectral density shows single peak distribution characteristics. Note that the varying frequency of v
y is consistent with the frequency of front wheel steering angle. They are both 0.4 Hz.Figures21 and 22 show the Lyapunov exponents of state variables. Figure 22 is the local enlargement of Figure 21. The algorithm employed in this work for determining Lyapunov exponents was proposed by Wolf et al. [16]. The Lyapunov spectra are (
0,0
,
0
,
-
,
-
,
-
,
-
), which prove that the system is not chaotic.


## 4.2. Amplitude of Front Wheel Steering AngleA
=
0.02 rad

Figures23–30 are the numerical results with A
=
0.02 rad. From the previous analysis, it is known that the system bifurcation occurs.Figure 23
v
x-v
y-ω phase space trajectories (A
=
0.02 rad, front-wheel-drive mode).Figure 24
v
x-v
y-ω phase space portrait onto v
y-ω plane (A
=
0.02 rad, front-wheel-drive mode).Figure 25
State variables (A
=
0.02 rad, front-wheel-drive mode).Figure 26
Part of state variables sequence (A
=
0.02 rad, front-wheel-drive mode).Figure 27
Poincare section of phase trajectory (A
=
0.02 rad, front-wheel-drive mode).Figure 28
Power spectral density of state variables (A
=
0.02 rad, front-wheel-drive mode).Figure 29
Lyapunov exponents (A
=
0.02 rad, front-wheel-drive mode).Figure 30
Lyapunov exponents (A
=
0.02 rad, front-wheel-drive mode).Figures23 and 24 are v
x-v
y-ω phase space trajectories. Compared with phase space trajectories, while A
=
0.015 rad, the phase space trajectories transform into bounded chaos.By the state variablesv
x, v
y, and ω in Figures 25 and 26, it can be concluded that, at the beginning of the simulation, the absolute value of longitudinal velocity v
x increases gradually and stabilizes in a certain range. The absolute value of lateral velocity v
y and yaw rate ω increases gradually and then begins to swing within a certain range. But when the simulation time at aboutt = 251 s (at this moment, the v
x = 38.16 m/s, ω
=
0.1383 rad/s, v
y
=
-
1.083 m/s), the first dramatic changes of system dynamic state suddenly take place.(1) The lateral velocity decreases quickly at the beginning and then begins to increase rapidly. Att = 257 s, lateral velocity reaches the minimum v
y = −38.72 m/s. Whent = 261.8 s, lateral velocity increased to v
y = 38.54 m/s. Whent = 264.5 s lateral speed turns back to the post scope.(2) At the beginning, yaw rate begins to rapidly increase and then decreases rapidly, after that increases rapidly again, and finally returns to the initial state range. Att = 257 s, yaw rate reaches the peak ω = 0.742 rad/s. Att = 259.6 s yaw rate reduces to ω = 0.4464 rad/s. When t
=
261.8 s, yaw rate reaches the peak again ω = 1.026 rad/s. Att = 265 s yaw rate turns back to the initial state range.(3) The longitudinal velocity begins to decrease rapidly firstly, then increases rapidly, and finally stabilizes within the original scope. Whent = 259.5 s, longitudinal velocity reaches the minimum v
x = −37.22 m/s. Whent = 264 s, longitudinal velocity increases to v
x = 34.51 m/s. After that, the longitudinal velocity increases slowly.When the simulation time at aboutt = 446 s (at this moment, the v
x = 41.5 m/s,   ω
=
-
0.05763 rad/s, v
y = 2.843 m/s), system dynamic state suddenly shows second dramatic change.(1) The lateral velocity begins to rapidly increase and then decreases quickly. Whent
=
449.8 s, the lateral velocity increases to v
y = 38.71 m/s. When t
=
456 s, lateral velocity quickly reduces to v
y = −38.82 m/s. Then after vibrating many times within negative scope, lateral velocity eventually stabilizes within the original domain at t
=
490 s.(2) The yaw rate shows repeated oscillation at first and eventually returns to the initial state scope. Whent = 449.7 s, yaw rate reaches ω
=
-
0.7104 rad/s. Whent = 453.4 s, yaw rate increases to ω
=
-
0.2618 rad/s. Yaw rate reaches its peak att = 456 s, ω
=
-
0.9067 rad/s. Then the continuous repeated vibrations occur. The maximum yaw rate reaches ω
=
0.7415 rad/s at t
=
477.3 s. And the minimum value achieves ω
=
-
0.7982 rad/s att = 470.1 s. Finally, yaw rate returns to the post state range att = 490 s.(3) The longitudinal velocity increases quickly again after decreases for three times consecutively. The minimum longitudinal velocityv
x = −37.39 m/s appears att = 452.9 s to 37.39 m/s. Whent = 459.4 s, longitudinal velocity increases to the maximum v
x = 34.04 m/s. When t
=
488 s, the longitudinal velocity goes back to the previous steady state. And then the longitudinal velocity increases slowly.Figure27 is the phase trajectory of the Poincare section captured in period of front wheel steering angle. It can be seen from the picture that each state variable shows a limited point set, which is the typical characteristics of chaotic motion.Figure28 is the power spectral density diagram of each state variable. It is obvious that the lateral velocity and yaw rate show the broadband multipeak distribution characteristics of typical chaos. Compared with the power spectral density, whileA = 0.015 rad, frequency range of longitudinal velocity and front and rear wheel rotational velocity is increased.Figures29 and 30 are the Lyapunov exponents of state variables.Figure29 is the local enlargement of Figure 30. When the simulation timet = 210 s, system of Lyapunov exponents spectrum turns from 0,0
,
-
,
-
,
-
,
- to (
+
,
0,0
,
-
,
-
,
-
,
-
), which is kept to the end of the whole simulation. This is the proof that the system enters the state of chaos.


## 4.3. Amplitude of Front Wheel Steering AngleA
=
0.04 rad

Figures31–38 are the simulation results with A
=
0.04 rad. It is clear that the system bifurcation has occurred, and the system has completely entered into chaos state.Figure 31
v
x-v
y-ω phase space trajectories (A
=
0.04 rad, front-wheel-drive mode).Figure 32
v
x-v
y-ω phase space portrait onto v
y-ω plane (A
=
0.04 rad, front-wheel-drive mode).Figure 33
State variables (A
=
0.04 rad, front-wheel-drive mode).Figure 34
Part of state variables sequence (A
=
0.04 rad, front-wheel-drive mode).Figure 35
Poincare section of phase trajectory (A
=
0.04 rad, front-wheel-drive mode).Figure 36
Power spectral density of state variables (A
=
0.04 rad, front-wheel-drive mode).Figure 37
Lyapunov exponents (A
=
0.04 rad, front-wheel-drive mode).Figure 38
Lyapunov exponents (A
=
0.04 rad, front-wheel-drive mode).The phase space trajectories in Figures31 and 32 show that the vehicle state has lost stability. In other words, the vehicle has entered into chaos. The state variables of time series in Figures 33 and 34 also show the severe oscillation. But its overall range is bounded. The Poincare section diagram in Figure 35 can also get the same conclusion that the Poincare section is the limited point set in bounded scope.In Figure36, the power spectral density of the state variables all presents the typical chaotic characteristics of broadband multimodal. Figures 37 and 38 are the Lyapunov exponents of state variables. It is obvious that, at the beginning of the simulation, the maximum Lyapunov exponent quickly stabilizes to the state in which maximum Lyapunov exponent is positive. It also shows that the system has completely entered into chaos.


## 5. Conclusion

(1) Under different driving conditions, vehicle steering and driving bifurcation characteristics exhibit obvious difference. So, the research on bifurcation characteristics without the driving mode is incomplete.(2) The quasiperiodic torus rupture chaos route is proposed for 5DOF vehicle dynamics system; namely, with the periodic front steering angle input, the stable quasiperiodic vehicle motion may turn into chaos. It is also proved that the essential characteristics of 5DOF vehicle dynamics system instability are the dynamic system chaotic motion.



---

*Source: 847258-2015-08-26.xml*