IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 8, NO. 2, JUNE 2007 195
Elucidating Vehicle Lateral Dynamics
Using a Bifurcation Analysis
Der-Cherng Liaw, Senior Member, IEEE, Hsin-Han Chiang, Member, IEEE, and Tsu-Tian Lee, Fellow, IEEE
Abstract—Issues of stability and bifurcation phenomena in ve-hicle lateral dynamics are presented. Based on the assumption of constant driving speed, a second-order nonlinear lateral dynamics model is obtained. Local stability and existence conditions for saddle-node bifurcation appearing in vehicle dynamics with re-spect to the variations in front wheel steering angle are then derived via system linearization and local bifurcation analysis. Bifurcation phenomena occurring in vehicle lateral dynamics might result in spin and/or system instability. A perturbation method is employed to solve for an approximation of system equi-librium near the zero value of the front wheel steering angle, which reveals the relationship between sideslip angle and the applied front wheel angle. Numerical simulations from an example model demonstrate the theoretical results.
Index Terms—Bifurcation analysis, perturbation method, vehicle’s lateral dynamics.
I. INTRODUCTION
I
N RECENT years, the study of vehicle lateral dynamics has attracted considerable attention (e.g., [1]–[9]). One of the major concerns in vehicle dynamics is safety. Due to the large number of traffic accidents occurring daily, the link between the nonlinear behavior of vehicle dynamics and the applied front wheel steering angle becomes a very important issue. Among the existing studies, the sliding mode approach has been used to design robust control laws for providing system stability with respect to large variations in system parameters such as axial velocity, vehicle mass, and the contact force between tire and road surface [3]. A 5-degree-of-freedom (DOF) vehicle model was used in [7] to design an extended Kalman filter for estimating the historic data of vehicle motion and tire force. Based on a linear model of vehicle lateral dynamics, linear control laws have been proposed in [4] and [5].Bifurcation theory and its corresponding analytical tech-niques have been recently well exploited (e.g., [10]–[14]). These methods have been successfully applied to the study and control of several engineering systems such as tethered satellite systems, jet engine compressors, longitudinal flight dynamics, Manuscript received October 24, 2005; revised July 10, 2006, September 19, 2006, September 20, 2006, and September 29, 2006. This work was supported by the Program for Promoting Academic Excellence of Universities under Grant 91X104 EX-91-E-FA06-4-4. The Associate Editor for this paper was R. W. Goudy.
D.-C. Liaw and H.-H. Chiang are with the Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: [email protected]; [email protected]).
T.-T. Lee is with the National Taipei University of Technology, Taipei 106, Taiwan, R.O.C. (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TITS.2006.888598
Fig. 1. Front-steering vehicle model.
and power systems (e.g., [15]–[18]). Although saddle-node bifurcation in vehicle dynamics was observed in [6] to link with system instability via a numerical example, no theoretical ana-lysis of possible bifurcation phenomena has been studied yet.
Instead of using a numerical approach, the main goal of this paper is to provide an analytical study of vehicle lateral dynamics. Based on previous successful applications to the an-alytical study of longitudinal flight dynamics, tethered satellite systems and jet engine compressors by Liaw et al. [15]–[18], the existences and corresponding stability conditions of system equilibrium will be analytically discussed in this paper.
This paper is organized as follows. Nonlinear dynamics of a vehicle system is recalled in Section II. It is followed by the analysis of existences and the corresponding stability conditions for system equilibrium. Analytical formulas will then be applied to the detection of possible occurrences of local bifurcation phenomena and the construction of corresponding existence conditions. Numerical studies with CarSim software [21] are also given in Section IV to demonstrate the analytical results.
II. NONLINEARVEHICLEDYNAMICS
Consider the vehicle’s steering dynamics as depicted in Fig. 1 (e.g., [19]). Here, we have front-wheel angle δfas system input
and both sideslip angle β and yaw rate γ as two system outputs for the steering characteristics. In addition, Lfdenotes the
dis-tance between the center of gravity (CG) and front-wheel axle, and Lris the distance between CG and rear-wheel axle,
respec-tively. Fyfland Fyrlare the cornering forces of the left front and
Fig. 2. Cornering characteristics of front and rear tires. Dotted line: high-friction road, and dashed line: low-high-friction road.
left rear tires, while Fyfrand Fyrrare the cornering forces for
the right front and right rear tires. Fxfland Fxrlare the traction
forces for the left front and left rear tires, while Fxfr and Fxrr
are the traction forces for the right front and right rear tires. We assume the vehicle body is symmetric about the longitu-dinal plane. Let Fyf = Fyfl+ Fyfr, Fyr= Fyrl+ Fyrr, Fxf =
Fxfl+ Fxfr, and Fxr= Fxrl+ Fxrr. The basic equations of
motion for steering dynamics with roll motion neglected were derived (e.g., [19]) and given as
m ( ˙ν− νβγ) = Fxf+ Fxr− Fyfsin δf (1) mν· ˙ β + γ = Fyf+ Fyr+ Fxfsin δf (2) and Iz˙γ = (LfFyf− LrFyr) cos β + LfFxfsin δf (3)
where m is the mass of the vehicle, Iz is the yaw moment
around z-axis, and ν is the longitudinal velocity.
In this paper, we focus on the characteristic analysis of lateral dynamics by assuming Fxf = 0, and the vehicle is not
accelerating or decelerating along the longitudinal direction, i.e., ˙ν = 0. Thus, (1) can then be neglected in this analysis. The steering dynamics for constant speed ν can then be reduced to a second-order model as given by
˙ β = 1 mν{Fyf+ Fyr} − γ (4) ˙γ = 1 Iz{Lf Fyf− LrFyr} cos β (5)
where Fyf is a function of β, γ, and δf, and Fyr is a function
of β and γ only. Examples of Fyf and Fyr are given in (21)
and (22).
III. STABILITY ANDBIFURCATIONANALYSIS Here, we will study the vehicle steering dynamics by using the second-order model given by (4) and (5) instead of (1)–(3). Details are given as follows.
TABLE I
VEHICLEPARAMETERS ANDVALUES
TABLE II
COEFFICIENTS OFMAGICFORMULA
A. Stability Analysis
Define x0= (β0, γ0)Tas an equilibrium point of system (4),
(5) for a given δf= δ0f. We then have Fyf β0, γ0, δf0 + Fyr β0, γ0, δ0f = γ0· mν (6) and LfFyf β0, γ0, δf0= LrFyr β0, γ0, δ0f (7) or cos β0= 0. (8)
Consider the condition as given in (8), we have β0= nπ +
π/2 for n = 0, 1, 2, . . .. It is clear that this condition cannot be
achieved for a vehicle. Thus, the equilibrium point x0 should satisfy the two conditions given in (6) and (7) only.
Let x = [β, γ]Tand ˜x = x− x0. Taking the linearization of
system (4), (5) at x = x0, we have ˙˜x = A˜x (9) where A = a1 a2 a3 a4 (10) with a1= 1 mν ∂ ∂β(Fyf+ Fyr) β0, γ0, δf0 (11) a2= 1 mν ∂ ∂r(Fyf+ Fyr) β0, γ0, δf0− 1 (12) a3= 1 Iz Lf ∂Fyf β0, γ0, δ0 f ∂β − Lr ∂Fyr β0, γ0, δ0 f ∂β cos β0 (13) and a4= 1 Iz Lf ∂Fyf β0, γ0, δ0 f ∂r − Lr ∂Fyr β0, γ0, δ0 f ∂r cos β0. (14)
LIAW et al.: ELUCIDATING VEHICLE LATERAL DYNAMICS USING A BIFURCATION ANALYSIS 197
Fig. 4. Equilibrium point valued at ν = 9 m/s. (a) Side-slip angle versus front-wheel angle. (b) Yaw rate versus front-wheel angle.
By applying the Routh–Hurwitz stability criterion, we then have the following stability results.
Lemma 1: The equilibrium point x0 of system (4), (5) is
asymptotically stable if a1+ a4< 0 and a1a4− a2a3> 0.
Here, the ais are given in (11)–(14). Moreover, the equilibrium
point x0is unstable if a
1+ a4> 0 and a1a4− a2a3< 0. Observation 1: It is known from the so-called “Magic
for-mula” that both values of a1and a4will generally be negative.
An example is given in Section III. Thus, the stability condition in Lemma 1 can then be reduced to where the equilibrium point
x0 of system (4), (5) will be stable if a
1a4− a2a3> 0 and
unstable if a1a4− a2a3< 0.
From Observation 1 above, it is clear that the system lin-earization of (4), (5) at x0 will in general not have a pair of
pure imaginary eigenvalues but might have zero eigenvalue. That means the lateral dynamics of a vehicle system will not
TABLE III
SADDLE-NODEBIFURCATION FORDIFFERENTSPEEDS
undergo Hopf bifurcation (e.g., [12]–[14]). However, it might have chance for the appearance of stationary bifurcation at some δf = δf0such that a1a4− a2a3= 0.
LIAW et al.: ELUCIDATING VEHICLE LATERAL DYNAMICS USING A BIFURCATION ANALYSIS 199
Fig. 5. Time responses with initials: β =−0.01 rad and γ = 0.1 rad/s. (a) Side-slip angle β versus time. (b) Yaw rate γ versus time.
B. Local Bifurcations
In the following, we will discuss the possibility of having a stationary bifurcation (e.g., [11] and [16]) for system (4), (5). Let x0be the equilibrium point such that a
1a4− a2a3= 0
with a1< 0 and a4< 0. This implies that the linearization
of system (4), (5) at x0 possesses one zero eigenvalue and
a stable eigenvalue λ = a1+ a4< 0. In order to study the
Fig. 6. Time responses with initials: β = 0.01 and γ =−0.1. (a) Side-slip angle β versus time. (b) Yaw rate γ versus time.
we recall the result regarding the existence condition of the so-called “saddle-node bifurcation” from (e.g., [10], [12], and [13]), as presented below.
Consider a class of nonlinear system with the approximation up to the second order in its state x as given by
˙x = L0x + Q0(x, x) + gµ. (15)
Here, the Jacobian matrix L0 possesses one zero eigenvalue
with remaining eigenvalues lying in the open left-half of the complex plane. The variable µ denotes the system parameter, and Q0(x, x) denotes the quadratic term in x. Moreover,
as-sume x = 0 is an equilibrium point of system (15) at µ = 0. We recall the following lemma (e.g., [10]) regarding the existence conditions of saddle-node bifurcation.
Lemma 2: The equilibrium point x = 0 of system (15) will
undergo saddle-node bifurcation from the origin at µ = 0 if 1) lg= 0;
2) lQ0(r, r)= 0.
TABLE IV
EXISTENCECONDITION OFTHEOREM1
Fig. 7. Saddle-node bifurcation point in (δf− ν) space.
Here, l and r denote the left and right eigenvectors correspond-ing to the zero eigenvalues of L0, respectively, with lr = 1.
Now, we apply Lemma 2 to the study of local bifurcation for system (4), (5). Define x0 as the equilibrium point of system (4), (5) with a1a4= a2a3. Let ˜x = x− x0 ∆= (x1, x2)T and µ = δf− δf0. Taking the Taylor series expansion of the system
(4), (5) at (x0, δ0
f), we then have
LIAW et al.: ELUCIDATING VEHICLE LATERAL DYNAMICS USING A BIFURCATION ANALYSIS 201
Fig. 8. Vehicle states (a) slip angle and (b) yaw rate before saddle-node point at ν = 72 km/h, δf =−9.09◦. Vehicle states (c) slip angle and (d) yaw rate after saddle-node point at ν = 72 km/h, δf=−9.14◦.
where O(2) denotes the remaining second-order term and other high-order terms g∆= g1 g2 = 1 mν ∂ ∂δf(Fyf+ Fyr) x0, δ0 f cos β0 Iz ∂ ∂δf (LfFyf− LrFyr) x0, δ0 f (17) and Q(˜x, ˜x) = q11x21+ q12x1x2+ q13x22 q21x21+ q22x1x2+ q23x22 . (18)
The Jacobian matrix A is the same as the one in (10) but with
a1a4= a2a3, and the values of qij are given in Appendix A.
From Observation 1, matrix A has one zero eigenvalue and a stable eigenvalue for a general vehicle. In the following, we assume a1< 0 and a4< 0 without loss of generality.
Define l and r as the left and right eigenvectors correspond-ing to the zero eigenvalues of A with lr = 1. We then have
l = a4 a1+ a4 − a2 a1+ a4 and r = 1 −a1 a2 T .
It is clear from the relation of a1a4= a2a3 that we can
rewrite r as r = 1 −a3 a4 T .
By applying Lemma 2 to system (16), we have
l· g = 1 a1+ a4 (a4g1− a2g2) (19) and lQ0(r, r) = a4 a1+ a4 − a2 a1+ a4 · q11+ q12 −a3 a4 + q13 −a3 a4 2 q21+ q22 −a1 a2 + q23 −a1 a2 2 = 1 a1+ a4 a4q11− a3q12+ a23 a4 q13 − a2q21+ a1q22− a21 a2 q23 . (20)
Fig. 9. Vehicle states (a) slip angle and (b) yaw rate before saddle-node point at ν = 108 km/h, δf=−4.89◦. Vehicle states (c) slip angle and (d) yaw rate after saddle-node point at ν = 108 km/h, δf=−4.93◦.
Theorem 1: The system (4), (5) will undergo saddle-node
bifurcation for the equilibrium point x0if the following
condi-tions hold:
1) a1a4= a2a3with a1< 0 and a4< 0;
2) a4g1= a2g2;
3) a4q11− a3q12+ (a23/a4)q13− a2q21+ a1q22−
(a21/a2)q23= 0.
C. Application to the Vehicle Dynamics With Magic Formula Type Cornering Force
There are many models for cornering force. In this paper, we adopt the well-known “Magic formula” mathematical model (e.g., [7]) for the nonlinear cornering forces Fyf and Fyras
Fyf = Dfsin Cftan−1 Bf(1− Ef)αf+ Eftan−1(Bfαf) (21) Fyr= Drsin Crtan−1 Br(1− Er)αr+ Ertan−1(Brαr) (22) where αf = β + tan−1 Lf ν r· cos β − δf and αr= β− tan−1 Lr ν r· cos β .
Here, αf and αrdenote the slip angle for the front and rear
tires, respectively. The approximation relationship between tire slip angles and cornering forces are depicted in Fig. 2.
It is clear from (21) that x = (β, r)T= (0, 0)T will make Fyf = Fyr= 0 when δ0f = 0. Thus, x0= (0, 0)T is an
equi-librium point for system (4), (5) for δ0
f = 0. This agrees with
the natural behavior of vehicle dynamics. In the following, we first consider to solve the approximation of equilibrium solution about x0= (0, 0)T.
It is known that θ≈ tan θ, sin θ ≈ θ, and cos θ ≈ 1 for
θ≈ 0. By setting ˙x = 0 to solve for the equilibrium solution x0= (β0, r0)Tof system (4), (5) near x0= (0, 0)T, we have,
from (21), that DfCfBf β0+Lf ν r 0−δ0 f + DrCrBr β0−Lr ν r 0 = mνr0 (23) LfDfCfBf β0+Lf ν r 0− δ0 f −LrDrCrBr β0−Lr ν r 0 = 0. (24)
LIAW et al.: ELUCIDATING VEHICLE LATERAL DYNAMICS USING A BIFURCATION ANALYSIS 203
Fig. 10. Vehicle states (a) slip angle and (b) yaw rate before saddle-node point at ν = 144 km/h, δf =−2.91◦. Vehicle states (c) slip angle and (d) yaw rate after saddle-node point at ν = 144 km/h, δf=−2.94◦.
By suitable manipulation, we can then rewrite (23) and (24) as
(Lf+ Lr)DfCfBf β0+Lf ν r 0− δ0 f = Lrmνr0 (25) and (Lf+ Lr)DrCrBr β0−Lr ν r 0 = Lfmνr0. (26)
The following observation can be made from (25) and (26).
Observation 2: In general, the values of constants Bf, Br, Cf, and Crare positive, while those of Dfand Drare negative.
Example values of these variables will be given in Section IV. Thus, it is not difficult to find from (25) and (26) that the values of β0and r0will have the same sign if
ν < νss =
−Lr(Lf+ Lr)DrCrBr Lfm
(27)
and the sign of the sideslip angle β0will be opposite from that of the yaw rate r0for large axial velocity ν.
Next, we consider the stability of system (4), (5) at the equilibrium point x0= (0, 0)Tby using Lemma 1. According
to the Magic formula as given in (21) and (22), the values of ai for the Jacobian matrix A at the equilibrium point
x0= (β0, γ0)T are calculated as those given in Appendix B.
For the case of x0= (0, 0)T, we have from Observation 2 a1= 1 mν(DfCfBf+ DrCrBr) < 0 (28) a2= 1 mν2(LfDfCfBf− LrDrCrBr)− 1 < 0 (29) a3= 1 Iz (LfDfCfBf− LrDrCrBr) > 0 (30) and a4= 1 Izν L2fDfCfBf+ L2rDrCrBr < 0. (31) It is clear from (28)–(31) that a1< 0, a2< 0, a3> 0, and a4< 0 for practical values of Bi, Ci, Di(i = f, r). One
ex-ample is given in Section IV. These agree with the discussions in Observation 1. The next stability result follows readily from Lemma 1.
Corollary 1: The equilibrium point x0= (0, 0)T for δ0 f is
asymptotically stable.
Since the equilibrium point x0= (0, 0)T is asymptotically
stable, the bifurcation phenomena will not emerge from it. According to (21) and (22), giand qij are also calculated and
given in Appendix B. Based on those values of ai, gi, and qij
given in Appendix B, it is not difficult to apply Theorem 1 to the study of existence conditions of saddle-node bifurcation for system (4), (5). Numerical examples are given in Section IV for finding such a possibility.
IV. NUMERICALSIMULATIONS
In this section, we present the numerical study of lateral vehicle dynamics as given in (4) and (5). The selected values of system parameters given in (21) and (22) are the same as those of [6]. Details are given in Tables I and II. In order to verify the study of this paper, the system parameters for low-friction roads are chosen such that the vehicle has a propensity to spin. This could conform the driving condition to that of traveling down hill at constant velocity with equivalent braking effect at zero throttle being applied.
As discussed in Section III, the applied front wheel angle δfis
treated as a bifurcation parameter. Computer code AUTO [20] is employed to do the numerical analysis since it can numeri-cally calculate the system equilibrium branches by the property of continuity, the eigenvalues of the Jacobian matrix at each equilibrium point, and then determine the corresponding system stability. It is known (e.g., [7], [13], and [14]) that a nonlinear system will have the property of stability exchange before and after the stationary bifurcation point and possess a new system equilibrium branch with respect to the variation of bifurcation parameter. This gives a guide to determine the occurrence of stationary bifurcation and used in AUTO software.
The bifurcation diagram of system (4), (5) with respect to the variation of δf is shown in Fig. 3. Note that in Fig. 3,
the solid line denotes the stable equilibrium point while the dashed line is for the unstable equilibrium point. As depicted in Fig. 3(a) and (b), the system equilibrium near the origin are all asymptotically stable for different driving speeds ν. This agrees with the analytical result given in Corollary 1. As discussed in Observation 2, the equilibrium values of β and γ will have the same sign if the value of velocity ν is less than νss ≈ 9.5 m/s.
Figs. 3 and 4 demonstrate such results. Saddle-node bifurcation emerging from the equilibrium solution of system (4), (5) is also observed in Fig. 3. The locations of the bifurcation points for different driving speeds, ν are obtained by using AUTO and given in Table III. It is clear from Fig. 3 that the system equilibrium changes stability at the saddle-node bifurcation point and the magnitude of δfcorresponding to the saddle-node
bifurcation becomes smaller as the velocity ν increases. Note that the equilibrium points within the range of δf
between the two saddle nodes are stable. It should be pointed out that the equilibrium of sideslip angle is divergent while the yaw rate is convergent for increasing δf in the bifurcation
diagram. Figs. 5 and 6, respectively, show the time responses for two initial conditions with ν = 20 m/s for different values of
δf. It can be seen that the vehicle is stable and ends at the value
corresponding to the equilibrium point x0 within the range |δf| < 0.0158 rad. It is observed in Figs. 5 and 6 that the system
becomes unstable when|δf| > 0.0158 rad. This is attributed by
the existence of saddle-node bifurcation and its corresponding stability property (e.g., [7], [11], and [12]).
The bifurcation analysis obtained in Section III is employed to verify the numerical results. As given in Table IV, the saddle-node bifurcation point for ν = 20 m/s does satisfy the existence conditions of Theorem 1. Thus, the analytical results presented in Section III will be very helpful in finding possible bifurcation scenarios and system stability. In addition, the location of
Fig. 11. Vehicle animation at ν = 72 km/h in CarSim (a) before saddle-node bifurcation point (δf=−9.09◦). (b) After saddle-node bifurcation point (δf=−9.14◦).
saddle-node bifurcation for values of δf with respect to the
variation in velocity ν is also given in Fig. 7.
Similar phenomena can be found by using the famous vehicle simulation software CarSim. An example of using the code CarSim to find the occurrence of saddle-node bifurcation is depicted in Figs. 8–10. It is observed from Figs. 8–10 that both slip angle and yaw rate will reach stable values before the saddle-node bifurcation point but become unstable after the bifurcation point for three different values of velocity. Note that all the simulations presented in Figs. 8–10 are obtained by sending the steering command at time t = 2 s. Computer animations of the vehicle’s behavior before and after the bi-furcation point are also obtained using CarSim and depicted in Fig. 11 to demonstrate the existence how the existence of saddle-node bifurcation might affect the vehicle’s stability. The location of saddle-node bifurcation in two-parameter space is also obtained as given in Fig. 12.
V. CONCLUSION
In this paper, we focused on the study of stability and non-linear behavior of vehicle lateral dynamics. This is achieved by applying the Routh–Hurwitz stability criterion and bifurcation theory to the second-order model of lateral vehicle dynamics. Without assuming small angles, the stability condition at a given constant velocity for steering systems is also derived, which depends heavily on the cornering force characteristics of
LIAW et al.: ELUCIDATING VEHICLE LATERAL DYNAMICS USING A BIFURCATION ANALYSIS 205
Fig. 12. Saddle-node bifurcation point in (ν− |δf|) plane for CarSim vehicle model.
the rear tires. Saddle-node bifurcation phenomenon is observed from the equilibrium solution of the nonlinear model, while it never appears in the linear bicycle model (e.g., [3]–[5]). It is found by this paper that the velocity plays a very important role in determining the location of bifurcation points, which is not discussed in the existing results (e.g., [6]). In fact, the value of the applied front wheel angle for the bifurcation points is found to be smaller as the velocity increases. This may provide a partial reason as to why the vehicle steering is hard to control and easily becomes unstable under high speeds.
APPENDIXA The values of qijare given as follows:
q11= 1 mν ∂2 ∂β2(Fyf+ Fyr) β0, r0, δ0f q12= 1 mν ∂2 ∂β∂r(Fyf+ Fyr) β0, r0, δ0f q13= 1 mν ∂2 ∂r2(Fyf+ Fyr) β0, r0, δf0 q21= 1 Iz ∂2 ∂β2LfFyf β0, r0, δ0f − ∂2 ∂β2LrFyr β0, r0, δf0cos β0 − 2 · ∂ ∂βLfFyf β0, r0, δf0 − ∂ ∂βLrFyr β0, r0, δ0fsin β0 q22= 1 Iz ∂2 ∂β∂rLfFyf β0, r0, δ0f − ∂2 ∂β∂rLrFyr β0, r0, δf0 cos β0 − ∂ ∂rLfFyf β0, r0, δf0 − ∂ ∂rLrFyr β0, r0, δf0sin β0 q23= 1 Iz ∂2 ∂r2LfFyf β0, r0, δ0f − ∂2 ∂r2LrFyr β0, r0, δf0cos β0. APPENDIXB
The values of aiand giare shown at the bottom of the page
and are continued on the next page, where ν, m, Iz, Lj, Bj, Cj,
Dj, Ej(j = f, r) are as defined in previous sections.
a1= 1 mν Dfcos Cftan−1 Bf(1− Ef)αf β0, r0, δf0+ Eftan−1 Bfαf β0, r0, δf0 · Cf 1+(Bf(1− Ef)αf(β0, r0, δ0f)+ Eftan−1(Bfαf(β0, r0, δf0))) 2 · Bf(1− Ef)+ EfBf 1+(Bfαf(β0, r0, δ0f))2 · 1− Lf νr 0sin β0 1+Lf νr0cos β0 2 + Drcos Crtan−1 Br(1− Er)αr(β0, r0)+ Ertan−1 Brαr(β0, r0) · Cr 1+(Br(1− Er)αr(β0, r0)+Ertan−1(Brαr(β0, r0)))2 · Br(1− Er)+ ErBr 1+(Brαr(β0, r0))2 · 1+ Lr ν r0sin β0 1+Lr ν r0cos β0 2
a2= 1 mν Dfcos Cftan−1 Bf(1− Ef)αf β0, r0, δf0+ Eftan−1 Bfαf β0, r0, δf0 · Cf 1+(Bf(1− Ef)αf(β0, r0, δ0f)+ Eftan−1(Bfαf(β0, r0, δ0f))) 2 · Bf(1− Ef)+ EfBf 1+(Bfαf(β0, r0, δ0f))2 · Lf ν cos β0 1+Lf νr0cos β0 2 +Drcos Crtan−1 Br(1− Er)αr(β0, r0)+ Ertan−1 Brαr(β0, r0) · Cr 1+(Br(1− Er)αr(β0, r0)+ Ertan−1(Brαr(β0, r0)))2 · Br(1− Er)+ ErBr 1+(Brαr(β0, r0))2 · − Lνrcos β0 1+Lr νr0cos β0 2 − 1 a3= cos β Iz LfDfcos Cftan−1 Bf(1− Ef)αf β0, r0, δ0f+ Eftan−1 Bfαf β0, r0, δ0f · Cf 1+(Bf(1− Ef)αf(β0, r0, δ0f)+ Eftan−1(Bfαf(β0, r0, δ0f))) 2 · Bf(1− Ef)+ EfBf 1+(Bfαf(β0, r0, δ0f))2 · 1− Lf ν r 0sin β0 1+Lf ν r0cos β0 2 − LrDrcos Crtan−1 Br(1− Er)αr(β0, r0) + Ertan−1 Brαr(β0, r0) · Cr 1 + (Br(1− Er)αr(β0, r0) + Ertan−1(Brαr(β0, r0)))2 · Br(1− Er) + ErBr 1 + (Brαr(β0, r0))2 · 1 + Lr ν r 0sin β0 1 +Lr ν r0cos β0 2 a4= cos β Iz LfDfcos Cftan−1 Bf(1− Ef)αf β0, r0, δ0f+ Eftan−1 Bfαf β0, r0, δf0 · Cf 1 + (Bf(1− Ef)αf(β0, r0, δ0f) + Eftan−1(Bfαf(β0, r0, δf0))) 2 · Bf(1− Ef) + EfBf 1 + (Bfαf(β0, r0, δ0f))2 · Lf ν cos β 0 1 +Lf νr0cos β0 2 − LrDrcos Crtan−1 Br(1− Er)αr(β0, r0) + Ertan−1 Brαr(β0, r0) · Cr 1 + (Br(1− Er)αr(β0, r0) + Ertan−1(Brαr(β0, r0)))2 · Br(1− Er) + ErBr 1 + (Brαr(β0, r0))2 · − Lνrcos β 0 1 +Lr νr0cos β0 2 g1= 1 mν Dfcos Cftan−1 Bf(1− Ef)αf β0, r0, δf0 · Cf 1 + (Bf(1− Ef)αf(β0, r0, δf0) + Eftan−1(Bfαf(β0, r0, δf0))) 2 · −Bf(1− Ef)− EfBf 1 + (Bfαf(β0, r0, δf0))2 g2= Lfcos β Iz Dfcos Cftan−1 Bf(1− Ef)αf(β0, r0, δ0f) · Cf 1 + (Bf(1− Ef)αf(β0, r0, δ0f) + Eftan−1(Bfαf(β0, r0, δf0))) 2 · −Bf(1− Ef)− EfBf 1 + (Bfαf(β0, r0, δ0f))2
LIAW et al.: ELUCIDATING VEHICLE LATERAL DYNAMICS USING A BIFURCATION ANALYSIS 207
ACKNOWLEDGMENT
The authors would like to thank the reviewers and the As-sociate Editor for their helpful and detailed comments, which have helped to improve the presentation of this paper.
REFERENCES
[1] A. Vahidi and A. Eskandarian, “Research advances in intelligent collision avoidance and adaptive cruise control,” IEEE Trans. Intell. Transp. Syst., vol. 4, no. 3, pp. 143–153, Sep. 2003.
[2] L. Li, F. Y. Wang, and Q. Zhou, “Integrated longitudinal and lateral tire/ road friction modeling and monitoring for vehicle motion control,” IEEE Trans. Intell. Transp. Syst., vol. 7, no. 1, pp. 1–19, Mar. 2006.
[3] J. Ackerman, J. Guldner, W. Sienel, R. Steinhauser, and V. I. Utkin, “Linear and nonlinear controller design for robust automatic steering,” IEEE Trans. Control Syst. Technol., vol. 3, no. 1, pp. 132–143, Mar. 1995. [4] H. Peng and M. Tomizuka, “Vehicle lateral control for highway automa-tion,” in Proc. Amer. Control Conf., San Diego, CA, May 23–25, 1990, pp. 788–794.
[5] H. Peng, T. Hessburg, M. Tomizuka, and W. Zhang, “A theoretical and experimental study on vehicle lateral control,” in Proc. Amer. Control Conf., Chicago, IL, Jun. 24–26, 1992, pp. 1738–1742.
[6] E. Ono, S. Hosoe, H. D. Tuan, and S. Doi, “Bifurcation in vehicle dynam-ics and robust front wheel steering control,” IEEE Trans. Control Syst. Technol., vol. 6, no. 3, pp. 412–420, May 1998.
[7] L. R. Ray, “Nonlinear state and tire force estimation for advanced vehicle control,” IEEE Trans. Control Syst. Technol., vol. 3, no. 1, pp. 117–124, Mar. 1995.
[8] J. E. Naranjo, C. Gonzalez, R. Garcia, T. Pedro, and R. E. Haber, “Power-steering control architecture for automatic driving,” IEEE Trans. Intell. Transp. Syst., vol. 6, no. 4, pp. 406–415, Dec. 2005.
[9] L. Beji and Y. Bestaoui, “Motion generation and adaptive control method of automated guided vehicles in road following,” IEEE Trans. Intell. Transp. Syst., vol. 6, no. 1, pp. 113–123, Mar. 2005.
[10] N. Kopell and L. N. Howard, “Bifurcations and trajectories joining critical points,” Adv. Math., vol. 18, pp. 306–358, 1975.
[11] E. H. Abed and J. H. Fu, “Local feedback stabilization and bifurcation control, II. Stationary bifurcation,” Syst. Control Lett., vol. 8, no. 5, pp. 467–473, May 1987.
[12] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag, 1983.
[13] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory. New York: Springer-Verlag, 1992.
[14] L. N. Howard, “Nonlinear oscillation,” in Nonlinear Oscillations in Biology, F. C. Hoppendteadt, Ed. Providence, RI: Amer. Math. Soc., 1979, pp. 1–67.
[15] D.-C. Liaw and E. H. Abed, “Stabilization of tethered satellites during station-keeping,” IEEE Trans. Autom. Control, vol. 35, no. 11, pp. 1186–1196, Nov. 1990.
[16] D.-C. Liaw and E. H. Abed, “Active control of compressor stall inception: A bifurcation-theoretic approach,” Automatica, vol. 32, no. 1, pp. 109– 115, 1996.
[17] D.-C. Liaw and C.-C. Song, “Analysis of longitudinal flight dynamics: A bifurcation-theoretic approach,” J. Guid. Control Dyn., vol. 24, no. 1, pp. 109–116, 2001.
[18] D.-C. Liaw, K.-H. Fang, and C.-C. Song, “Bifurcation analysis of power systems with tap changer,” in Proc. IEEE ICNSC, Tucson, AZ, Mar. 19–22, 2005, pp. 283–288.
[19] J. Y. Wong, Theory of Ground Vehicles. Hoboken, NJ: Wiley, 2001. [20] E. J. Doedel, “AUTO: A program for the automatic bifurcation analysis of
autonomous systems,” Congressus Numerantium, vol. 30, pp. 265–284, 1981.
[21] CarSim User Manual, Mech. Simul. Corp., Ann Arbor, MI, 2001, Version 5.11.
Der-Cherng Liaw (S’86–M’90–SM’02) received
the B.S. degree in control engineering from National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C., in 1982, the M.S. degree in electrical en-gineering from National Taiwan University, Taipei, Taiwan, in 1985, and the Ph.D. degree in electrical engineering from the University of Maryland, Col-lege Park, in 1990.
During 1990–1991, he was a Postdoctoral Fellow with the Institute of Systems Research, University of Maryland. Since August 1991, he has been with the NCTU, Hsinchu, where he is currently a Professor of electrical and control en-gineering. His research interests include nonlinear control systems, spacecraft control, singular perturbation methods, bifurcation control, jet engine control, flight control, electric power systems and power electronics, radio frequency identification (RFID), and implementation issues.
Dr. Liaw served as a Designated Assistant to the Associate Editor for the IEEE TRANSACTIONS ONAUTOMATICCONTROLduring 1990–1992. He was also a member of Editorial Board of 1993 American Control Conference.
Hsin-Han Chiang (S’02–M’03) received the B.S.
degree in electrical and control engineering from National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C., in 2001, where he is currently work-ing toward the Ph.D. degree in electrical and control engineering.
His research interests include intelligent systems and control theory, fuzzy systems and control, auto-mated vehicle control, and intelligent transportation systems.
Tsu-Tian Lee (M’87–SM’89–F’97) was born in
Taipei, Taiwan, R.O.C., in 1949. He received the B.S. degree in control engineering from the National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C., in 1970 and the M.S. and Ph.D. degrees in electrical engineering from the University of Oklahoma, Norman, in 1972 and 1975, respectively. In 1975, he was appointed as an Associate Pro-fessor and, in 1978, as a ProPro-fessor and Chairman of the Department of Control Engineering, NCTU. In 1981, he became a Professor and Director of the Institute of Control Engineering, NCTU. In 1986, he was a Visiting Professor and, in 1987, a Full Professor of electrical engineering with the University of Kentucky, Lexington. In 1990, he was a Professor and Chairman of the Department of Electrical Engineering, National Taiwan University of Science and Technology (NTUST), Taipei, Taiwan, R.O.C. In 1998, he became a Professor and Dean of the Office of Research and Development, NTUST. In 2000, he was appointed as a Chair Professor with the Department of Electrical and Control Engineering, NCTU. Since 2004, he has been with the National Taipei University of Technology, Taipei, where he is currently the President.
Prof. Lee was elected as a Fellow of the Institution of Electrical Engineers in 2000. He became a Fellow of the New York Academy of Sciences in 2002. He has served as a member of the Technical Program Committee and a member of the Advisory Committee for many IEEE sponsored international conferences. He is now the Vice President for Conferences of the IEEE Systems, Man, and Cybernetics Society. He received the Distinguished Research Award from the National Science Council, Taiwan, R.O.C., in 1991–1998; the Academic Achievement Award in Engineering and Applied Science from the Ministry of Education, Taiwan, R.O.C., in 1997; the National Endow Chair from the Ministry of Education, Taiwan, R.O.C., in 2003 and 2006, respectively; and the TECO Science and Technology Award from TECO Technology Foundation in 2003.
