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JMST Advances

, Volume 1, Issue 1–2, pp 13–21 | Cite as

Integrated chassis control with AFS, ARS and ESC under lateral force constraint on AFS

  • Seongjin YimEmail author
  • Young Hee Jo
Letter
  • 128 Downloads

Abstract

This paper presents an integrated chassis control with active front steering (AFS), active rear steering (ARS) and electronic stability control (ESC) under lateral force constraint on front wheels. The control yaw moment is calculated using sliding mode control. A weighted pseudo-inverse-based control allocation (WPCA) is used for yaw moment distribution. On low-friction road, AFS has little effect on control performance since the lateral tire forces of front wheels are easily saturated. To overcome the problem, the lateral force generated by AFS is limited to its maximum, and the braking force of ESC and the angle of ARS, obtained from WPCA, are applied. To check the effectiveness of the proposed method, simulation was performed on the vehicle simulation package, CarSim. From simulation, it was verified that the proposed method can enhance the maneuverability and lateral stability, if the lateral forces on front wheels are saturated.

Keywords

Integrated chassis control Active front steering Active rear steering Electronic stability control Lateral force saturation 

List of symbols

ay

Lateral acceleration (m/s2)

Cf, Cr

Cornering stiffness of front/rear tires (N/rad)

Fyf, Fyr

Lateral tire forces of front and rear wheels (N)

Fyfc, Fyrc

Lateral tire force generated by AFS and ARS (N)

Fxi

Longitudinal tire forces generated by ESC (N)

g

Gravitational constant (= 9.81 m/s2)

H, D

Effectiveness matrix in WPCA

Iz

Yaw moment of inertial (kg·m2)

Kγ

Gain in sliding mode control

KB

Pressure–force constant (N·m/MPa)

lf, lr

Distance from C.G. to front and rear axles (m)

m

Vehicle total mass (kg)

ΔMB, ΔMN

Control yaw moment (Nm)

PB

Brake pressure of ESC (MPa)

x, y

Vector of tire forces in WPCA

rw

Radius of a wheel (m)

tf, tr

Track width of front and rear axles (m)

vx, vy

Longitudinal and lateral velocities of a vehicle (m/s)

V

Vehicle speed (m/s)

W, V

Weighting matrix in WPCA

αf, αr

Tire slip angles of front and rear wheels (rad)

β

Side-slip angle (rad)

ε

Weight on a particular tire force in WPCA

δf, δr

Front and rear steering angles (rad)

Δδf

Corrective steering angle of AFS (rad)

γ, γd

Real and reference yaw rates (rad/s)

η

Tuning parameters on yaw rate error and side-slip angle

μ

Tire–road friction coefficient

ρ

Vector of variable weights in WPCA

1 Introduction

Generally, vehicle stability control (VSC) is designed to enhance maneuverability and lateral stability of a vehicle [1]. The maneuverability means that a vehicle follows the driver’s intention. Driver’s intention is represented by the reference yaw rate, algebraically calculated from the steering wheel angle. So, vehicle stability control improves the maneuverability by making the error between the real and reference yaw rates zero. Lateral stability means that the vehicle has small side-slip angle because the large side-slip angle stands for loss of stability. So, vehicle stability control maintains the lateral stability by reducing the side-slip angle within a certain level. Generally, the side-slip angle is restricted within 3° by vehicle stability control [1].

Vehicle stability controller has a two-level control structure: upper- and lower-level controllers [2]. The upper-level controller calculates control yaw moment, needed to stabilize a vehicle, using vehicle models such as 2-DOF bicycle model or 3-DOF planar model. Controller design methodologies such as LQ optimal control, sliding mode control and fuzzy control have been applied to it [3, 4, 5]. Lower-level controller determines the tire forces of several actuators, needed to generate the control yaw moment calculated from the upper-level controller. This is called yaw moment distribution [2].

In the lower-level controller, several actuators can be used for yaw moment distribution. The most representative actuator is electronic stability control (ESC), developed in the early 1990s [6]. ESC uses braking in generating the control yaw moment. ESC has known to be effective for vehicle stability, since it has been developed and commercialized. As a result, installation of ESC became mandatory in the late 2000s. Other devices used for yaw moment generation are active front steering (AFS), active rear steering (ARS), and torque vectoring devices (TVD) [7, 8, 9]. These devices have been adopted solely or together with ESC. If there are multiple actuators for yaw moment generation, it is called integrated chassis control (ICC). In this paper, the integrated chassis control with AFS, ARS and ESC is considered [10].

The lower-level controller of ICC should determine the corrective steering angles of AFS and ARS and the braking force of ESC to generate the control yaw moment. Up to date, several methods have been proposed for yaw moment distribution. Typical method is to apply optimization algorithm to determine the tire forces generated by ESC, AFS and ARS. The unified chassis control (UCC) with optimum distribution was proposed for yaw moment distribution [11]. In the method, the quadratic objective function was set to minimize the tire forces of ESC. The equilibrium condition between the control yaw moment and the tire forces was set as the equality constraint. The friction circle on tire forces was set as the inequality constraint. The constrained quadratic programming problem was solved algebraically by applying KKT optimality condition. Another optimization-based method for yaw moment distribution is the weighted pseudo-inverse-based control allocation (WPCA) [1, 10, 12]. In WPCA, the quadratic objective function with tire forces generated by ESC, AFS and ARS was set. The equilibrium condition between the control yaw moment and the tire forces was set as the equality constraint. This problem is the quadratic programming with single equality constraint, which can be algebraically solved. In this paper, the weighted pseudo-inverse-based control allocation (WPCA) is adopted for yaw moment distribution. With the variable weights of WPCA, it is easy to represent several actuator configurations used for yaw moment distribution [1, 10].

When applying WPCA for yaw moment distribution, the smaller weight on AFS makes ICC use only AFS. If only AFS is used for yaw moment distribution, the smaller speed reduction and the enhanced ride comfort can be achieved because the braking of ESC is not used [11]. However, the sole use of AFS can make the lateral tire force of front wheels easily saturated because the corrective steering angle of AFS is added to that of driver. Under the situation, AFS cannot generate the lateral tire force needed to generate the control yaw moment [13]. So, it is necessary to limit the lateral tire force of AFS, and to compensate the loss of the control yaw moment using ESC and ARS.

For the purpose, several researches have done up to date. Yim proposed the WPCA-based scheme to cope with the saturation of lateral tire forces on front wheels [14]. In the other research, Yim proposed the method by combining the UCC and WPCA to cope with the problem [15]. In these researches, ESC and AFS were used as an actuator to generate the control yaw moment. ARS or TVD has not been used for yaw moment distribution up to date under that condition.

Recent advances of ARS can make the performance of the vehicle stability control be enhanced [8]. Especially, the lateral stability, represented by the side-slip angle, can be drastically improved by ARS. So, ARS is adopted as an actuator to generate the control yaw moment in this paper. Nagai et al. proposed ICC with ARS and braking-based DYC [16]. Yim proposed the ICC with ESC, AFS and ARS [10, 17, 18]. However, there have been little researches on ICC with ESC, AFS and ARS under the saturation of lateral tire forces on front wheels.

This paper proposes a method which copes with the saturation of lateral tire forces on front wheels, caused by the excessive application of AFS. Contrary to the previous works, the ICC proposed in this paper uses ESC, AFS and ARS. The proposed method compensates the loss of the control yaw moment, caused by the saturated lateral tire force of AFS, by re-applying WPCA with ESC and ARS if the lateral tire forces of front wheels are saturated. Since WPCA algebraically calculates the optimum solution, it is easy to implement WPCA in real vehicles.

This paper is organized as follows: Sect. 2 presents the design procedure of the integrated chassis controller. New method to compensate the saturated lateral tire forces of front wheels is proposed in Sect. 3. To validate the proposed method, simulation is performed in Sect. 4. Finally, Sect. 5 concludes this paper.

2 Design of the integrated chassis controller

In this section, the integrated chassis controller with ESC, AFS, and ARS is designed. Direct yaw moment control and WPCA are adopted for control yaw moment generation and yaw moment distribution in the upper- and lower-level controllers, respectively. Variable weights in WPCA are defined for several purposes.

2.1 Design of an upper-level controller

The 2-DOF bicycle model is used to design a yaw moment controller. Figure 1 shows the 2-DOF bicycle model. This model describes the yaw and the lateral motions of a vehicle, assuming that the longitudinal velocity vx is constant [2].
Fig. 1

2-DOF bicycle model

The equations of motion for the 2-DOF bicycle model are obtained as follows:
$$\begin{aligned} mv_{x} (\dot{\beta } + \gamma ) = F_{yf} \cos \delta_{\text{f}} + F_{yr} \hfill \\ I_{z} \dot{\gamma } = l_{\text{f}} F_{yf} \cos \delta_{\text{f}} - l_{\text{r}} F_{yr} + \Delta M_{B} . \hfill \\ \end{aligned}$$
(1)
In (1), the lateral tire forces Fyf and Fyr are assumed to be linearly proportional to the tire slip angle for small α, as shown in (2) [2]. The tire slip angles αf and αr of front and rear wheels are defined as the difference between the direction of wheel velocity and the steering angle, as given in (3), which can be obtained through the approximation \(\tan^{ - 1} (\theta ) \approx \theta\).
$$F_{yf} = - C_{\text{f}} \alpha_{\text{f}} ,\quad F_{yr} = - \;C_{\text{r}} \alpha_{\text{r}} ,$$
(2)
$$\left\{ \begin{array}{l} \alpha_{\text{f}} = \frac{{v_{y} + l_{\text{f}} \gamma }}{{v_{x} }} - \delta_{\text{f}} = \beta + \frac{{l_{\text{f}} \gamma }}{{v_{x} }} - \delta_{\text{f}} \quad \hfill \\ \alpha_{\text{r}} = \frac{{v_{y} - l_{\text{r}} \gamma }}{{v_{x} }} = \beta - \frac{{l_{\text{r}} \gamma }}{{v_{x} }} \hfill \\ \end{array} \right.,$$
(3)
The reference yaw rate, generated by driver’s steering input, can be algebraically calculated from the formula, given in Eq. (4), under the assumption that the lateral tire force is linear [2].
$$\gamma_{d} = \frac{{C_{\text{f}} \cdot C_{\text{r}} \cdot (l_{\text{f}} + l_{\text{r}} ) \cdot v_{x} }}{{C_{\text{f}} \cdot C_{\text{r}} \cdot (l_{\text{f}} + l_{\text{r}} )^{2} + m \cdot v_{x}^{2} \cdot (l_{\text{r}} \cdot C_{\text{r}} - l_{\text{f}} \cdot C_{\text{f}} )}} \cdot \delta_{\text{f}} .$$
(4)
There are two typical objectives in vehicle stability control. The first is the maneuverability, which means the yaw rate tracking. The second is the lateral stability, which means the small side-slip angle. To improve these objectives, it is necessary for a yaw moment controller to make a vehicle follow the reference yaw rate and reduce the side-slip angle. To design a yaw moment controller, a sliding mode control is adopted in this paper. To achieve these objectives, the error surface is defined as Eq. (5). In Eq. (5), η is the parameter used to tune the trade-off between the yaw rate error and the side-slip angle. For the error surface to have a stable dynamics, the condition (6) should be satisfied [4].
$$s = (\gamma - \gamma_{d} ) + \eta \cdot \beta ,$$
(5)
$$\dot{s} = - \;K_{\gamma } s\quad (K_{\gamma } > 0).$$
(6)
By differentiating Eq. (5) and combining it with Eqs. (1) and (6), the control yaw moment ΔMB is obtained as Eq. (7).
$$\begin{aligned} \Delta M_{B} = I_{z} \cdot \dot{\gamma }_{d} + I_{z} \cdot \eta \cdot \left( {\frac{{F_{yf} \cos \delta_{\text{f}} + F_{yr} }}{{mv_{x} }} - \gamma } \right) \hfill \\ \;\;\;\; - l_{\text{f}} F_{yf} \cos \delta_{\text{f}} + l_{\text{r}} F_{yr} - I_{z} \cdot K_{\gamma } \cdot (\gamma - \gamma_{d} + \eta \cdot \beta ). \hfill \\ \end{aligned}$$
(7)

2.2 Design of a lower-level controller

Once the control yaw moment ΔMB is computed in the upper-level controller, it should be distributed into braking forces of ESC and steering angles of AFS and ARS. In this paper, a WPCA is adopted to distribute the control yaw moment into the tire forces generated by ESC, AFS and ARS [10].

Figure 2 shows the geometric relationship between the tire forces and the control yaw moment when the control yaw moment is positive [10]. In Fig. 2, Fx1, Fx2, Fx3 and Fx4 are the longitudinal braking forces generated by ESC, and Fyfc and Fyrc are the lateral tire forces generated by AFS and ARS, respectively. These tire forces should be determined to generate the control yaw moment ΔMB. A WPCA is used to determine the tire forces [1].
Fig. 2

Coordinate system corresponding to tire forces

Equation (8) shows the geometric relation between the tire forces and control yaw moment, given in Fig. 2. The elements in the vector H in (8) are given in (9).
$$\underbrace {{\left[ {\begin{array}{*{20}c} {a_{1} } & {a_{2} } & {a_{3} } & {a_{4} } & {a_{5} } & {a_{6} } \\ \end{array} } \right]}}_{{\mathbf{H}}}\underbrace {{\left[ {\begin{array}{*{20}c} {F_{yfc} } \\ {F_{yrc} } \\ {F_{x1} } \\ {F_{x2} } \\ {F_{x3} } \\ {F_{x4} } \\ \end{array} } \right]}}_{{\mathbf{q}}} = \Delta M_{B} ,$$
(8)
$$\left\{ \begin{array}{l} a_{1} = 2l_{\text{f}} \cos \delta_{\text{f}} ,\quad a_{2} = - \;2l_{\text{r}} \cos \delta_{\text{r}} \hfill \\ a_{3} = - l_{\text{f}} \sin \delta_{\text{f}} + \frac{{t_{\text{f}} }}{2}\cos \delta_{\text{f}} ,\quad a_{4} = - \;l_{\text{f}} \sin \delta_{\text{f}} - \frac{{t_{\text{f}} }}{2}\cos \delta_{\text{f}} \hfill \\ a_{5} = l_{\text{r}} \sin \delta_{\text{r}} + \frac{{t_{\text{r}} }}{2}\cos \delta_{\text{r}} ,\quad a_{6} = l_{\text{r}} \sin \delta_{\text{r}} - \frac{{t_{\text{r}} }}{2}\cos \delta_{\text{r}} \hfill \\ \end{array} \right. .$$
(9)
The objective function of WPCA is defined as follows:
$$\begin{aligned} J = \frac{{\rho_{1} F_{yfc}^{2} + \rho_{3} F_{x1}^{2} }}{{(\mu F_{z1} )^{2} }} + \frac{{\rho_{1} F_{yfc}^{2} + \rho_{4} F_{x2}^{2} }}{{(\mu F_{z2} )^{2} }} \hfill \\ + \ \frac{{\rho_{2} F_{yrc}^{2} + \rho_{5} F_{x3}^{2} }}{{(\mu F_{z3} )^{2} }} + \frac{{\rho_{2} F_{yrc}^{2} + \rho_{6} F_{x4}^{2} }}{{(\mu F_{z4} )^{2} }} \hfill \\ = {\mathbf{q}}^{T} {\mathbf{Wq}} \hfill \\ \end{aligned} ,$$
(10)
where \({\mathbf{W}} = {\text{diag}}\left[ {\frac{1}{{\xi_{1}^{2} }} + \frac{1}{{\xi_{2}^{2} }},\frac{1}{{\xi_{3}^{2} }} + \frac{1}{{\xi_{4}^{2} }},\frac{1}{{\xi_{1}^{2} }},\frac{1}{{\xi_{2}^{2} }},\frac{1}{{\xi_{3}^{2} }},\frac{1}{{\xi_{4}^{2} }}} \right]{\varvec{\uprho}}\), ρ = [ρ1ρ2ρ3ρ4ρ5ρ6]T, and ξi= μFzi.

In Eq. (10), the vertical tire forces should be estimated because these cannot be easily measured. The vertical tire forces can be estimated with the longitudinal and lateral accelerations, as given in the previous work [11].

In Eq. (10), ρ is the vector of fictitious variable weights ρi. In this paper, the variable weight ρ is used for several purposes [1, 10]. Firstly, ρ is used to capture several actuator combinations such as ESC, AFS, ARS, ESC + AFS, ESC + ARS, ESC + AFS + ARS, and AFS + ARS. Secondly, ρ is used to limit excessive tire slip ratio and slip angle when ESC, AFS and ARS are applied. Thirdly, ρ is used to improve the maneuverability and lateral stability with simulation based tuning. The detailed roles of ρ will be explained later.

The optimization problem is the quadratic programming with an equality constraint. Applying the Lagrange multiplier technique to this problem, the optimum solution can be easily obtained as Eq. (11). From (11), the optimal solution can be easily calculated from an algebraic formula without matrix operations. So, it is easy to implement this procedure. After obtaining the optimum solution xopt, each tire force should be converted into the braking pressure PB, active front steering angle Δδf and active rear steering angle δr. The conversion from the tire force into PB, Δδf and δr will be explained in Sect. 2.3.
$${\mathbf{x}}_{\text{opt}} = {\mathbf{W}}^{ - 1} {\mathbf{H}}^{\text{T}} ({\mathbf{HW}}^{ - 1} {\mathbf{H}}^{\text{T}} )^{ - 1} \Delta M_{B} .$$
(11)

In this paper, four actuator configurations, ESC, AFS, ARS and ESC + ARS, are considered. For these actuator combinations, a particular constraint on yaw moment distribution is needed. For example, if only ESC is available and the control yaw moment is positive, then the longitudinal braking forces of left wheels, i.e., Fx1 and Fx3 in Fig. 2, should be generated and those of right wheels, i.e., Fx2 and Fx4, should not be generated.

To capture these actuator configurations, the vector of variable weights, ρ, in Eq. (10) is introduced [1, 10]. If the variable weight ρi in ρ is decreased, then the corresponding tire force Fxi or Fyfc or Fyrc is increased, and vice versa. Using this fact, the distribution scheme can be set for each actuator combination. Let us assume that all the variable weights in ρ are set to 1e−4. In this situation, if only ESC is available and the control yaw moment ΔMB is positive, the braking pressure can be applied to the left wheels. For this purpose, ρ1, ρ2, ρ4 and ρ6 should be set to a high value, e.g., 1, as shown in Eq. (12). As a result of this setting, only Fx1 and Fx3 can be generated from WPCA. Following the fact, the sets of variable weights for actuator configurations, ESC, AFS, ARS and ESC + ARS, can be determined, as given from (12) to (15) [10].

ESC
$$\begin{aligned} {\varvec{\uprho}} = \left[ {\begin{array}{*{20}c} 1 & 1 & {\varepsilon_{1} } & 1 & {\varepsilon_{2} } & 1 \\ \end{array} } \right]\quad {\text{if}}\;\Delta M_{B} > 0 \hfill \\ {\varvec{\uprho}} = \left[ {\begin{array}{*{20}c} 1 & 1 & 1 & {\varepsilon_{1} } & 1 & {\varepsilon_{2} } \\ \end{array} } \right]\quad {\text{if}}\;\Delta M_{B} < 0 \hfill \\ \end{aligned} ,$$
(12)
AFS
$$\begin{aligned} {\varvec{\uprho}} = \left[ {\begin{array}{*{20}c} {\varepsilon_{1} } & 1 & 1 & 1 & 1 & 1 \\ \end{array} } \right]\quad {\text{if}}\;\Delta M_{B} > 0 \hfill \\ {\varvec{\uprho}} = \left[ {\begin{array}{*{20}c} {\varepsilon_{1} } & 1 & 1 & 1 & 1 & 1 \\ \end{array} } \right]\quad {\text{if}}\;\Delta M_{B} < 0 \hfill \\ \end{aligned} ,$$
(13)
ARS
$$\begin{aligned} {\varvec{\uprho}} = \left[ {\begin{array}{*{20}c} 1 & {\varepsilon_{1} } & 1 & 1 & 1 & 1 \\ \end{array} } \right]\quad {\text{if}}\;\Delta M_{B} > 0 \hfill \\ {\varvec{\uprho}} = \left[ {\begin{array}{*{20}c} 1 & {\varepsilon_{1} } & 1 & 1 & 1 & 1 \\ \end{array} } \right]\quad {\text{if}}\;\Delta M_{B} < 0 \hfill \\ \end{aligned} ,$$
(14)
ESC + ARS
$$\begin{aligned} {\varvec{\uprho}} = \left[ {\begin{array}{*{20}c} 1 & {\varepsilon_{1} } & {\varepsilon_{2} } & 1 & {\varepsilon_{3} } & 1 \\ \end{array} } \right]\quad {\text{if}}\;M_{B} > 0 \hfill \\ {\varvec{\uprho}} = \left[ {\begin{array}{*{20}c} 1 & {\varepsilon_{1} } & 1 & {\varepsilon_{2} } & 1 & {\varepsilon_{3} } \\ \end{array} } \right]\quad {\text{if}}\;M_{B} < 0 \hfill \\ \end{aligned} .$$
(15)

In the actuator configuration of ESC + ARS, the variable weights, ε1, ε2 and ε3, correspond to the use of ARS and ESC on the front and rear wheels, respectively. If a particular variable weight increases, then the corresponding actuator will be less used. For instance, if ε1 increases, lesser AFS will be used for yaw moment distribution.

2.3 Determination of corrective angles of AFS and ARS

The optimum solution, i.e., optimum tire forces, from WPCA should be converted into PB, Δδf and δr. The braking pressure PB at each wheel can be easily calculated with (16). The corrective steering angles of AFS and ARS are not easy to calculate via a certain algebraic formula. They can be calculated with the definition of the lateral tire forces and the tire slip angles, as given in (2) and (3), respectively [2]. For the calculation, the first step is to calculate the tire slip angle using (17) from (2). The second step is to represent the tire slip angle with Δδf and δr, as given in (18). By combining (17) and (18), Δδf and δr are calculated as (19). When using (19), it is necessary to measure or estimate the side-slip angle β. For the purpose, the previously proposed method is adopted [19].
$$P_{{{\text{B}}i}} = \frac{{r_{\text{w}} }}{{K_{\text{B}} }}F_{xi} \quad (i = 1,2,3,4)$$
(16)
$$\alpha_{\text{f}} = - \frac{{F_{yfc} }}{{C_{\text{f}} }},\quad \alpha_{\text{r}} = - \frac{{F_{yrc} }}{{C_{\text{r}} }}$$
(17)
$$\left\{ \begin{array}{l} \alpha_{\text{f}} = \beta + \frac{{l_{\text{f}} \gamma }}{{v_{x} }} - (\delta_{\text{f}} + \Delta \delta_{\text{f}} )\quad \hfill \\ \alpha_{\text{r}} = \beta - \frac{{l_{\text{r}} \gamma }}{{v_{x} }} - \delta_{\text{r}} \quad \hfill \\ \end{array} \right.$$
(18)
$$\left\{ \begin{array}{l} \Delta \delta_{\text{f}} = \frac{{F_{yfc} }}{{C_{\text{f}} }} + \beta + \frac{{l_{\text{f}} \gamma }}{{v_{x} }} - \delta_{\text{f}} \hfill \\ \delta_{\text{r}} = \frac{{F_{yrc} }}{{C_{\text{r}} }} + \beta - \frac{{l_{\text{r}} \gamma }}{{v_{x} }} \hfill \\ \end{array} \right.$$
(19)

3 Compensation of the saturated lateral tire forces of front wheels

As given in (13), the sole use of AFS in ICC makes smaller speed reduction and better ride comfort due to no braking. However, the lateral tire forces of front wheels are easily saturated due to excessive steering of AFS and drivers’. As a result, the optimum front lateral tire force Fyfc obtained by WPCA cannot be generated at front wheels. Figure 3 shows the fact [13]. As shown in Fig. 3, it is not effective to use AFS if the tire slip angles of front wheels are over 6°. Moreover, Fyfc cannot be generated at low-friction roads because the radius of friction circle, μ·Fz, is reduced. Figure 4 shows the variation of the lateral tire force according to that of tire–road friction coefficient [20]. As shown in Fig. 4, the lateral tire force is significantly reduced as the tire–road friction becomes smaller. So, it is necessary to limit Fyfc to the maximum of the front lateral tire force, and to compensate the saturated front lateral tire force by ESC and ARS.
Fig. 3

Characteristic of lateral tire force with respect to tire slip angle

Fig. 4

Lateral tire forces with respect to vertical tire forces and slip angles at μ = 0.87

The maximum of the lateral tire force is calculated with (20). In (20), q1(Fz, μ) is the maximum value of the lateral tire force obtained statically from a carpet plot or experimental data for given Fz and μ regardless of the tire slip angle, as given in Fig. 4, and q2(Fx, Fz, μ) is the maximum value of the lateral tire force dynamically calculated from the definition of friction circle, as given in (21). As given in (20), the maximum lateral tire force, Fy,max, is calculated as the minimum of q1 and q2.
$$F_{y,\hbox{max} } = \hbox{min} \{ q_{1} (F_{z} ,\mu ),q_{2} (F_{x} ,F_{z} ,\mu )\}$$
(20)
$$q_{2} (F_{x} ,F_{z} ,\mu ) = \sqrt {(\mu F_{z} )^{2} - F_{x}^{2} }$$
(21)
The lateral tire force Fyfc generated by AFS should not exceed its maximum Fy,max. If Fyfc exceeds Fy,max, Fyfc is set to the constant, Fy,max. As a result, (8) and (10) are changed to (22) and (23) in WPCA, respectively. The solution of the optimization problem with objective function (23) and the equality constraint (22) is easily obtained as (24). This method is called constrained WPCA (CWPCA) [14].
$$\underbrace {{\left[ {\begin{array}{*{20}c} {a_{2} } & {a_{3} } & {a_{4} } & {a_{5} } & {a_{6} } \\ \end{array} } \right]}}_{{\mathbf{D}}}\underbrace {{\left[ {\begin{array}{*{20}c} {F_{yrc} } \\ {F_{x1} } \\ {F_{x2} } \\ {F_{x3} } \\ {F_{x4} } \\ \end{array} } \right]}}_{{\mathbf{y}}} = \Delta M_{B} - a{}_{1} \cdot F_{yfc} = \Delta M_{N}$$
(22)
$$\begin{aligned} L = \frac{{\rho_{3} F_{x1}^{2} }}{{(\mu_{1} F_{z1} )^{2} }} + \frac{{\rho_{4} F_{x2}^{2} }}{{(\mu_{2} F_{z2} )^{2} }} + \frac{{\rho_{2} F_{yrc}^{2} + \rho_{5} F_{x3}^{2} }}{{(\mu_{3} F_{z3} )^{2} }} + \frac{{\rho_{2} F_{yrc}^{2} + \rho_{6} F_{x4}^{2} }}{{(\mu_{4} F_{z4} )^{2} }} \hfill \\ \;\;\; = {\mathbf{y}}^{\text{T}} {\mathbf{Vy}} \hfill \\ \end{aligned}$$
(23)
$${\mathbf{y}}_{\text{opt}} = {\mathbf{V}}^{ - 1} {\mathbf{D}}^{\text{T}} ({\mathbf{DV}}^{ - 1} {\mathbf{D}}^{\text{T}} )^{ - 1} \Delta M_{N}$$
(24)

4 Simulation

In this section, simulations are performed to check the performance of the proposed method. The simulation scenario is a closed-loop steering with a driver model. The simulation was performed on the vehicle simulation package, CarSim. The vehicle model used in simulation was the small-sized SUV model as provided by CarSim. Table 1 shows the parameters of the 2-DOF bicycle model, which were referred from the small-sized SUV model in CarSim [15]. The path followed by the vehicle is the moose test track. The steering input is generated by the driver model, given in CarSim. The preview time of this driver model is set to 0.75 s, which means an inexperienced driver [15]. The vehicle speed and the tire–road friction coefficient are set to 80 km/h and 0.6, respectively. Under the condition, an uncontrolled vehicle loses its stability. The maximum AFS and ARS angles are bounded to 10° and 5°, respectively. The actuators of the brake of ESC, AFS, and ARS were modeled as a first-order system with the time constants of 0.12, 0.05 and 0.05, respectively. ABS, provided in CarSim, is used to prevent the locking of a wheel in excessive braking.
Table 1

Parameters and values of a small-sized SUV model in CarSim

m

1146.0 kg

I z

1302.1 kg m2

C f

36000 N/rad

C r

50,000 N/rad

l f

0.88 m

l r

1.32 m

v x i

80 km/h

r w

0.398 m

K B,front

150 N m/MPa

K B,rear

70 N m/MPa

To compensate the saturated lateral tire force of the front wheels, ESC and ARS are used. Regarding the actuator configuration, four cases, CASE1, CASE2, CASE3, and CASE4, are considered in simulation. CASE1 uses only AFS for yaw moment distribution without considering the saturation of the front lateral tire forces. CASE2, CASE 3 and CASE4 use ESC, ARS and ESC + ARS for yaw moment distribution with considering the saturation of the front lateral tire forces, respectively. WPCA is used for CASE1, and CWPCA is used for CASE2, CASE3 and CASE4. Simulation with four cases can make it clear which actuator combination is the best for yaw moment distribution under the saturation of the front lateral tire forces.

Figure 5 shows the simulation results for each case. Figure 6 shows the applied braking pressures and AFS/ARS angles of each case. Figure 7 shows the front lateral tire forces for all cases. In Fig. 6, the legends FL, FR, RL and RR represent the front left, front right, rear left and rear right wheels, respectively. In Fig. 7, Fyf,max is the maximum lateral tire forces on the front wheels, calculated by (20). In this paper, the yaw rate error or the maneuverability is regarded as satisfactory if it is less than 0.08 rad/s or 4.58 °/s, as given by FMVSS 126, and the side-slip angle or the lateral stability is regarded as satisfactory if it is less than 3° [21]. Simulation results are summarized in Table 2.
Fig. 5

Simulation results for each case

Fig. 6

Control inputs for each case

Fig. 7

Lateral tire forces of front wheels for CASE2, CASE3 and CASE4

Table 2

Summary of the simulation results for all cases

Case

Maximum |γ − γd| (°/s)

Maximum |β| (°)

Minimum vx (km/h)

CASE1

6.0

3.5

64

CASE2

3.9

2.2

60

CASE3

3.8

2.3

64

CASE4

3.8

2.3

62

As shown in Fig. 5, the controlled vehicles did not lose its stability. Coordinated control with CASE2, CASE3 and CASE4, considering the saturation of the front lateral tire forces, gave satisfactory results in terms of the yaw rate error and side-slip angle. Especially, the side-slip angles were significantly reduced by the proposed method. This is caused by the fact that the front lateral tire forces were not saturated by AFS, and that ESC and ARS compensated the loss of control yaw moment. As shown in Fig. 7, the front lateral tire forces generated by AFS were limited to its maximum Fyf,max by CWPCA for CASE2, CASE3, and CASE4. As a result, the corrective steering angles of AFS were reduced, as shown in Fig. 6a. Every case with CWPCA has nearly identical yaw rate errors angles and side-slip angles, as given in Table 2. However, the vehicle speeds are different from one another, as shown in Fig. 5c. This is caused by the fact that the brake pressures of ESC are applied in CASE2 and CASE4, as shown in Fig. 6c. On the other hand, CASE3 shows the largest vehicle speed because there were no braking pressures of ESC by virtue of ARS. So, CASE3 shows the best performance in yaw moment distribution. The effect of the coordination between ESC and ARS is clear as given in Fig. 6b, c. In other words, the ARS angle of CASE4 is smaller than that of CASE3 by virtue of the braking pressure of ESC. If one tries to maintain the vehicle speed under the control, it is desirable to use CASE 3. On the other hand, if one tries to reduce the ARS angle, it is desirable to use CASE4.

5 Conclusions

In this paper, the integrated chassis control with ESC, AFS and ARS, designed to compensate the saturated lateral tire force of front wheels with WPCA, was proposed. With WPCA, the control yaw moment, calculated from the upper-level controller, was distributed into the tire forces generated by ESC, AFS and ARS. Several actuator configurations are represented by the variable weights of WPCA. Especially, only AFS is adopted for enhanced ride comfort and smaller speed reduction. Constrained WPCA was proposed to cope with the saturated lateral tire force of front wheels caused by excessive steering. ESC and ARS were used to compensate the loss of the control yaw moment in CWPCA. From simulation results, it was checked that the proposed method can limit the corrective steering angle of AFS, and increase the braking force of ESC or steering angle of ARS using CWPCA.

Notes

Acknowledgements

This study was supported by Seoul National University of Science and Technology.

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Copyright information

© The Korean Society of Mechanical Engineers 2019

Authors and Affiliations

  1. 1.Department of Mechanical and Automotive EngineeringSeoul National University of Science and TechnologySeoulKorea

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