# Robust Shared Control for Four-Wheel Steering Considering Driving Comfort and Vehicle Stability

^{*}

## Abstract

**:**

## 1. Introduction

_{∞}control method. Furthermore, the mixed H

_{2}/H

_{∞}optimal control strategy was adopted into the 4WS system to improve handling stability performance in [28]. To balance energy saving and vehicle stability, the hierarchical chassis coordinated control strategy was proposed in [29]. The expert PID and model predictive control (MPC) were integrated to achieve the speed tracking and path tracking in the upper-level. In the low level, the mutant particle swarm optimization algorithm was used to track the desired steering angle of each wheel. In fact, the front and rear wheels can be regarded as four agents from the perspective of the game theory. Therefore, Zhu et al. [30] designed novelty a quadratic differential game-based 4WS controller to optimize the stability performance under the J-turn maneuver. The Pareto game optimal method was also applied into 4WS system to improve the execution efficiency of four steering motors in [31].

- To understand the driver’s continuous steering behavior, a driver model with adaptive preview distance is proposed. Meanwhile, to solve the model mismatch problem caused by vehicle parameter perturbation, the fuzzy shared driver-vehicle dynamics model is constructed for steering control.
- The shared steering control method for 4WS is proposed to suppress parametric uncertainties caused by time-varying driver characteristics, cornering stiffness, and vehicle speed. Moreover, constraints on the driver-vehicle system and actuators are considered by using the robust invariant set to enhance the safety of EVs.

## 2. Mathematical Model of 4WS Vehicle System

#### 2.1. 4WS Vehicle Dynamics Model

_{f}and δ

_{r}are the desired front and rear wheel angles, respectively. δ

_{i}(i = fl, fr, rl, rr) represent the steering angles of each wheel. As shown in Figure 1, according to Newton’s second law, the single-track dynamic equation of the vehicle can be derived as [3]:

_{x}and v

_{y}mean the longitudinal vehicle speed and lateral vehicle speed, respectively. r and m are the yaw rate and the vehicle mass, respectively. F

_{yi}(i = fl, fr, rl, rr) denotes the tire lateral forces. F

_{yf}= F

_{yfl}+ F

_{yfr}. F

_{yr}= F

_{yrl}+ F

_{yrr}. l

_{f}and l

_{r}represent front wheelbase and rear wheelbase, respectively. I

_{z}is the yaw moment of inertia of the vehicle. M

_{z}can be calculated by the following equation:

_{xi}(i = fl, fr, rl, rr) denotes the tire longitudinal forces. Under the premise that the tire slip angle is small at high speed, the tire lateral of front wheels and rear wheels can be approximated as follows:

_{f}and α

_{r}are tire slip angles, calculated by [3]:

_{x}can be regarded as the constant value in specific condition. Therefore, substituting Equations (3)–(5) and (7) into Equation (2), the control-oriented dynamics are rewritten as follows:

#### 2.2. Driver Model

_{near}and the far vision angle θ

_{far}. θ

_{far}is used to predict the upcoming road, while θ

_{near}is adopted to correct vehicle position.

_{far}and θ

_{near}are expressed by [6]:

_{far}is the distance between the far point and CoG. l

_{s}is the distance between the near point and CoG. Note that l

_{s}= 0.4 l

_{far}[6]. y

_{l}and φ

_{L}are defined as look-ahead path-tracking error and heading error, respectively, which are expressed as:

_{ref}=1/R

_{ref}is curvature value of the given desired path, which can be obtained in real time with on-board sensors.

_{p}and K

_{c}can be defined feedforward and feedback gains, respectively, to adjust the path-tracking errors. The PD control is used to represent the compensatory behavior of the driver for reducing the tracking error, in which τ

_{L}is a differential constant. Considering the brain’s delay, the steering decision behavior is simulated. Then, the arm neuromuscular system is approximated as 1/(1 + τ

_{d}

^{2}s) [8,9,10]. In order to facilitate controller design, e

^{−τd1s}is approximated as 1/(1 + τ

_{d}

_{1}s). Note that T

_{d}= τ

_{d}

_{1}+ τ

_{d}

^{2}represents the driver’s total delay time, and a

_{0}= τ

_{d}

_{1}τ

_{d}

^{2}/T

_{d}

^{2}. a

_{0}= 0.21 is selected as a constant for simplification. Therefore, a simplified driver model is built for control purposes as follows:

_{fd}represents the front wheel steering output from the driver model, R

_{g}means the transmission coefficient of the steering-by-wire system.

#### 2.3. Driver-Vehicle Model

#### 2.4. Validation of the Driver-Vehicle Model

_{p}K

_{c}τ

_{L}T

_{d}]

^{T}. More details of the identification algorithm can be referred to in [6]. Thirteen drivers, denoted as Driver 1 to Driver 13, were invited to participate in the tests, driving along an identical reference trajectory by using a driving simulator. The reference trajectory is depicted in Figure 4. The 13 drivers’ personal information is given in Appendix A.

_{p}and K

_{c}are relatively dispersed, reflecting the different driving styles and skills of these drivers, while τ

_{L}and T

_{d}, which represent driver’s reflection time, are relatively concentrated. In this paper, the average values of these drivers’ characteristic parameters are applied as the default parameters of the driver-vehicle model for the controller design.

## 3. Robust Shared Control Design for 4WS Vehicle

#### 3.1. T-S Fuzzy Description of Parameter Uncertainty

_{x}in Equation (13), the closed-loop system (13) is nonlinear. Therefore, we linearize the variable 1/v

_{x}based on the following equation:

_{x}. The constants v

_{0}and v

_{1}are obtained as and ${v}_{0}=\left({v}_{\mathrm{max}}+{v}_{\mathrm{min}}\right)/2$, respectively. Equation (15) is brought into Equation (13). Then, the T-S fuzzy method is used to handle the time-varying parameters l

_{s}, C

_{f}, C

_{r}, and Δv.

- If l
_{s}is S, C_{f}is S, C_{r}is S, and v_{x}is S, then l_{smin}, C_{fmin}, C_{rmin}, and v_{xmin}replace l_{s}, C_{f}, C_{r}, and v_{x}of A_{1}, B_{u}_{1}, and B_{w}_{1}in Equation (18); - If l
_{s}is S, C_{f}is S, C_{r}is S, and v_{x}is B, then l_{smin}, C_{fmin}, C_{rmin}, and v_{xmax}replace l_{s}, C_{f}, C_{r}, and v_{x}of A_{2}, B_{u}_{2}, and B_{w}_{2}in Equation (18); - If l
_{s}is S, C_{f}is S, C_{r}is B, and v_{x}is S, then l_{smin}, C_{fmin}, C_{rmax}, and v_{xmin}replace l_{s}, C_{f}, C_{r}, and v_{x}of A_{3}, B_{u}_{3}, and B_{w}_{3}in Equation (18); - …
- If l
_{s}is B, C_{f}is B, C_{r}is B, and v_{x}is S, then l_{smax}, C_{fmax}, C_{rmax}, and v_{xmin}replace l_{s}, C_{f}, C_{r}, and v_{x}of A_{15}, B_{u}_{15}, and B_{w}_{15}in Equation (18); - If l
_{s}is B, C_{f}is B, C_{r}is B, and v_{x}is B, then l_{smax}, C_{fmax}, C_{rmax}, and v_{xmax}replace l_{s}, C_{f}, C_{r}, and v_{x}of A_{16}, B_{u}_{16}, and B_{w}_{16}in Equation (18).

Rule No. | Premise Variables | |||
---|---|---|---|---|

l_{s} | C_{f} | C_{r} | v_{x} | |

1 | S | S | S | S |

2 | S | S | S | B |

3 | S | S | B | S |

4 | S | S | B | B |

… | … | … | … | … |

15 | B | B | B | S |

16 | B | B | B | B |

#### 3.2. Robust Shared Controller Design for 4WS

_{∞}robust control theory, the optimized objective function is designed as follows [41]:

_{i}in each fuzzy system can be expressed as:

_{i}is the gain vector to be calculated.

^{−1}.

#### 3.3. Robust Positive Invariant Set Constraint Design

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

_{i}(i = 1, 2, …, 16), and $\mathsf{{\rm Y}}$ that satisfy the following optimization solution issue:

## 4. Driver-in-the-Loop Experiment

#### 4.1. Experiment Settings

_{1}indicates that the steering input of the driver is opposite to that of the controller, i.e., the conflict occurs. As J

_{1}decreases, the level of conflict between humans and machines increases. Conversely, a significantly positive value of J

_{1}indicates a high degree of cooperation. The metrics for driving comfort, vehicle stability, and path tracking performance are denoted as J

_{2}, J

_{3}, and J

_{4}, respectively [6]:

_{i}(i = 1, 2, …, 7) are weight coefficients. Note that J

_{2}is defined as the driving comfort of the vehicle with the shared steering control. Large and constant changes of the driver steering angle and the assistance steering angle may lead to an uncomfortable driving experience.

#### 4.2. Performance Analysis of the Proposed Shared Controller

_{1}index for evaluating human-machine conflict. When compared to the RC and MPC, the FRC can realize a remarkable reduction in steering conflicts, surpassing 59%.

_{1}is positive among both controllers for Driver III due to the conservative behavior. The cooperative performance of Driver III is obviously superior to Drivers I and II with RC. However, as shown in Table 4, experienced Driver I has a better cooperative performance with the proposed controller considering driver-automation interaction level.

_{2}for all three drivers experience an improvement of more than 70% when compared to the RC and MPC (seen Table 4). Notably, Driver II exhibits the lowest driving comfort, primarily due to their aggressive driving behavior, which hinders effective human-machine cooperation. Conversely, Driver III’s driving comfort fares relatively better because their conservative driving style allows the controller to have a more dominant role in control authority. However, there exists a large steering jitter when driving in and out of the curve (see Figure 11c). This causes rapid changes of assistance steering inputs with RC, which is not beneficial to driving comfort and safety. Nonetheless, the proposed controller can improve the driving comfort of Driver III because it considers the driver’s characteristics. Since Driver I has excellent driving skills, the driving comfort is better with both controllers. In summary, the proposed controller can improve driving comfort and reduce human-machine conflicts.

_{3}of the FRC surpasses that of the RC and MPC.

_{4}with FRC can be enhanced by 18% compared with RC. Driver I has the best path-tracking ability, with the maximum path tracking error smaller than 0.3 m. Instead, the path-tracking ability of Driver II is the worst with both controllers. However, the FRC yields a better lane-keeping performance than RC. The conservative Driver III has a better path-tracking ability. This is because the control authority is dominated by the controllers in this case.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

B | body width |

l | wheelbase |

δ_{f} | desired front wheel angle |

δ_{r} | desired rear wheel angle |

δ_{i} (i = fl, fr, rl, rr) | steering angles of each wheel |

v_{x} | longitudinal vehicle speed |

v_{y} | lateral vehicle speed |

r | yaw rate |

m | vehicle mass |

F_{yi} (i = fl, fr, rl, rr) | tire lateral forces |

CoG | center of gravity |

l_{f} | distance between CG and front axle |

l_{r} | distance between CG and rear axle |

I_{z} | yaw moment of inertia of the vehicle |

${M}_{z}$ | vehicle yaw moment of inertia |

F_{xi} (i = fl, fr, rl, rr) | tire longitudinal forces |

${\alpha}_{f}$ | front tire slip angle |

${\alpha}_{r}$ | rear tire slip angle |

$\beta $ | sideslip angle |

θ_{near} | the near point from the near vision angle |

θ_{far} | the far point from the far vision angle |

l_{far} | distance between the far point and CoG |

l_{s} | distance between the near point and CoG |

y_{l} | look-ahead lateral error |

φ_{L} | heading error |

ρ_{ref} | road curvature of the reference trajectory |

K_{p} | feedforward gain |

K_{c} | feedback gain |

τ_{L} | differential constant |

T_{d} | the driver’s total delay time |

δ_{fd} | front wheel steering angle exerted by the driver model |

R_{g} | steering ratio |

$y$ | measured output |

$Q$ | weighting matrix |

γ | the vehicle stability |

z | performance output |

$\Delta v$ | variation of v_{x} |

$J$ | objective performance function |

$\mathbb{Q}$ | weighting matrix |

u_{i} | control law |

K_{i} | state feedback gain of each fuzzy system |

$\vartheta \left(x\right)$ | Lyapunov function |

ρ | positive scalar |

w | system disturbance |

C_{f} | Equivalent Cornering Stiffness of front tire |

C_{r} | Equivalent Cornering Stiffness of rear tire |

## Appendix A

Driver No. | Age | Driving Years |
---|---|---|

1 | 28 | 5 |

2 | 38 | 10 |

3 | 45 | 13 |

4 | 23 | 1 |

5 | 39 | 11 |

6 | 57 | 29 |

7 | 29 | 3 |

8 | 25 | 1 |

9 | 27 | 2 |

10 | 37 | 7 |

11 | 43 | 11 |

12 | 51 | 17 |

13 | 46 | 16 |

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**Figure 6.**Results of the validation for the driver-vehicle model. (

**a**): comparison of steering wheel angle between driver and model; (

**b**): comparison of yaw rate between driver and model.

**Figure 9.**(

**a**) The steering phase of the drivers. (

**b**) Trajectories of the vehicles driven by the drivers.

**Figure 11.**The steering wheel angle under FRC and RC. ${\tilde{\delta}}_{d}^{A}$ and ${\tilde{\delta}}_{c}^{A}$ are the steering wheel angles of the driver and FRC, respectively. ${\delta}_{d}^{B}$ and ${\delta}_{c}^{B}$ are the steering wheel angles of the driver and RC, respectively. (

**a**) Driver I; (

**b**) Driver II; (

**c**) Driver III.

**Figure 13.**Lateral path-tracking error under FRC and RC. (

**a**) Driver I; (

**b**) Driver II; (

**c**) Driver III.

Symbol | Meaning | Value |
---|---|---|

m | Vehicle total mass | 1705 kg |

M_{z} | Vehicle yaw moment of inertia | 3048 kg·m^{2} |

l_{f} | Distance between CG and front axle | 1.035 m |

l_{r} | Distance between CG and rear axle | 1.665 m |

C_{f} | Equivalent Cornering Stiffness of front tire | 103,130 N/rad |

C_{r} | Equivalent Cornering Stiffness of rear tire | 73,854 N/rad |

Parameter | K_{P} | K_{c} | τ_{L} | T_{d} | Driving Style |
---|---|---|---|---|---|

Driver I | 2.70 | 1.50 | 0.18 | 0.18 | Experienced |

Driver II | 3.60 | 2.50 | 0.18 | 0.18 | Aggressive |

Driver III | 1.90 | 1.10 | 0.19 | 0.19 | Conservative |

Driver Type | Controller Type | J_{1} | J_{2} | J_{3} | J_{4} |
---|---|---|---|---|---|

Driver I | FRC | 7.5 | 1.2 | 5.2 | 2.0 |

RC | −6.3 | 6 | 6.8 | 2.5 | |

MPC | −3 | 8 | 7.0 | 2.2 | |

Driver II | FRC | −5.3 | 4.1 | 7.5 | 2.4 |

RC | −40 | 40 | 8.4 | 3.2 | |

MPC | −36 | 42 | 8.6 | 2.7 | |

Driver III | FRC | 5.5 | 1.8 | 4.4 | 2.1 |

RC | 4.2 | 3.2 | 5.6 | 2.3 | |

MPC | 5.1 | 5 | 5.8 | 2.6 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, C.; Liu, H.; Dang, M.
Robust Shared Control for Four-Wheel Steering Considering Driving Comfort and Vehicle Stability. *World Electr. Veh. J.* **2023**, *14*, 283.
https://doi.org/10.3390/wevj14100283

**AMA Style**

Zhang C, Liu H, Dang M.
Robust Shared Control for Four-Wheel Steering Considering Driving Comfort and Vehicle Stability. *World Electric Vehicle Journal*. 2023; 14(10):283.
https://doi.org/10.3390/wevj14100283

**Chicago/Turabian Style**

Zhang, Chuanwei, Haoxin Liu, and Meng Dang.
2023. "Robust Shared Control for Four-Wheel Steering Considering Driving Comfort and Vehicle Stability" *World Electric Vehicle Journal* 14, no. 10: 283.
https://doi.org/10.3390/wevj14100283