## 1. Introduction

With the intensification of interest in environmental protection and energy issues, electric vehicles have ushered in significant development opportunities. Compared with traditional centralized drive electric vehicles, the torque of each wheel of an electric vehicle driven by several in-wheel motors, also commonly called independent-wheel-drive electric vehicle (IWDEV), can be independently controlled. Differential drive assist steering (DDAS) is a novel power steering technology based on the unique advantages of independent-drive of the electric vehicle driven by several in-wheel motors [

1]. It uses the different driving force of two-side front wheels to generate steering assistance, which can substitute the traditional power steering system, such as hydraulic power steering (HPS) system or electric power steering (EPS) system. The reason is that the DDAS technology has the advantages of more compact structure and lower cost. Specifically, on the one hand, the DDAS system does not need an add-on actuator, like the steering motor of the EPS system. On the other hand, due to having the same actuators, the in-wheel motors, with the driving system the controller of the DDAS system can be integrated into the vehicle driving controller. Hence, the DDAS technology appeals to many researchers’ attention and interest.

Although the concept of DDAS has only been proposed in recent years, many scholars have conducted much research and achieved some progress. Wang [

1] first proposed the concept of DDAS technology, conducted a theoretical analysis and proposed a steering-wheel-torque direct control strategy based on an anti-windup PID algorithm. The steering assist feasibility, steering return ability and driving torque coordination of the DDAS were studied in depth and verified by software simulations and real vehicle tests. Zhao [

2] further studied a coupling control strategy of force and displacement for a DDAS system to improve the steering maneuverability and handling stability of EVs with motorized wheels. By analyzing the key factors that affect the interaction between vehicle and driver, the optimum hand wheel torque of a DDAS system is designed and achieved by the torque difference between two front wheels based on H

_{2}/ H∞ control method, and its effectiveness was verified by a MatLab/Simulink software (MathWorks, Natick, Massachusetts, USA) simulation. Hu [

3] studied the lane keeping control for four-wheel independently actuated autonomous vehicles only when the active-steering motor entirely fails and designed an adaptive multivariable super-twisting control strategy and verified its effectiveness by CarSim (Mechanical Simulation Corporation, Ann Arbor, Michigan, USA) and Simulink co-simulations. Kuslits [

4] proposed a full state feedback control system for scenarios at higher speeds, whereas a simple angle tracking controller can be used for a DDAS system in scenarios at lower speed. The effectiveness of the strategy was verified through simulations. Peng [

5] developed a coordinated steering control strategy with a hierarchical structure for a multi-axle independent-drive electric vehicle, which is steered simultaneously by traditional mechanical steering and differential drive steering and verified the effectiveness of the strategy through simulations. Wang [

6] also designed a hierarchical coordinated controller for the DDAS and vehicle stability control based on the phase plane theory. Various typical simulations on roads with different adhesion characteristics showed the effects on expanding the working range of DDAS systems and simultaneously mitigating the additional influence of the DDAS on vehicle stability. Römer [

7] studied the potential of independent-wheel-drive influencing the driver’s steering torque using a control technique based on classical EPS control plans, and compared the energy saving potential of DDAS system with the conventional EPS system. The energy saving potential was proved through realistic driving cycles experiments which included the Karlsruhe motorway, the Herzogenaurach highway and a trip through the city of Karlsruhe. It was concluded that the DDAS system can save up to 121.93 Wh/100 km of energy, or approximately 0.95% in lateral acceleration ranges below 4.5 m/

${\mathrm{s}}^{2}$, and about 0.43% (55.2 Wh/100 km) in mean value compared to a conventional EPS system. Interestingly, if the tractive energy of an independent-wheel-drive electric vehicle is considered, the comprehensive energy conservation generated by the optimization of the two-side differential drive torque, as DDAS does, can reach up to 4% without any loss of vehicle stability [

8].

A review of these references shows that a DDAS system can be used to substitute for a traditional power steering system, such as EPS and as a result, the energy consumption of these traditional add-on assistance steering systems is removed. Although the aspect of energy conservation is not the core purpose of our research, it can be concluded that the existing studies of DDAS system have proved the apparent engineering application value of this novel technology and its feasibility for providing steering assistance and the coordination issue with other chassis control system have been well studied in the above existing literature. However, other important aspects, such as the steering assistance quality issue of DDAS system, are also important factors determining whether this technology can be finally applied in practice. Unfortunately, the steering assistance quality of the DDAS system, which reflects smooth steering hand force with less interference and good tracking performance with ideal steering force characteristic, has not been widely studied until now. To better understand this requirement of the technology, the basic principles and assistance characteristics of a DDAS system and some previous experimental results have to be reviewed and discussed first.

Figure 1 shows the working principle diagram of a DDAS system.

As shown in

Figure 1, the DDAS system maintains the traditional mechanical steering mechanism but removes the power assist steering actuators, such as a hydraulic cylinder or an electric motor. Since the longitudinal driving force of the left and right steering wheels of the IWDEV is independent and controllable, the torque generated by the front wheels around the respective kingpin can be unequal. Here we define this torque as the driving steering torque. At the same time, because the two steering wheels are connected by the steering trapezium and have a fixed geometric motion relationship, therefore, the driving steering torque will drive the two steering wheels to turn to the side with a small driving force. In theory, the controller of the DDAS system controls the outer steering wheel to increase the driving torque properly and the inner steering wheel to reduce the driving torque equally, which can ensure that the driving steering torque generated by the inner steering wheel and the outer steering wheel on the steering rack are exactly equal to the required steering power torque. It is obvious that the DDAS system can realize the power steering function without changing the total driving torque. Compared with a traditional EPS system, the DDAS system can achieve the same power steering effect without needing a power assist steering actuator. The DDAS system saves the part of energy used to drive the power assist steering actuator, so the DDAS system must be more energy-saving compared with a traditional EPS system. Since the energy consumption of the power assist steering actuator is small, the energy saving of the DDAS system is limited, but it still plays an important role in improving the driving range of pure electric vehicles. DDAS system has the same actuator, two front in-wheel motors, as the driving system of IWDEV, and the DDAS electric control unit (ECU) is also integrated into the driving controller. Consequently, a DDAS system has advantages over other power assist steering systems in layout and cost.

However, it should be noted that the DDAS system is an indirect power steering system, that is, the steering assistance provided by the system is achieved by indirectly acting on the mechanical steering rack through changing the tire forces of two-side steerable driving wheels. The steering assistance generated by DDAS system can be expressed as follows:

where

T_{1} and

T_{3} are the driving torques of the left and right front wheels,

${r}_{\sigma}$ is the scrub radius,

${r}_{w}$ the tire rolling radius,

${N}_{L}$ is the transmission ratio of the rack translation to the knuckle arm angular displacement,

${I}_{w}$ is the moment of inertia of the wheel about its central axis,

${\omega}_{1}$ and

${\omega}_{3}$ are the rotational velocity of left and right front wheels. According to Equation (1), it can be seen that the assistance provided by the DDAS system is related to the wheel rotational dynamic characteristics and suspension parameters. During the operation of the vehicle, tires may work in a nonlinear range, and the scrub radius of the wheels is also constantly changing. In addition, the steering wheel torque/angle sensor noise may also have a great impact on the control of a DDAS system.

Based on the review of the characteristics of DDAS technology, its control issue has also been studied in many published references. Most of the researchers applied conventional control algorithms, such as classical open-loop look-up table control plan [

7] like EPS does, anti-windup PID control plan [

6] and fuzzy adaptive PID control plan [

1] to the control issue of DDAS system and the their control effects on steering assistance and returnability performance look good in the corresponding simulations, but the control effects of these classic linear control methods on the steering assistance qualities, such as road feeling, the steering wheel torque control stability and robustness against system parameters variation and sensor noise in real applications, are considered to be unacceptable. To better understand the lack of competence of the traditional linear PID control plan with fixed control parameters,

Figure 2 shows a real world double-lane-change road test result of the steering wheel torque of an IWDEV that is controlled by a conventional anti-windup PID controller based on a DDAS system published in reference [

1]. It can be seen that the PID controller has poor tracking performance to the reference steering wheel torque though the steering assistance function is achieved. This means the smooth road feeling and accurate hand force feedback cannot be fully achieved in the real application of PID controllers for DDAS system. In addition, because the nonlinear mathematical models of tire dynamics, steering system and suspension system are difficult to establish accurately, the changing laws of these interferences are difficult to identify. Thus, despite having better robustness and optimality, some advanced controllers that depend on the accurate model of the controlled system with interference observer or estimator, such as H infinite control, linear quadratic regulator (LQR) control, etc. may be not easy or suitable to apply to the DDAS system, too.

In summary, though the driver’s steering effort can be obviously reduced by the DDAS system, it can be seen from the above analysis that the selected control strategies and control algorithms may highly impact its effect on steering assistance quality. This performance will ultimately decide whether this novel power assistance steering technology can be actually applied in a real car. In this paper, having good robustness in nonlinear control issues, the use of the active disturbance rejection control (ADRC) method is attempted for this purpose. As an improved form of PID controller, the ADRC approach combines the advantages of the PID controller and some robust algorithms. It is relatively easy to implement, robust against possible system interferences and one does not need to know an accurate controlled system model [

9].

Compared with the existing literature, the main purpose or main contribution of this paper is that we try to pay more attention on the improvement of the steering assistance quality of the DDAS system before its real application, and firstly attempt to apply the ADRC control approach to improve the steering assistance quality of the DDAS system, in order to make the driver have a better road feeling, and achieve a smooth steering force with less interference caused by possible sensor noise and model parameter changes.

The structure of this paper is as follows: Firstly, the independent-wheel-drive electric vehicle model with four degrees of freedom mechanical steering system is established, and then the ADRC controller model of DDAS is designed for the steering-wheel-torque direct control strategy. Secondly, aiming at solving the problem that the parameters of the ADRC controller are numerous and difficult to set, a simulated annealing algorithm is used to optimize the parameters offline. Finally, typical driving conditions are selected for simulation and experimental verification, which verify the effectiveness of the control method proposed in this paper.

## 4. Controller Parameter Optimization Based on Simulated Annealing Algorithm

Compared with the PID controller, though the ADRC controller has the advantages of better robustness, simple structure and easy implementation without knowing the accurate mathematical model of the controlled system, it also has the disadvantages of needing more control parameters and complicated parameter tuning [

26], which severely limits its further industrial application. At the same time, these parameters have a great impact on the performance of the controller, so appropriate method selection to set the values of each parameter has to be done first.

According to the theory of ADRC, some of the parameters are determined empirically, and once these parameters are determined, no correction is needed. For example, ${\alpha}_{1}$, ${\alpha}_{2}$, ${\alpha}_{3}$, ${\alpha}_{4}$ and ${\varphi}_{1}$, ${\varphi}_{2}$, ${\varphi}_{3}$, ${\varphi}_{4}$ are the parameters of the nonlinear function fal, which affect the change trend of the nonlinear function, but they usually do not change with the change of the controlled system. Therefore, the ranges of ${\alpha}_{3}$ and ${\alpha}_{4}$ in the nonlinear state error feedback are generally $0<{\alpha}_{3}<1$, ${\alpha}_{4}>1$, so in this paper, ${\alpha}_{3}$ and ${\alpha}_{4}$ are chosen as fixed values, 0.95 and 1.25, respectively, ${\alpha}_{1}$ and ${\alpha}_{2}$ in the third-order extended state observer are chosen as fixed values, 0.5 and 0.25, respectively. The values of ${\varphi}_{1}$, ${\varphi}_{2}$, ${\varphi}_{3}$ and ${\varphi}_{4}$ have a great influence on the nonlinearity of the controller. After multiple simulations, the value of ${\varphi}_{1}$, ${\varphi}_{2}$, ${\varphi}_{3}$ and ${\varphi}_{4}$ are chosen as 0.01 which is ten times of the sampling step. The value of speed factor R in this paper is 10.

In summary, in addition to the empirically determined parameters, the other parameters which need to be specifically set are the following six parameters

${\gamma}_{01}$,

${\gamma}_{02}$,

${\gamma}_{03}$,

${\gamma}_{1}$,

${\gamma}_{2}$ and

${b}_{0}$. Generally, there is no relationship between the six parameters

${\gamma}_{01}$,

${\gamma}_{02}$,

${\gamma}_{03}$,

${\gamma}_{1}$,

${\gamma}_{2}$,

${b}_{0}$ and the parameters

${\alpha}_{1}$,

${\alpha}_{2}$,

${\alpha}_{3}$,

${\alpha}_{4}$,

${\varphi}_{1}$,

${\varphi}_{2}$,

${\varphi}_{3}$,

${\varphi}_{4}$ mentioned above. At the same time,

${\gamma}_{01}$,

${\gamma}_{02}$ and

${\gamma}_{03}$ in the extended state observer are mainly related to sampling step size [

23], which can be designed separately. In addition,

${\gamma}_{1}$ and

${\gamma}_{2}$ in nonlinear state error feedback are also important parameters of the controller and

${b}_{0}$ is an important parameter to characterize the difference of different systems. Due to the fact that there is a certain mutual influence between these parameters, and manual adjustment is too complicated, offline optimization to set the values of these six parameters is implemented. In this optimization process, three parameters

${\gamma}_{01}$,

${\gamma}_{02}$,

${\gamma}_{03}$ are optimized first, and then the rest three parameters are optimized.

There are many existing optimization algorithms, such as genetic algorithm, simulated annealing algorithm and particle swarm optimization, etc. Among them, the simulated annealing algorithm has the advantages of simple description, flexible use, high operational efficiency and less constraint on initial conditions [

27]. Therefore, the simulated annealing algorithm is chosen as the optimization algorithm.

In order to implement the optimization, the objective function of the optimization problem according to the needs of this paper should be determined first. The target of this paper is to design a better DDAS controller, which is to control the actual steering wheel torque to follow the ideal steering wheel torque in real time by controlling the front wheels driving torque difference. Therefore, the objective function is defined as follows:

In order to speed up the optimization process, the initial values of each parameter are determined at first by multiple simulations as shown in

Table 1. Then the relevant optimization program is coded in MatLab software, in which the

sim function is used to call the simulation model. The simulation condition selects the sinusoidal steering angle input at 30 km/h vehicle speed, and the road surface adhesion coefficient is high adhesion, which is 0.8. As an example, the iterative optimization process of the three parameters

${\gamma}_{01}$,

${\gamma}_{02}$ and

${\gamma}_{03}$ in extended state observer is shown in

Figure 10.

As shown in

Figure 10, after around 870 generations, the fitness function basically reaches the optimal value. The final six optimized parameters are shown in

Table 1 below: