**1. Introduction**

The vehicles independently driven by in-wheel motors removes the transmission system of traditional vehicles and the drive torque of each wheel is independently controllable. Besides, the information such as the motor torque and speed can accurately feedback in real-time, so that the transmission efficiency of the vehicle is greatly improved and the layout design becomes more flexible. More importantly, the driving form has significant advantages in terms of stability control, active safety control and energy saving control [1,2], which is a huge attraction for multi-axle heavy vehicles. However, battery technology has always been one of the key issues limiting the development of pure electric vehicles [3]. For heavy vehicles, both the demand and consumption of energy are greater, which means the energy problem is more serious. In the case that the existing battery core technology cannot be solved temporarily, it is necessary to adopt an energy-saving control strategy for the electric vehicle, especially the multi-axle heavy-duty electric vehicle [4].

At present, the energy-saving driving control strategy for electric vehicles is mainly based on three aspects: motor control energy saving, energy feedback and traction control energy saving. The energy-saving of the motor is mainly based on the motor efficiency characteristic curve [5,6], aiming at the optimal system efficiency, and changing the actual working point of each motor by adjusting the front and rear axle torque distribution coefficients to avoid working in the low-efficiency zone, but this method is often only for straight-line driving conditions. Energy feedback mainly refers to regenerative braking technology, which hopes to maximize the recovery of braking energy by using different control strategies during vehicle braking [7–9]. In terms of traction control energy saving, the drive torque and braking torque of each wheel can be controlled independently for electric vehicles. By properly distributing the torque of each wheel, for example, taking the minimum sum of the tire utilization ratios of the driving wheels as the control target [10–12], so as to reduce the energy consumption rate or increase the power of the vehicle [13]. Generally, the optimization method is to turn the torque distribution formula according to vehicle dynamics into the parameter optimization problem under certain constraints [14–16]. However, this kind of method has great limitations in optimizing a multidimensional system.

At present, most of energy-saving control researches are aimed at the straight-line driving conditions evaluated by driving cycles [17] and there are relatively few studies on the vehicle energy-saving control for steering conditions. Compared with two-axle independent drive vehicles, only the two-dimensional optimal torque distribution control between the front and rear axles and between the left and right wheels is needed [18]. Multi-axle electric vehicles need to optimize the multidimensional independent space vector. Meanwhile, there are dynamic and kinematic connections between the wheels, which cannot be solved by traditional optimization algorithms.

The deep deterministic policy gradient (DDPG) [19,20] is an algorithm that improves on the basis of the deep Q network (DQN) [21,22] to solve continuous action problems. In reality, the vehicle is an extremely complex system, and the external environment is dynamic, complex and unknown, which means that it is difficult to simplify it into a fixed expression for quantitative analysis. The DDPG algorithm is highly adaptable and can be optimized for the black-box system in a dynamic environment, which is suitable for solving the practical problems of continuous action.

In the current paper, the four-axle (8 × 8) independent drive electric vehicle is taken as an example to study the torque distribution problem in the steering condition, and a 23-DOF (Degree of Freedom) vehicle dynamics model was built by MATLAB/Simulink (R2015a, MathWorks, Natick, MA, USA). After completing the relevant code of the DDPG algorithm, the data interaction between the algorithm and the vehicle model was realized, and the model was trained enough times through off-line simulation comparing energy consumption of the vehicle under the same conditions, so as to prove that the proposed control algorithm can effectively reduce energy consumption by reasonably distributing the drive torque of each wheel. Under the conventional steering condition and using the motor efficiency map of the current paper, energy consumption of the vehicle can be reduced by up to 5%.

### **2. Dynamics Model and Energy Analysis**

### *2.1. Model Overview*

As the number of axles increases, the dynamics of multi-axle vehicles becomes more complicated. Theoretically, the more the degrees of freedom of the vehicle are considered, the better the simulation effect will be, but the more parameters are actually required to be input, which will affect the results when relevant parameters cannot be obtained. In order to more accurately simulate the impact of vehicle systems and environment on the vehicle during driving, the classical 2-DOF linear model is not used in the vehicle dynamics model. Instead, based on the vehicle system dynamics theory, the differential equations of dynamics and kinematics are derived respectively about vehicle body, wheel and other systems. The suspension part is assumed to be static balance problem, and the tire part

is analyzed by "Magic Formula". Finally, the related physical quantities between each system are used to connect the parts into a whole, as shown in Figure 1. Meanwhile, the way of modeling is also suitable for two-axle vehicles, and the simulation accuracy is higher. Based on the dynamics and kinematics equations of each system, the vehicle dynamics model is established by using MATLAB/Simulink. Taking into account 6-DOF of the vehicle body, including longitudinal, lateral, vertical, yaw, pitch, roll, as well as the vertical runout and rotation freedom of each wheel, and steering wheel angle, a total of 23-DOF. In addition, the vehicle adopts the steer-by-wire technology, which can realize all-wheel steering. In the model, according to the fixed relationship between the steering wheel angle and the deflection angle of the right wheel of the first axle and Ackerman steering principle, the S-Function module is built to calculate the actual deflection angle of each wheel, which is directly input into the vehicle dynamic model. The main parameters of the vehicle are shown in Table 1.

**Figure 1.** Vehicle dynamics model architecture.


For electric vehicles with in-wheel motors, due to the complete decoupling of each wheel, in order to achieve electronic differential control, torque control mode is usually adopted for each in-wheel motor [23]. As shown in Figure 2, the drive control architecture is adopted. The total drive torque of the vehicle is obtained by the output of the PID (Proportion Integration Differentiation) controller, and the input of the controller is the deviation of the target speed and the actual speed. In general, the driving torque is evenly distributed to each wheel, so that the speed of wheel will follow according to its stress state. The average distribution mode can ensure the normal driving of vehicles, but it is not the optimal distribution method. Therefore, the optimal distribution mode of drive torque should be proposed, which is the main research content of the current paper.

**Figure 2.** Vehicle drive control.

### *2.2. Motor and Battery Model*

As a high-speed rotating component, the speed characteristic of the motor also determines its high-speed response [24]. In general, the instantaneous response speed of the motor is tens of times faster than that of the wheel, so it can be simplified to a second-order response system [25], whose transfer function is as follows.

$$G(\mathbf{s}) = \frac{T\_{mi}}{T\_{mi}^\*} = \frac{1}{2\xi^2 \mathbf{s}^2 + 2\xi \mathbf{s} + 1},\tag{1}$$

where *Tmi* is the actual input electromagnetic torque of each in-wheel motor, *Tmi\** is the desired input electromagnetic torque of each in-wheel motor, ξ denotes the damping ratio, which is related to the parameters of the drive motor. According to the response characteristics of PMSM, the value of ξ is 0.001.

At the same time, the motor efficiency map model is adopted. According to the speed and torque of the motor, the working efficiency can be obtained to calculate the corresponding energy loss. The efficiency map of the in-wheel motor used is shown in Figure 3.

**Figure 3.** Drive motor efficiency map.

For the battery model, in order to accurately compare the energy consumption, the ampere-hour integral method is adopted to estimate the battery *SOC* [26]. The formula is as follows.

$$SOC = SOC\_0 - \frac{1}{C\_{\rm N}} \int \eta l dt = SOC\_0 - \frac{1}{C\_{\rm N}} \int \eta \frac{P}{U} dt,\tag{2}$$

where *SOC*<sup>0</sup> is the initial state of charge and discharge, CN denotes the battery rated capacity, *I* is the instantaneous current of the battery, η represents the Coulomb efficiency coefficient, *P* is the actual working power of the battery, and U is the battery voltage. Generally, without considering the influence of temperature, the battery voltage will decrease with the decrease of *SOC*, but when the battery consumption is between 10% and 90%, the battery voltage variation is relatively small. In order to avoid the impact of the battery voltage change on the *SOC* drop, it is assumed that the battery consumption is always within this range, that is, the battery voltage remains constant.

## *2.3. Analysis of Steering Energy Consumption*

When the vehicle enters the steering condition from the straight driving and the accelerator pedal opening is constant, the vehicle speed will decrease, which indicates that the vehicle driving resistance has increased. The movement of the vehicle is the result of the force from the ground to the vehicle body through the tire. Generally, the force between the tire and the ground is decomposed into longitudinal force and lateral force, and the motion of the vehicle is the result of the combined action. That is, the combined force of the longitudinal force and the lateral force causes the vehicle to generate steering motion. The direction of the resultant force is affected by factors such as drive torque, steering angle, and tire side-slip angle, and in the case of the same drive torque and steering angle, its direction is determined by the tire side-slip angle. When the vehicle turns, the tire force is shown in the Figure 4 below.

**Figure 4.** Tire force decomposition diagram.

As shown in Figure 4, δ<sup>1</sup> represents the wheel deflection angle, α is the tire side-slip angle, *Fx* and *Fy* denotes the tire longitudinal force and lateral force. Due to δ<sup>1</sup> and α, the lateral force of the wheel will produce a reaction force along the longitudinal axis of the vehicle body, which increases the driving resistance. This explains why the speed of the vehicle will decrease when cornering and the opening of accelerator pedal remains the same, and it also means that if the vehicle wants to maintain the original speed, it needs to consume more energy. By establishing a single-track linear model and assuming that the vehicle moves in a uniform circular motion, the longitudinal force balance equation of the vehicle can be derived as follows.

$$\sum\_{i=1}^{4} F\_{xi} = F\_f + F\_a + \mathfrak{m} \frac{\mu^2}{\rho} (\frac{l\_4}{\mathcal{L}} \sin a\_1 + \frac{l\_3}{\mathcal{L}} \sin a\_2 + \frac{l\_2}{\mathcal{L}} \sin a\_3 + \frac{l\_1}{\mathcal{L}} \sin a\_4), \tag{3}$$

where *Fxi* is the longitudinal force of each axle, *Ff* is rolling resistance, *Fa* denotes air resistance, m is the total mass of the vehicle, *u* represents the longitudinal velocity, ρ denotes the curvature radius, *li* is the horizontal distance from *i*th axle to the center of mass, L represents the distance between 1st axle and 4th axle, α*<sup>i</sup>* is the side-slip angle of *i*th axle. On the left side of the equation is the sum of longitudinal force of each axle and the first two terms on the right are the conventional driving resistance of vehicles. Therefore, the last term is the additional steering resistance caused by the tire slid-slip when the vehicle is steering [27,28], which denoted by *Faf*. If the drive torque of each wheel is changed, the drive force of the outboard wheels is increased and the drive force of the inboard wheels is decreased, then Equation (3) changes as follow.

$$\sum\_{i=1}^{4} F\_{xi} = F\_f + F\_d + F\_{af} - \sum\_{i=1}^{4} \frac{BF\_\Delta}{\mathcal{L}} \sin \delta\_{i\prime} \tag{4}$$

where *B* is the wheel base, *F*<sup>Δ</sup> denotes the change in the drive force, δ*<sup>i</sup>* is the deflection angles of the wheels. With other conditions unchanged, the smaller additional steering resistance, the smaller driving force required by the vehicle, and the less energy consumption. Then it can be seen from Equations (3) and (4) that under certain condition the increase of *F*Δ is conducive to the reduction of driving resistance. However, as it increases, the tire side-slip angle also increases, which will lead to the increase of the additional steering resistance, so it is not a monotonous change for the total driving resistance. Besides, the speed and deflection angles of wheels also affect the tire side-slip angle, so it is necessary to find the optimal torque distribution ratio at different speeds and steering angle, so as to make the driving resistance of the vehicle minimum.

In addition, the torque distribution of each wheel will also affect the actual working efficiency of the motor. Therefore, the total energy consumption of the vehicle should be taken as the optimization goal, and efficiency of all in-wheel motor is taken into account to achieve dynamic optimization.
