© The Author(s), published by EDP Sciences, 2025

1 Background

In order to meet the needs and efficiency of contemporary transportation, heavy transport trucks and high-power transport trucks have developed rapidly. Heavy transport trucks and high-power transport trucks are mainly used in the transportation of large goods, and their driving and braking processes have significant energy changes and low energy efficiency, resulting in a lower range. Therefore, improving the power control strategy of transport vehicles and further enhancing their energy utilization efficiency can enable them to travel longer distances under the same electrical conditions [1, 2].

As the load capacity of goods increases, transport truck also changes from single axle drive to multi-axle drive, and from single power source drive to multi-power source drive. During transport-truck operations, energy consumption is high. An unlimited increase in battery capacity will lead to an increase in vehicle weight, which is obviously not the best way to increase driving range. Therefore, designing and developing better power control strategies is becoming increasingly important. Under certain energy source conditions, by designing more optimal power optimization control strategies, while meeting the requirements of driving power and economy, the efficiency of energy utilization of the vehicle can be improved, and the range of the vehicle can be further increased [3].

Manne et al. [4] analyzed the speed curve of the driving cycle, accurately controlled the driving torque, and reduced the fuel consumption of heavy vehicles. Cong et al. [5] analyzed the shifting mode of dual motor electric vehicles and proposed the optimal shifting torque distribution pattern under acceleration and braking conditions. Pataparambath and Subramaniam [6] developed a dual torque control strategy for 48v-P3 hybrid electric vehicles to improve the energy transfer efficiency of the wheels. S. Shakya [7] used fuzzy control theory to design a torque control system for electric vehicles that is insensitive to parameter changes and disturbances. Laguidi et al. [8] proposed a dual motor drive control method for electric vehicles based on nonlinear PI control technology to reduce wheel slip. Verbruggen et al. [9] used particle swarm optimization algorithm to optimize gear shifting, thereby further improving the driving efficiency of cars. Samuel et al. [10] studied the driving control strategy of plug-in hybrid vehicles using multi-objective optimization methods. Ryosuke et al. [11] proposed a four-wheel drive vehicle driving force distribution method based on wheel energy loss rate to improve off-road vehicles and optimized the driving force distribution. Kang et al. [12] proposed control modes and expected dynamic torque control strategies for the 4WD series hybrid electric vehicles with front and rear drive axles. Moinfar et al. [13] studied the effects of several tractor drive systems [rear wheel drive (RWD), four-wheel drive (4WD), and front wheel drive (FWD)] on tractor performance and fuel usage. Through reviewing the current research status of automotive drive control strategies, it has been found that whether it is a single-motor or multi-motor drive, how to reasonably control the torque output of each wheel drive and coordinate their work is the key to achieving optimal drive control. For heavy-duty transfer vehicles, the driving system is no longer a single-motor single-axis drive, but a multi-motor multi-axis drive.

2 Theory and modeling

2.1 Motor map diagram

The same motor is used for the front and rear drive motors in this article, and the selected drive motor map is shown in Figure 1. From the graph, it can be seen that the driving efficiency of the front/rear motor is related to the motor speed and the current motor speed, but the driving efficiency is different for each torque and torque. Therefore, to improve the driving efficiency of dual motor multi-axis drive autonomous vehicles, it is necessary to make the front/rear motors work in areas with higher motor driving efficiency as much as possible and avoid appearing in areas with lower efficiency.

Figure 1 Drive motor map.

2.2 Optimization model and strategy for torque distribution in dual motor multi-axis drive

A dual-motor torque allocation optimal control strategy is proposed for the motion control process of a dual-motor multi-axis drive autonomous vehicle, considering the road adhesion coefficient and aiming for optimal driving efficiency [14]. The torque distribution control scheme for dual-motor, multi-axis drive autonomous vehicles is illustrated in Figure 2.

Figure 2 Optimal torque distribution scheme for dual motor multi-axis drive autonomous vehicles.

For dual-motor multi-axis drive autonomous vehicles; in order to achieve optimal working efficiency, it is necessary to establish the optimal driving efficiency objective function for dual-motor multi-axis drive autonomous vehicles. This article defines the utilization efficiency η_opt of dual drive motor drive as:$$\begin{array}{c}{\eta }_{\mathrm{opt}}=\frac{(T_{t1}+T_{t2}){\eta }_1{\eta }_2}{T_{t1}{\eta }_1+T_{t1}{\eta }_2}\end{array},$$(1)the efficiency η₁ of the front motor that drives the rotation of the first and second bridges is when the speed is n and the torque is T_t1. The driving efficiency η₂ of the rear motor that drives the rotation of the third and fourth axles is when the speed is n and the torque is T_t2.

When driving normally, assuming the total driving torque required for a dual motor multi-axis drive autonomous vehicle is T_t, the output torque of the front motor driving the first and second axles to rotate is T_t1, and the output torque of the rear motor driving the third and fourth axles to rotate is T_t2, then there are:$$\begin{array}{c}{T}_t=T_{t1}+T_{t2}\end{array}.$$(2)

To simplify the model, assume the distribution coefficient of the rear motor is α, there are:$$\begin{array}{c}{T}_{t2}={\alpha T}_t.\end{array}$$(3)

For α, there are:$$\begin{array}{c}0\le \alpha \le 1.\end{array}$$(4)

For the front drive motor, there are:$$\begin{array}{c}{T}_{t1}={(1-\alpha )T}_t\end{array}.$$(5)

The driving motor efficiency is a function of the torque and speed. Therefore, for the determined reduction ratio and transmission efficiency of the dual-motor multi-axis drive autonomous vehicle, under the condition of fixed speed, the working efficiency of the motor is adjusted by adjusting the torque of the front/rear motor, so that the front/rear motor works in the high efficiency area. Assuming a dual motor drive efficiency objective function η_opt, the optimal value is Max(η_opt):$$\begin{array}{c}\mathrm{Max}\left({\eta }_{\mathrm{opt}}\right)=\mathrm{Max}\left(\frac{({\alpha T}_t+{(1-\alpha )T}_t){\eta }_1{\eta }_2}{{\alpha T}_t{\eta }_1+{(1-\alpha )T}_t{\eta }_2}\right) \end{array}.$$(6)The constraint conditions are:$$\begin{array}{c}\{\begin{array}{c}{T}_{t2}={\alpha T}_t\\ T_{t1}={\left(1-\alpha \right)T}_t\\ \begin{array}{c}{\eta }_1=f\left(n,T_{t1}\right)\\ {\eta }_2=f\left(n,T_{t2}\right)\\ \begin{array}{c}0\le \alpha \le 1\\ 0\le T_t\le T_{t\max 1}+T_{t\max 2}\\ \begin{array}{c}0\le T_{t1}\le T_{t\max 1}\\ 0\le T_{t1}\le T_{t\max 2}\end{array}\end{array}\end{array}\end{array}. \end{array}$$(7)

In the above equation, T_tmax1 is the maximum output torque of the front axle motor at a speed of n. The maximum output torque T_tmax2 of the rear axle motor at a speed of n.

When a dual-motor multi-axis drive autonomous vehicle is driving, the speed and torque constantly change. To obtain the optimal torque distribution for the front and rear motors of the vehicle under various torque and speed conditions, equations (6) and (7) are optimized. The optimal distribution diagram of the front and rear motors under different speeds and torques is shown in Figure 3.

Figure 3 Diagram of torque distribution coefficients for front and rear motors based on motor map characteristics.

2.3 Optimization control strategy for energy recovery of dual motor multi-axis braking

For dual motor-driven autonomous vehicles and meeting the above braking requirements, the motor map feature is added to further optimize the regenerative braking energy recovery [15–17]. The optimal braking distribution strategy is designed based on the feedback motor map characteristics, while considering the driving torque distribution optimization strategy principle diagram of the motor map and road adhesion coefficient [18–20]. The optimal braking distribution strategy is shown in Figure 4.

Figure 4 Schematic diagram of driving torque allocation optimization strategy considering motor map characteristics and road adhesion coefficient.

An optimal braking torque distribution model is built based on the feedback motor map characteristics. Assuming for a dual motor multi-axis drive autonomous vehicle, the braking torque between the first and second axes is the same, and the braking torque between the third and fourth axes is the same [21, 22]. At this point, the feedback braking force relationship between the front and rear motors is as:$$\begin{array}{c}\{\begin{array}{c}{T}_{b1}={\alpha }_b{T}_{b\mathrm{total}}\\ T_{b2}=(1-{\alpha }_b{)T}_{b\mathrm{total}}\end{array}. \end{array}$$(8)

Among them, T_b1 and T_b2 respectively represent the braking torque of the front and rear motors, and T_btotal are the total feedback braking torque. The braking energy efficiency of the double feedback motor is:$$\begin{array}{c}{\eta }_{b\mathrm{opt}}=\frac{{(T}_{b1}+T_{b2})n_{b1}{n}_{b2}}{T_{b1}{n}_{b1}+T_{b2}{n}_{b2}} \end{array}.$$(9)

The optimal value Max(η_bopt) of braking energy efficiency is set as:$$\begin{array}{c}\mathrm{Max}\left({\eta }_{b\mathrm{opt}}\right)=\mathrm{Max}\left(\frac{{(T}_{b1}+T_{b2})n_{b1}{n}_{b2}}{T_{b1}{n}_{b1}+T_{b2}{n}_{b2}}\right) \end{array}.$$(10)

The constraint conditions are:$$\begin{array}{c}\{\begin{array}{c}{T}_{b1}\le T_{b1\max }\left(n_{b1}\right)\\ T_{b2}\le T_{b2\max }\left(n_{b2}\right)\\ \begin{array}{c}{\eta }_{b1}=f(T_{b1},n_{b1})\\ {\eta }_{b2}=f(T_{b2},n_{b2})\\ \begin{array}{c}{P}_{b_1}=\frac{T_{b1}{n}_{b1}{\eta }_{b1}}{9550}\\ P_{b_2}=\frac{T_{b2}{n}_{b2}{\eta }_{b2}}{9550}\\ \begin{array}{c}{\alpha }_{\min }\le {\alpha }_b\le {\alpha }_{\max }\\ P_{b_1}+P_{b_2}\le P_{b_{\mathrm{total}}}\end{array}\end{array}\end{array}\end{array}. \end{array}$$(11)

Among them, T_bf and T_br respectively represent the braking feedback torque of the front and rear motors. α_b is distribution coefficient of braking torque for front and rear motors. $P_{b_{\mathrm{total}}}$ is maximum power of feedback braking. T_bfmax (n_bf) and T_brmax (n_br) respectively represent the maximum torque that the motor can provide. α_min and α_max are the upper and lower bounds, respectively.

The optimization model was obtained for different rotational speeds and braking torque requirements, as shown in Figure 5.

Figure 5 Optimal model for braking torque distribution of double feedback motor.

An optimal braking control strategy for a dual motor multi-axis drive autonomous vehicle is designed. In order to improve the efficiency of vehicle braking energy recovery and meet braking safety requirements, priority should be given to using feedback motors to generate braking force for braking. Developing specific control strategies is as follows:

(1) While z_b < 0.2 the distribution of braking force can be:$$\begin{array}{c}\{\begin{array}{c}{F}_{\mathrm{rm}1}={\alpha }_{b1}\min \left(G{z}_b,\min \left(\frac{i_g{(T}_{\mathrm{rm}1}\left(n\right)+T_{\mathrm{rm}1}\left(n\right))}{{\eta }_1{r}_e},T_{\mathrm{batt}1}\right)\right)\\ F_{\mathrm{rm}2}=(1-{\alpha }_{b1})\min \left(G{z}_b,\min \left(\frac{i_g{(T}_{\mathrm{rm}2}\left(n\right)+T_{\mathrm{rm}2}\left(n\right))}{{\eta }_2{r}_e},T_{\mathrm{batt}2}\right)\right)\\ \begin{array}{c}{F}_{\mathrm{bf}}=\frac{\mathrm{Gzb}}{L}-F_{\mathrm{rm}1}\\ F_{\mathrm{br}}=\frac{\mathrm{Gzb}}{L}-F_{\mathrm{rm}2}\end{array}\end{array} \end{array}.$$(12)

(2) While 0.2 < z_b < 0.5 the distribution of braking force can be:$$\begin{array}{c}\{\begin{array}{c}{F}_{\mathrm{rm}1}={\alpha }_{b1}\min \left(F_{\mathrm{fI}},\min \left(\frac{i_g{(T}_{\mathrm{rm}1}\left(n\right)+T_{\mathrm{rm}1}\left(n\right))}{{\eta }_1{r}_e},T_{\mathrm{batt}1}\right)\right)\\ F_{\mathrm{rm}2}=(1-{\alpha }_{b1})\min \left(F_{\mathrm{rI}},\min \left(\frac{i_g{(T}_{\mathrm{rm}2}\left(n\right)+T_{\mathrm{rm}2}\left(n\right))}{{\eta }_2{r}_e},T_{\mathrm{batt}2}\right)\right)\\ \begin{array}{c}{F}_{\mathrm{bf}}=F_{\mathrm{fI}}-F_{\mathrm{rm}1}\\ F_{\mathrm{br}}=F_{\mathrm{rI}}-F_{\mathrm{rm}2}\end{array}\end{array} \end{array}$$(13)

(2) While z_b > 0.5 the distribution of braking force can be:$$\begin{array}{c}\{\begin{array}{c}{F}_{\mathrm{bf}}=G{z}_b(b+z_b{h}_g)/L\\ F_{\mathrm{br}}=G{z}_b(a-z_b{h}_g)/L\\ \begin{array}{c}{F}_{\mathrm{rm}1}=0\\ F_{\mathrm{rm}2}=0\end{array}\end{array} \end{array}.$$(14)

Among them, T_batt1 and T_batt2 are the maximum allowable charging torque of the battery. F_rm1 and F_rm2, respectively, represent the feedback braking force of the front and rear motors. F_bf and F_br are the mechanical braking forces of the front and rear axles.

3 Results and analysis

The maximum speed is 49.75 km/h. The C-WTVC cycle condition is a standard developed based on WP29 to detect the economic and power performance of heavy-duty vehicles under instantaneous cycle conditions, which are divided into urban conditions, highway conditions, and high-speed conditions. The following diagram shows the schematic diagram of the C-WTVC cycle operating condition and the C-WTVC cycle operating condition taking a factor of 0.5.

Based on Figure 6, due to the dual motor multi-axis drive, the autonomous vehicle’s maximum speed exceeds 49 km/h. Therefore, this study uses half of the speed value of the C-WTVC cycle condition as the expected cycle condition for simulation, as indicated by the red line in Figure 6. The simulated speed following and speed error values are shown in Figure 7.

Figure 6 Schematic diagram of C-WTVC cycle operating conditions.

Figure 7 C-WTVC speed tracking results and error graph. (a) Speed following result graph, (b) speed following error.

For the C-WTVC cycle simulation model, the driving torque can be obtained as shown in Figure 8. From the graph, it can be seen that the torque distribution is mainly distributed between 0.5 and 1, with only a small portion distributed between 0.5 and 0.6.

Figure 8 C-WTVC speed tracking results and error graph. (a) Driving torque demand diagram, (b) drive torque distribution diagram.

As a comparison, using the average torque distribution of dual motors as a comparative experiment, the energy consumption of C-WTVC driving conditions can be obtained, as shown in the table below. From Table 1, it can be seen that the optimal driving control strategy, based on the motor map characteristics, can achieve the goal of reducing driving energy consumption.

For the braking process, the C-WTVC braking torque distribution and energy recovery change curve are shown in Figure 9.

Figure 9 C-WTVC braking torque distribution and energy recovery variation curve. (a) Braking torque distribution diagram, (b) energy recovery change curve.

From Figure 9, it can be seen that during the braking process, the motor provides a portion of regenerative feedback braking force, while the rest is provided by pneumatic braking. The energy consumption table for the C-WTVC braking condition is shown Table 2, with an energy recovery rate of 32.8%.

CHTC-HT is a standard officially implemented in May 2020 for the inspection of the power and economy of heavy-duty vehicles. The following diagram shows the schematic diagram of CHTC-HT cycle operating conditions and CHTC-HT cycle operating conditions taking half of the values.

Based on Figure 10, due to the dual motor multi-axis drive, the autonomous vehicle’s maximum speed exceeds 49 km/h. Therefore, this study used half of the speed value of the CHTC-HT cycle as the expected cycle condition for simulation, as indicated by the red line in Figure 10. The simulated speed following and speed error values are shown in Figure 11.

Figure 10 Schematic diagram of CHTC-HT cycle operating conditions.

Figure 11 Speed tracking results and error diagram of CHTC-HT cycle operating conditions. (a) Speed following result graph, (b) speed following error.

For the CHTC-HT cycle simulation model, the driving torque can be obtained as shown in Figure 12. From the graph, it can be seen that the torque distribution is mainly distributed as 1, with only a small portion distributed between 0.5 and 1.

Figure 12 CHTC-HT speed tracking results and error chart. (a) Driving torque demand diagram, (b) Drive torque distribution diagram.

As a comparison, using the average torque distribution of dual motors as a comparative experiment, the energy consumption of the CHTC-HT driving condition can be obtained, as shown in Table 3. From the table, it can be seen that the optimal driving control strategy based on motor map characteristics can achieve the goal of reducing driving energy consumption.

For the braking process, the CHTC-HT braking torque distribution and energy recovery change curve are shown in Figure 13.

Figure 13 CHTC-HT braking torque distribution and energy recovery variation curve. (a) Braking torque distribution diagram, (b) energy recovery change curve.

From Figure 13, it can be seen that during the braking process, the motor provides a portion of regenerative feedback braking force, while the rest is provided by pneumatic braking. The energy consumption table for the CHTC-HT braking condition is shown in Table 4, with an energy recovery rate of 36.7%.

4 Conclusion

Aiming at the dual motor multi-axis drive autonomous vehicle, a dual-motor multi-axis drive autonomous vehicle driving torque distribution model and driving torque distribution strategy are proposed. A dual feedback motor torque distribution optimization model is designed, and a braking torque distribution strategy that meets the optimal braking energy recovery is designed. Establishing simulation experiments for C-WTVC and CHTC-HT cycle conditions, the results show that the power optimization control strategy based on motor map characteristics has good energy utilization efficiency, further improving driving energy efficiency and increasing braking energy recovery efficiency by 10%. It can further increase the range of vehicles.

Funding

This work was supported by the High-level professional group of automotive manufacturing and testing technology in vocational colleges in Guangdong Province (Grant No. GSPZYQ202102), Research Project of Guangdong Provincial Department of Education (Grant Nos. 2022KQNCX188 or ZXKY2022039), and Research Project of Guangdong Polytechnic (Grant No. XJKY202329).

References

All Tables

All Figures

	Figure 1 Drive motor map.
In the text

	Figure 2 Optimal torque distribution scheme for dual motor multi-axis drive autonomous vehicles.
In the text

	Figure 3 Diagram of torque distribution coefficients for front and rear motors based on motor map characteristics.
In the text

	Figure 4 Schematic diagram of driving torque allocation optimization strategy considering motor map characteristics and road adhesion coefficient.
In the text

	Figure 5 Optimal model for braking torque distribution of double feedback motor.
In the text

	Figure 6 Schematic diagram of C-WTVC cycle operating conditions.
In the text

	Figure 7 C-WTVC speed tracking results and error graph. (a) Speed following result graph, (b) speed following error.
In the text

	Figure 8 C-WTVC speed tracking results and error graph. (a) Driving torque demand diagram, (b) drive torque distribution diagram.
In the text

	Figure 9 C-WTVC braking torque distribution and energy recovery variation curve. (a) Braking torque distribution diagram, (b) energy recovery change curve.
In the text

	Figure 10 Schematic diagram of CHTC-HT cycle operating conditions.
In the text

	Figure 11 Speed tracking results and error diagram of CHTC-HT cycle operating conditions. (a) Speed following result graph, (b) speed following error.
In the text

	Figure 12 CHTC-HT speed tracking results and error chart. (a) Driving torque demand diagram, (b) Drive torque distribution diagram.
In the text

	Figure 13 CHTC-HT braking torque distribution and energy recovery variation curve. (a) Braking torque distribution diagram, (b) energy recovery change curve.
In the text