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The in-wheel motor is increasingly used in electric vehicles due to the significantly improved controllability, response capability, and energy recovery efficiency based on this technology. However, the independent control of in-wheel motors will lead to braking torque distribution problems, especially in a situation where anti-lock braking systems (ABS) are triggered, which may cause the braking energy to be unrecoverable without the coordinated control between anti-lock and RB for two in-wheel motor-driven electric vehicles based on the RB efficiency map. Control-oriented wheel dynamics and slip ratio models of the system are generated. A sliding mode intervention of regenerative braking (RB) control. This paper presents an integrated algorithm to realize the control-based ABS controller is designed to prevent the wheels from locking and to maintain the slip ratio within a desired level, and the stability and robustness of the controller to uncertainties and disturbances are discussed. Moreover, the braking strength of the driver is calculated and divided into different modes to derive a dynamic braking torque distribution to improve the energy recovery efficiency. The hardware-in-the-loop simulation results show that the recovered energy of the proposed strategy under ABS-triggered maneuver is increased by 52.9% than that of the Proportional, Integral, and Derivative controller and can effectively improve the braking performance and stability.

Many studies imply that almost one-half of the driving energy is dissipated during braking, especially in urban driving circumstances with frequent acceleration and deceleration manipulations^{[1,2]}. The energy consumption during braking processes in different driving cycles is shown in ^{[3]}, where UDDS and ECE stand for urban dynamometer driving schedule and Economic Commission of Europe, respectively. Therefore, regenerative braking systems (RBS), which could recapture kinetic energy and greatly reduce power consumption^{[5]}, are widely researched in electric or hybrid vehicles^{[4–6]}, and brake-by-wire systems^{[7,8]} that support the application of regenerative braking (RB) are investigated as well^{[7–10]}. Xu ^{[11]}. Guo ^{[12]}. However, the above strategies do not take the vehicle braking stability into consideration, which will cause the vehicle instability in limited conditions. Biao ^{[13]}. The torque distribution ratio is determined by the braking strength and related constraint conditions, and the energy recapture efficiency of the proposed strategy was raised by about fifty percent, but the stability effect of the vehicle was not reflected in the verification part.

Energy consumption during braking processes

UDDS: Urban dynamometer driving schedule; ECE: Economic Commission of Europe. | |||

Total energy consumption, kJ | 28241 | 1814 | 395 |

Braking energy consumption, kJ | 13432 | 938 | 207 |

Rate, % | 47.6 | 51.7 | 52.3 |

The in-wheel motor, which is located directly in the wheels, will significantly enhance controllability and response speed by removing all the intermediate mechanical connections. Additionally, it acquires high efficiency both in motor and generator modes^{[14]}. Therefore, the control strategies for in-wheel motor-driven electric vehicles (IWMDEV) have been widely studied. Karabacak ^{[15]}. Gang and Zhi proposed an RB control method for four IWMDEV (4IWMDEV) under urban scenes using motor efficiency maps, and the simulation results validated the effectiveness of the proposed method under urban driving cycles^{[16]}. A feedback hierarchical controller for 4IWMDEV is presented by Chen ^{[17]}. However, in addition to improving recovery efficiency, the braking performance and stability should also be considered, especially under heavy braking strength request conditions, which will cause the braking torque distribution problem^{[18–21]}.

Dadashnialehi ^{[22]}. Wang and He presented an improved linear quadratic Gaussian controller to derive braking torques for ABS control, and a varying charge voltage in steps control is designed to recover more energy^{[23]}. Yang ^{[24]}. Although these methods integrate the RBS and ABS control together to improve the braking performance and recover more energy, rarely do these methods address the stability problem during braking situations. He ^{[25]}. Tang ^{[26]}. Pei ^{[27]}. However, the braking intension is rarely considered during the coordinated control of energy recovery and braking stability.

The above studies indicate that the problem of how to simultaneously improve the braking stability and regeneration efficiency for 2IWMDEVs considering braking intension is still not resolved. This paper aims to present an integrated control method to combine ABS and RBS control, even in emergency braking situations, and synthetically improve braking performance, recovery efficiency, stability, and road adaptability of 2IWMDEVs. The main contributions of the paper include a robust SMC-based ABS controller to make full use of the road conditions under different adhesion coefficients and an RBS strategy to distribute the regenerative and hydraulic braking torques to recapture more energy.

The rest of the paper is organized as follows: In the second section, a SMC-based ABS controller is proposed for 2IWMDEVs, including wheel dynamics models, slip ratio models, ABS controllers, and stability and robustness analysis of the controller. The distribution method of regenerative and frictional braking torques is designed in the third section. In the fourth section, the RB efficiency map is generated through bench experiments. Hardware-in-the-loop (HIL) simulations are conducted in the fifth section to validate the proposed algorithm. Finally, within the same section, conclusions are made, and a future work outlook is provided.

An electric vehicle equipped with two front in-wheel motors and RBSs is applied in the paper, and the two rear wheels are driven by a traction motor.

Wheel dynamic.

The following assumptions are made.

● The tire rolling resistance and vehicle aerodynamics are negligible during braking processes.

● Tire self-aligning torque is negligible.

● The chamber angle of each wheel is zero.

The dynamic equations are as follows.

where

Combining Equations (1), (2), and (3) yields

where the slip ratio could be expressed as follows during braking situations

Taking the derivative of Equation (5) with respect to time and substituting Equation (4) into it, we have

The curve of

where

Substituting Equation (7) into Equation (6), after some manipulations, we have

ABS controllers are designed to maintain the slip ratios of wheels at a particular level, where both the longitudinal and lateral adhesion forces reach their maximum value. To this end, we define the sliding surface in terms of slip ratio tracking errors as

It is apparent that if we find a control law to make

where

Differentiating Equation (9) with respect to time and combining Equation (8) yield

Since

The Lyapunov function is designed as follows:

Thus,

Substituting Equation (8) into Equation (15), after some manipulations, we have

Substituting Equation (13) into Equation (16) yields

The stability of the proposed ABS controller has been proved, and the system will move to the surface according to Equation (17). Assume that the initial slip ratio tracking error is finite, and then,

Nonlinear characteristics, disturbances, and uncertainties always exist in practice control systems, e.g., motor torques, estimated vehicle velocity, and sensor noise. Therefore, if the controller is to be applied in practice, it is essential to analyze its robustness.

In the actual motor control process, it is generally difficult to achieve real-time precise implementation of the desired motor torque listed in Equation (13), and the torque tracking error may reduce the control performance or even cause the controller instability. Moreover, the additive disturbance will affect the design value of convergence factor

Replacing

Note that the first derivative of the Lyapunov function should be negative definite to ensure the system stability; the torque disturbance should satisfy the following inequality constraints.

The above equation means that the factor of convergence

The actual vehicle velocity is normally obtained from the transmission output shaft speed sensor or estimated based on the wheel speed sensor. However, the above methods will cause velocity error, which leads to slip ratio error

For the controller, the actual slip ratio is expressed as

From the above equation,

Sensor noise has a significant influence on the stability of a closed-loop system, and the signal fluctuation will degrade measurement accuracy and affect state parameter calculation; e.g., the fluctuation of wheel velocity signals will influence the slip ratio calculation according to Equation (5). High-frequency filters^{[7]} are adopted to estimate the actual value, decrease measurement noise, and guarantee the system stability.

With two front in-wheel motors, the regenerative and frictional blending brake is related, and the optimal objective is to improve energy recovery efficiency and braking stability. To this end, the distribution of the regenerative and frictional braking torques should be first discussed. As shown in ^{[7]}, and then the requested braking force ^{[28]}, and the novel electro-hydraulic actuator generates braking force fast and has a fast and good pressure-tracking performance.

Braking torque control schematic.

To improve tire-road friction coefficient (TRFC) utilization efficiency and directional stability during braking situations, both the front and rear wheels should be locked simultaneously, and the desired braking forces of the front and rear axles are expressed as follows

where

With manipulations of Equation (23), we have

As shown in

(A) When

where

(B) When

where

(C) When

where

An optimal energy recovery method should be designed based on an efficiency map by minimizing the transfer loss during blending brake, and the motor-to-battery efficiency is calculated by the following expression.

where

Test bench.

RB efficiency map.

A HIL simulation system equipped with vehicle-manipulator systems, steering systems, and hydraulic braking system is developed to evaluate the performance of the proposed integrated algorithm.

HIL simulation system.

Algorithm block diagram.

Vehicle model and controller parameters

Vehicle mass ( |
1,855 kg |

Rotational inertia of the front/rear tire ( |
1.5 kg |

Effective wheel rolling radius ( |
316 mm |

Height of the center of gravity ( |
530 mm |

Distance between the front and rear axles ( |
2,490 mm |

Front axle to the center of gravity ( |
1,100 mm |

Rear axle to the center of gravity ( |
1,390 mm |

Effective radius of the front brake disc ( |
105 mm |

Piston radius of the front wheel cylinder ( |
27 mm |

Friction coefficient of the front brake pad ( |
0.4 |

Maximum RB torque ( |
1,200 N |

Convergence factor ( |
0.02 |

Boundary layer thickness ( |
0.1 |

A steering-braking maneuver with a 75 km/h initial forward velocity is performed on a high-

Velocities of vehicles and wheels. (A) Front left wheel; (B) Front right wheel; (C) Rear left wheel; (D) Rear right wheel.

Slip ratio of the wheels.(A) Front left wheel; (B) Front right wheel; (C) Rear left wheel; (D) Rear right wheel.

Vehicle sideslip angular velocity and yaw rate. (A) Sideslip angular velocity; (B) Yaw rate.

Distribution of the regenerative braking energy.

Comparison of total recaptured energy.

A driver-in-the-loop simulation on the split surface with a 70 km/h initial forward velocity is conducted to evaluate the effectiveness of the proposed algorithm in non-ideal scenarios, and the TRFCs of the left and right sides are 0.3 and 0.8, respectively.

Simulation results on the split surface. (A) Braking pressure; (B) Velocities; (C) Regenerative braking; (D) Hydraulic braking.

With the parameter values listed in

An integrated algorithm is proposed for ABS and RBS coordinated control of 2IWMDEVs to improve vehicle braking stability and energy recovery efficiency, including both RB control on the front two wheels and anti-lock braking control on all four wheels. HIL simulations are carried out to verify the effectiveness of the integrated algorithm, and the results show that the ABS controller can prevent wheels from being locked to ensure braking performance and vehicle stability. With the distribution strategy of braking torques among in-wheel motors and friction brakes considering the required braking torque and strength, the recovered energy of the proposed strategy under ABS-triggered maneuver is increased by 52.9% than that of the PID controller. Therefore, the presented integrated algorithm can not only maintain the slip ratio within the desired range and braking stability but also achieve excellent energy recapture efficiency, which is significant to ensure safety and economy of the 2IWMDEVs.

Our future work will concentrate on adaptive threshold value design for braking modes division, experimental validation of the proposed algorithm through in-vehicle field and road tests with the consideration of uncertainties, and modification of the proposed algorithm to adapt to 4IWMDEVs.

The authors would like to thank the anonymous reviewers for their valuable comments.

Made substantial contributions to the conception and design of the study and performed data analysis and interpretation: Yong J, Dong Y, Zhang Z

Performed data acquisition and provided administrative, technical, and material support: Zhang Z, Feng N, Li W

Not applicable.

This work was supported by the National Natural Science Foundation of China (No. 52002009), Beijing Natural Science Foundation (No. 3222003), and the State Key Laboratory of Automotive Safety and Energy under Project No. KF2010.

All authors declared that there are no conflicts of interest.

Not applicable.

All the authors have agreed to publish the paper.

© The Author(s) 2024.