Development and Control of a Magnetorheological Damper-Based Brake Pedal Simulator for Vehicle Brake-by-Wire Systems

Performance indexes of the designed MRDBBPS are listed as follows: the maximum pedal displacement was 120 mm, the maximum pedal force was 500 N, and the maximum pedal velocity is 500 mm/s. Figure 2 shows the force diagram of the brake pedal. In the figure, \(P_{1}\) represents the initial position of the pedal, and \(P_{2}\) is the position after the pedal rotates at an angle of \(\alpha\). \(F\) is the pedal force, and r is the vertical distance from the action center of the pedal force to the rotational centerline of the pedal shaft. In the design, \(r = 150\;{\text{mm}}\).

In the preliminary design, without considering the spring force and friction resistance, the relationship between the pedal force \(F\) and the damping torque of the MR damper \(T_{{\text{M}}}\), and that between the pedal velocity \(v\) and the rotational speed of the driven gear \(n\) are described bywhere \(i\) is the transmission ratio of the driving gear and driven gear; \(i = 4\). By substituting the values of the performance indexes and \(i\) into Eqs. (1), (2), the required maximum damping torque of the MR damper is calculated as \({\text{T}}_{{{\text{M}}_{\max } }} = 18.75\;{\text{N}} \cdot {\text{m}}\), the maximum rotational speed of the driven gear is \(n_{\max } = 12{7}\;{\text{r/min}}\), and the maximum rotation angle of the pedal is 45.84°.

As described above, a disk-type MR damper shown in Figure 3 is designed to provide the braking reaction for the BPS. Without current supplied to the coil, the MRFs, filled in the working gap between the disk and the housing, exhibits Newtonian fluid characteristics, resulting in a very small damping torque. When the coil is energized, a magnetic field is generated in the working gap. The ferromagnetic particles in the MRFs attract to form a solid-like structure perpendicular to the disk instantly, and a large damping torque is produced. In this design, a single-disk form is utilized for the MR damper. According to Ref. [28], the total damping torque \(T_{{\text{M}}}\) can be expressed aswhere \(\tau_{y}\) is the field-induced shear stress of the MRFs, \(\eta\) is the apparent viscosity of the MRFs, \(R_{2}\) and \(R_{1}\) are the effective outer and inner radii of the working gap, respectively, \(h\) is the working gap thickness, and \(\omega\) is the angular velocity of the shaft; \(\omega { = }n{\uppi }/30\).

The material properties of the MRFs significantly affect the operating performance of the MR damper. In this study, a type of MRFs (model SG-MRF2035), provided by Ningbo Shangong Monitoring and Control Engineering Co., Ltd., is selected. Its apparent viscosity is \(\eta { = 24}0\;{\text{MPa}}\cdot{\text{s}}\). Referring to the relationship between the field-induced shear stress \(\tau_{y}\) and the magnetic flux density B measured in our previous study [39], the following equation is derived using the polynomial fitting method

Eq. (4) indicates that \(\tau_{y}\) is fully determined by B. For a coil current of 1.1 A, a two-dimensional electromagnetic simulation of the MR damper is conducted. The distribution of magnetic flux density is shown in Figure 4, which shows that the maximum flux density appears at the housing near the coil and its value is approximately 1.57 T, which is lower than the saturation flux density of the selected material (ISO C20e). Moreover, the flux density distributes approximately uniformly along the radial direction of the working gap with an average value of 0.44 T. Combining Eqs. (1) to (4), the total damping torque is calculated as \(T_{{\text{M}}} = 22.9\;{\text{N}}\cdot{\text{m}} > T_{{{\text{M}}_{\max } }} = 18.75\;{\text{N}}\cdot{\text{m}}\), which means the torque output of the MR damper satisfies the design requirements.

As mentioned earlier, the main function of the torsion spring is to provide a reaction force to the pedal when released. However, while the pedal is pressed, the torsion spring also produces a resistance expressed aswhere \(T_{{\text{s}}}\) is torsional resistance torque of the torsion spring; \(E\) is the elastic modulus of the material for the torsion spring, \(E{ = }206\;{\text{GPa}}\); \(d_{{\text{s}}}\) is the wire diameter, \(d_{{\text{s}}} { = }3\;{\text{mm}}\); \(D_{{\text{s}}}\) is the mean diameter of the torsion spring, \(D_{{\text{s}}} { = }29\;{\text{mm}}\); \(n\) is the effective number of the wire turns, \(n{ = }4\); and \(\alpha_{{\text{s}}}\) the torsional angle of the spring.

By substituting the above parameter values into Eq. (5), \(T_{{\text{s}}}\) can be rewritten as

According to Figure 2, the relationship between the pedal displacement \(s\) and the torsional angle of the spring \(\alpha_{{\text{s}}}\) is expressed as

Thus, the relationship between \(T_{{\text{s}}}\) and \(s\) can be written as

The maximum torsional resistance torque provided by the torque spring is calculated as 1.8 N·m by substituting the value of maximum pedal displacement into Eq. (8). It accounts for approximately 7.8% of the total damping force of the MR damper; thus it is ignored in this study.

Without coil current applied to the MR damper, the pedal force is the sum of the Coulomb friction resistance generated by bearings and rotary rings and the viscous resistance of the MRFs. It is measured at different pedal velocities and the results are plotted in Figure 5(a). As observed, the value of the pedal force is considerably small, basically fluctuating at approximately around 16 N. Figure 5(b) presents the variation in the pedal force with time for a coil current of 0.4 A. The pedal force is observed to increase slightly with the pedal velocity, which means that the pedal velocity has only a slight effect on the pedal force. As the coil current changes from 0→1.1→0 A, the relationship curves between the pedal force and the coil current are gained in Figure 5(c). The two curves do not coincide with each other in the current increasing and decreasing stages. Specifically, the pedal force at the current decreasing stage is higher than that at the current increasing stage. We can infer that the pedal force generated by the MRDBBPS has an apparent hysteresis effect. The main reason is that MR fluids are composed of soft magnetic particles and other materials. In the process of repeated magnetization, the change of magnetic induction intensity of magnetic material always lags behind its magnetic field intensity, which will lead to the obvious hysteresis effect of pedal force. Moreover, the measured pedal force at a coil current of 1.1 A is 552 N, higher than the required value of 500 N, which means that the MRDBBPS prototype satisfies the force requirement.

Figure 6 depicts the relationship between the pedal force and pedal displacement, known as BPCC, for a CHB system in an A0-class car, which is obtained by testing the pedal characteristics of the hydraulic braking system of a passenger car [40]. In the figure, stage OA is for the elimination of the pedal clearance, during which the pedal force increases inconspicuously with the pedal displacement. However, the pedal force increases faster with the pedal displacement at stage AB because it acts on the vacuum booster. In stage BC, the pedal force acts directly on the brake cylinder and its value increases rapidly. In general, the pedal force increases non-linearly with the pedal displacement, and the increasing trend accelerates as the pedal displacement increases.

As Figure 5(c) shows, the pedal force exhibits an apparent hysteresis when the coil current is not 0. In other words, two pedal force values map to the same current value, which negatively affects the control accuracy. Figure 7 shows the change in the pedal force with the coil current in the range of 0→0.4→0→0.8→0→1.1→0 A. Generally, the pedal forces are equal without hysteresis when the current reduces to 0 from different values. Based on this feature, an RTZ algorithm is proposed to avoid the hysteresis of the pedal force.

The basic concept of the proposed RTZ algorithm is described as follows: when the driver steps on or maintains the pedal in an applied position, the pedal velocity \(v \ge 0\), and a coil current is applied to the MR damper in accordance with the real-time pedal displacement. In contrast, if the driver releases the pedal, the pedal velocity \(v < 0\), and an RTZ operation should be performed to the coil current. The algorithm can ensure that the coil current is applied from 0 every time the pedal is pressed to effectively avoid the hysteresis effect and improve the control accuracy and repeatability.

The RTZ algorithm requires the coil current to be capable of changing to the desired value in real-time. However, in practice, the response lag of the coil circuit also affects the real-time performance of the current output. Normally, the rheological time of MRFs is negligible, within a few milliseconds [41, 42]. While using a constant voltage source U as the power supply of the coil, the coil circuit can be equivalent to a series circuit composed of an inductor L and a resistor R. The following equation can be obtained

Subsequentlywhere \(\tau\) is the response time of the coil current, \(\tau = \frac{L}{R}\). For the first-order inertial link, the response time refers to the time required for the current to attain 63.2% of its stable value. In this design, the response time is calculated as \(\tau { = }90\;{\text{ms}}\) according to the coil parameters. Since the brake pedal operates in a rapid movement state, a relatively large time delay negatively affects the accurate control of the coil current.

Therefore, a real-time current-tracking algorithm is proposed with its principle presented in Figure 8. At the initial moment, the current is \(I_{0}\). When \(I_{1}\) is applied, \(5\tau\) is required for the current to attain a stable value. However, the required time for the current to attain \(I_{1}\) abruptly reduces to \(\tau_{1}\) when \(I_{2}\) is applied. Since \(\tau_{1} \ll 5\tau\), a rapid response of the current is achieved, and the following equation can be obtained

Then

Since the cycle of the pulse-width modulation (PWM) signal provided by the selected control card is 20 ms, the required time for the current to increase from \(I_{0}\) to \(I_{1}\) should be less than 20 ms. To avoid overshoot, the value of \(\tau_{1}\) is set as 20 ms.

Substituting the values of \(\tau\) and \(\tau_{1}\) into Eq. (12) yieldswhere \(\Delta I\,{ = }\,I_{1} - I_{0}\).

As shown in Figure 9, the controller is designed by considering the pedal displacement as the input and the duty ratio of the PWM signal as output. First, the equation of the relationship between the pedal force F and the pedal displacement s is fitted based on the BPCC in Figure 6. Subsequently, the relationship between the coil current I and the pedal force F is fitted by polynomial curve based on the least square method referring to the experimental data in Figure 5(c). During this step, the RTZ algorithm is integrated to determine whether the coil current must be reset to zero. Finally, because I is regulated by changing the duty ratio of the PWM signal d, the relationship between d and I is fitted accordingly.

Considering the hysteresis effect, the equation of the relationship between I and F can be adjusted by introducing the RTZ algorithm as follows:where \(\xi\) is the coefficient of the RTZ algorithm. A pedal velocity of \(v \ge 0\) means that the driver steps on or holds the pedal in the applied position, then \(\xi { = }1\). If the pedal speed \(v < 0\), it means the driver releases the pedal, then \(\xi { = }0\).

Figure 10 exhibits the block diagram of the RTZ algorithm-based real-time current-tracking controller (RTZRC). When the pedal is pressed, the encoder collects the values of s and v. If \(v < 0\), the coil current returns to zero. Otherwise, the theoretical current \(I_{1}\) is calculated with reference to Eq. (14), and it is transferred to the real-time current-tracking algorithm. Combining it with the initial current \(I_{0}\) detected by the current sensor, the actual current \(I_{2}\) and the duty ratio of the PWM signal d are calculated in turn. Finally, the corresponding coil current is supplied to the MR damper by the PWM signal, and the actual pedal force is measured by the pedal force sensor.

The experimental system, shown physically in Figure 11, is established by integrating the MRDBBPS prototype into the self-developed AMRBTB. As shown in Figure 11(a), the entire system is primarily composed of three subsystems including the AMRBTB, a data acquisition and control system, and the MRDBBPS prototype. In more detail, the AMRBTB orienting to an A0-class quarter-car is developed using the single-ended inertial flywheel simulation mode. It primarily includes a motor, a flywheel set, a torque/speed sensor, an MR brake, as well as several couplings and supports. The main design parameters of the AMRBTB are listed in Table 1. For the data acquisition and control system shown in Figure 11(b), a main control card (model 74HC244) is used, which has a total of 16 input and output interfaces: 8 for analog-to-digital (AD) signal acquisition, 4 for counting, and 4 for the PWM signal output.

As mentioned above, to realize an accurate simulation of the BPCC is the key index to measure the performance of the MRDBBPS prototype and the proposed control algorithm. Figure 12 exhibits a comparison of the BPCC tracking effect without and with the proposed control algorithm. To comprehensively assess the control performance under various conditions, the experiments are carried out at three levels of pedal velocity: low, medium, and high. The results indicate that the actual pedal force is apparently lower than the ideal pedal force, primarily because of the response lag of the coil induction circuit without the proposed control algorithm. The mean absolute error is used to evaluate the force tracking effect. The mean absolute error of force tracking at low, medium and high pedal velocity is 17.19 N, 19.90 N, 30.49 N without control algorithm respectively. As shown in Figure 12(b), with the help of the proposed control algorithm, the tracking effect turns much better, even at a high pedal velocity, and the mean absolute error of force tracking with control algorithm is 8.19 N, 8.90 N, 10.49 N. The phenomenon in return verifies the effectiveness of the proposed control algorithm in Section III.

In order to further verify the effectiveness of the designed control algorithm, we have chosen the conventional PID algorithm for the pedal force tracking comparison experiment, and PID parameters are tuned based on the designer’s experiences. As shown in Figure 13(a), comparing the experimental results of the three algorithms, both the PID algorithm and the proposed RTZRC algorithm have good tracking effects compared with the uncontrolled algorithm. However, an apparent hysteresis occurs without the control algorithm and the current difference becomes very large. As shown in Figure 13(b), When using RTZRC algorithm, the current is observed to approach the desired value, the current difference is maintained within 0.04 A and the mean absolute error of current difference is less than 0.016 A. The mean absolute error of current difference is slightly larger by PID algorithm, about 0.049 A. As a consequence, the RTZRC algorithm designed in this study has a good tracking effect on pedal force and coil current.

Figure 14 shows the variations in the pedal displacement, pedal force, and pedal velocity during the intermittent braking process. In the three experiments, labeled A, B and C, the pedal is initially pressed for different displacements, and then released, followed by a re-pressed again. The variation in the pedal force is observed to still satisfy a good tracking effect of the BPCC, and the results verifies the effectiveness of the RTZRC algorithm.

In this study, three groups of braking experiments with different intensities are conducted, including two emergency braking experiments (I and II), and a normal braking experiment (III). Figure 15 presents the changes in pedal displacement, velocity, and force under different braking intensities. The variation curves of the pedal parameters are observed to be basically consistent between experiments I and II, which indicates good repeatability for the MRDBBPS prototype. In addition, the maximum values of three pedal parameters in an emergency braking are greater than these in normal braking.

Figure 16 shows changes in the braking torque of the MR brake and vehicle speed during braking. Similarly, the changing curves under emergency braking I are coincident with those under emergency braking II. The results indicate that the MRDBBPS prototype has a stable control effect on the AMRBTB under the same conditions. Moreover, the vehicle speed is observed to decline gradually under the action of the braking torque provided by the MR brake. For normal braking III, because of a small maximum pedal displacement, the braking torque is small, and the vehicle speed decreases gradually. The braking times are approximately 5.5 s and 6.8 s for the emergency and normal braking, respectively.

Figure 17 depicts the braking parameter variations of the AMRBTB while the brake pedal is pressed. As observed, the coil current, braking torque, and vehicle speed all change in accordance with the pedal displacement. For the first step on the brake pedal with a small displacement, the vehicle speed declines gradually. After the brake pedal is released and re-pressed with a larger displacement, the declining trend of the vehicle speed becomes faster. The results verify that the MRDBBPS prototype achieves an effective and favorable control of the AMRBTB.