Abstract
Road friction information is very important for vehicle active braking control systems such as ABS, ASR, or ESP. It is not easy to estimate the tire/road friction forces and coefficient accurately because of the nonlinear system, parameters uncertainties, and signal noises. In this paper, a robust and effective tire/road friction estimation algorithm for ABS is proposed, and its performance is further discussed by simulation and experiment. The tire forces were observed by the discrete Kalman filter, and the road friction coefficient was estimated by the recursive least square method consequently. Then, the proposed algorithm was analysed and verified by simulation and road test. A sliding mode based ABS with smooth wheel slip ratio control and a threshold based ABS by pulse pressure control with significant fluctuations were used for the simulation. Finally, road tests were carried out in both winter and summer by the car equipped with the same threshold based ABS, and the algorithm was evaluated on different road surfaces. The results show that the proposed algorithm can identify the variation of road conditions with considerable accuracy and response speed.
1. Introduction
The adhesion information between road and tire is very important for vehicle active safety control systems such as ABS (antilock braking system), ASR (acceleration slip regulation), and ESP (electronic stability program). Therefore, plenty of researches on tire-road friction coefficient estimation and identification have been and are being focused. For instance, it is well known that ABS is an active safety control system by utilizing the potential road-tire adhesion to improve vehicle deceleration, handling, and stability during braking [1]. In order to utilize the available tire forces on variable roads as much as possible, the identification of peak tire-road friction coefficient and its optimal slip ratio is a critical task for guaranteeing the performance of ABS [2].
Unlike normal vehicle dynamic information, it is hard to measure tire-road friction directly. A cutting-edge approach is the intelligent tire which monitors the tire adhesion information in real time by the sensors such as ultrasonic approach [3], optical position detection [4], and piezoelectric film sensor [5] mounted inside of tires. However, these researches are still in their early stage and unsuitable for wide application due to the newly equipped sensors and the increased cost. Therefore, more researches are focused on the methods and algorithms for tire-road adhesion information estimation. As the friction phenomenon between a tire and road is affected complexly by different factors and it has significant nonlinear and dynamic properties such as viscousness and hysteresis, it is not easy to analyze the interaction between a pneumatic tire and road. Various kinds of dynamic model based methods are utilized to solve this problem [6]. For these methods, available sensors on cars are used optimally to estimate tire/road information based on the wheel dynamic or vehicle dynamic state variables. The wheel dynamic based method described in [7], the vehicle dynamic based method in [8], and the self-aligning torque based method in [9] are three typical kinds of these solutions.
The vehicle dynamic based method can be used to estimate road/tire friction forces by observers based on vehicle dynamic information. In practice, the mathematical models of vehicle system are simplified for analysis and calculation; the differences between actual vehicles and these models might lead to errors of the estimation results [10]. The transient characteristics and ever-changing parameters of the vehicle in motion might be another reason that affects the estimation results [11]. At the same time, as signals from the sensors are vulnerable to the noise pollution, there will be deviations on the observed results [12]. Therefore, the observer must have good robustness for tolerating differences between models and vehicles, variations of model parameters, and the signal errors. For this reason, Kalman filter is widely used for states observer design because of its robustness performance. It has been proven to be effective in extracting the useful information from the noised signals [13–15].
With regard to the road friction coefficient, the observer method is not a good choice. The coefficient is not the state variable of a vehicle or a wheel. The nonlinear tire model should be written into Jacobian matrix expressions, and the extended Kalman filter is necessary. Using this method, the dimension of the state variable will be very large which results in heavy computation task. Thus, this way is seldom applied. The road friction coefficient is often treated as a parameter of different kinds of tire models and detected by some kinds of parameter identification methods [16]. The relatively simple theoretical tire models such as tire brush model and dugoff model are suitable for this application [17, 18]. However, the road friction coefficient is determined by the interaction of tire and the road, and the tire/road friction coefficient changes along with the variations of tire parameters, such as inflation pressure, tread material, and wear [19]. At the same time, vehicle dynamic such as load transfer caused by the motion of vehicle can also affect the tire/road adhesion coefficient, combined with the influence of dynamic characteristics of pneumatic tire, such as relaxation and hysteresis. Thus, it is always difficult to achieve a precise tire/road friction coefficient estimation result even when the accurate tire forces are obtained [20]. In order to obtain better estimation results of tire-road friction coefficient, identification methods with good robustness are also necessary to reduce the interference of disturbance signals and converge to the actual value quickly.
Recursive unbiased estimation algorithms can perform a good unbiased estimation result [21], and the recursive least square (RLS) method is one of them. Not only is the RLS method convenient for computer calculation and storage but it also has a fast response speed and strong antijamming capability by adding the forgetting factor [22, 23]. It also could deal with the nonlinear model such as tire model after linearization processing. Thus, based on the previous observation of tire forces by Kalman filter, it is a good approach by combining the RLS method to estimate road friction coefficient.
The characteristics of the practical actuators are another problem in the road friction estimation for active brake control. Nowadays, almost all of the hydraulic active brake control systems are equipped with high-speed switching valves (HSV). ABS is a typical example that applies the pulse control mode by switching the HSV valves. Thus, considerable fluctuation caused by ABS control will be observed on wheels dynamic state variables such as wheel slip ratio and brake pressure, which are important for friction estimation. In this case, the negative influences on the estimation from the nonlinear characteristics of the vehicles and tires will be amplified. Therefore, the effects on the estimation performance caused by the fluctuation from actuators should be also considered, which are not been fully discussed yet.
In this paper, a tire-road friction estimation algorithm for ABS was proposed with the combination of a Kalman filter and a linearized RLS estimator. The tire forces are observed by the discrete Kalman filter using the available vehicle signals based on a planar vehicle model, and the tire-road friction coefficient, which is a parameter of tire brush model, is identified based on linearized RLS method. The proposed estimation algorithm was evaluated by both CarSim and Matlab/Simulink cosimulation and road tests. First, it was simulated using a sliding-mode-control (SMC) based ABS control algorithm that controlled wheel speeds and brake pressures more smoothly with almost no noise being shown. Then, a typical threshold based ABS algorithm was used and the performance estimation was evaluated by fluctuant input signals. Finally, road tests were carried out in both winter and summer by the car equipped with the same threshold based ABS, and the performance of the algorithm was discussed on different road surfaces.
2. The Tire/Road Friction Estimation Algorithm
The sketch of the proposed system is shown in Figure 1. The system consists of a calculator, a Kalman filter based observer, and a recursive least square (RLS) based estimator. First, the tire longitudinal and lateral forces are estimated by the observer. Then, the road adhesion coefficient is estimated by these forces together with the vertical wheel loads, the slip ratios, and the side slip angles calculated by the calculator. All of the descriptions of the arguments in this paper can be referred to in Notations. The signals of individual wheel speed and brake torque or brake pressure , the yaw rate , the longitudinal and lateral accelerations and , and velocities and of target vehicle are necessary for the estimation algorithm. Based on these signals, the wheel vertical load , slip angle , slip ratio , longitudinal and lateral forces and , and road adhesion coefficient can be estimated by the algorithm proposed in this paper.
2.1. The Tire Force Observation Based on Kalman Filter
The schematic diagram of the vehicle model used for the tire forces estimation is shown in Figure 2, whose aerodynamic resistance and tire rolling resistance are neglected and only tire adhesion forces are considered. The dynamic analysis of a single wheel is shown in Figure 3. In these models, the longitudinal forces are positive for driving, whose directions are the same as shown in the figures, and negative for braking otherwise.
The dynamic equations of the planar vehicle model shown in Figure 2 are where is the longitudinal force of each wheel, is the lateral force of each wheel (where the , , , and represent the front left, front right, rear left, and rear right wheel, resp., hereinafter inclusive), and and are the lateral forces of the front and rear axle, respectively. is the steering angle of the front wheels, is the mass of the vehicle, and are the longitudinal and lateral accelerations of the vehicle, respectively, is the yaw rate of the vehicle, is the moment of inertia of the vehicle, and are the distances from the center of mass of the vehicle to the front axle and rear axle, respectively, and is the wheel base.
The moment equilibrium equation for the single wheel model shown in Figure 3 is is the moment of inertia of the wheel, is the rotation speed of the wheel, is the braking torque, and is the effective rolling radius. Furthermore, in Figure 3, is the vertical force of each wheel. is the vertical load from the mass and is the longitudinal force acts on the wheel from the axle.
Based on (1) to (3), the following states space equations can be achieved:
The system state is an 11-dimensional vector , and the measurement variable is a 7-dimensional vector . The system process noise is an 11-dimensional zero-mean white noise vector, the measurement noise is a 7-dimensional zero-mean white noise vector, and and are mutually unrelated.
The discrete Kalman filter method based on linear discrete system is widely used in engineering because of its advantages in computer aid calculation and the less requirements of data storage. The discretization of the continuous vehicle system is necessary for the discrete Kalman filter approach. In this paper, the zero-order hold (ZOH) method is utilized to discretize the continuous vehicle model (4), and the discrete system state space equation is
Based on (8), the discrete Kalman filter estimation process is as follows. Prediction: State estimation: Filter gain matrix calculation: Step prediction error variance matrix calculation: The estimated error variance matrix calculation: where the matrices and represent the process noise and measurement noise covariance matrix, respectively. As long as the initial values and are given, according to the measured values of the th step, the state estimation of the th step can be recursively calculated, and the corresponding tire force included in could be achieved.
2.2. Information from the Calculator
The “practical” slip ratios , the side slip angles , and the vertical tire loads are calculated by the calculator which is shown in Figure 1: where is the gravitational acceleration and is the height of center of mass.
2.3. The Combined Longitudinal and Lateral Tire Brush Model
Plenty of researches have been carried on exploring the dynamic phenomenon and transient properties of the tire-road forces, and various tire friction models, including the theoretical and empirical models, were proposed [24]. The empirical models could describe tire characteristic precisely, but their expressions are complicated and difficult for parameters fitness, while the theoretical models have simpler expression with less parameters and are convenient for parameters fitness in real time. Therefore, the theoretical tire models are widely used in road identification generally. The tire brush model is one of theoretical tire models and it is widely used for estimation or control purposes because of its simplicity and qualitative correspondence to the experimental tire behaviors such as the friction ellipse influence and tire nonlinear characteristics, for example, tire force saturation [7].
The expression of tire brush model for the braking action is [25] where , are the longitudinal and lateral stiffness of tire, respectively, and is the tire-road friction coefficient.
Generally, the magic formula is considered the accurate model for road friction illustration. Figure 4 shows the comparison of the longitudinal tire forces calculated by the tire brush model and the magic formula model at a tire vertical load of 4000 N. It can be seen that the result of tire brush model is close to that of the magic formula model within the range of zero to optimal wheel slip ratio, while its precision decreases at large wheel slip ratio comparing with magic formula. As the tire brush model is precise enough around the optimal wheel slip ratio, which is the ABS control target, it could meet the requirements of road identification as long as the wheel slip ratio could be controlled in the optimal range by ABS.
The parameters , , and in tire brush model are needed to be identified. According to [2], the values of and which depend on tire properties such as tire size, tread width, tread stiffness, inflation pressure, and load tend to be static for a short period; therefore, they could be assigned as the fixed value in a short time. In this paper, the value of tire-road friction coefficient is estimated by the usage of the RLS method, which is described in the following section.
2.4. Tire-Road Friction Coefficient Estimation
Recursive least squares (RLS) method is an iterative algorithm which could estimate the parameters recursively by minimizing the sum of the squares of the difference between observed data and computed data [26]. However, RLS method is only available for linear systems. As the tire brush model is not only a nonlinear function but also a segmented function, it is hard to transform the tire brush model to a linear representation directly. Therefore, in order to utilize the RLS to estimate tire model parameters, an approximate linearization method was adopted.
First, tire brush model in nonlinear form could be written as follows: Consider that is the observed values of tire forces estimated by Kalman filter, is the expression of tire brush model, is tire brush model parameter vector, and is the observed noise.
can be written approximately as follows: where
Define a variable :
Bring the expression of into (20); can be written approximately as follows:
After approximately linearizing the nonlinear tire brush model, the RLS method is applicable for the tire-road friction coefficient estimation. Equation (21) can be used as the linear observation model and the cost function is as follows: where is a forgetting factor. The least-squares (LS) solution is classically obtained by zeroing the gradient of the cost function , which gives the LS estimation:
The parameter vector can be updated for each new observation and thus be estimated online. For this, the error covariance matrix is computed recursively by using the Sherman-Morrison formula [23]:
By putting (24) and (25) into (23) and setting the initial and , the RLS algorithm is obtained:
When the wheel slip ratio is very small, the algorithm might result in fluctuant and unexpected estimation [27]. In order to avoid incorrect estimation, a slip ratio threshold was set, and the estimation results are only updated when the slip ratio is larger than and held as the latest valid estimation value at a slip ratio smaller than . Consider According to [25], the relationship between “calculative” slip ratio and “practical” slip ratio for braking maneuver is as follows:
The performance of the algorithm is sensitive to the preset threshold . If is too small, there might be fluctuant and unexpected estimation when the actual wheel slip ratio is pretty low, while, if is set too large, low sensitivity might occur and the estimated will not be updated in time. As the slip ratio for ABS trigger is around 6% in the ABS algorithm used for the verification, with integrated consideration of reliability and sensitivity, threshold of 5% was chosen in this paper:
3. Verification by Simulink and CarSim Cosimulation
The algorithm was validated by the Matlab/Simulink and CarSim cosimulation platform, and a sedan vehicle model was chosen. The road was set as a -jump road whose friction coefficient is shown in Table 1. The vehicle was braked at an initial speed of 120 km/h, and the algorithm runs at a loop time of 0.001 s, which acted on both the discrete Kalman filtering and RLS estimator. The parameters used in this paper can be referred to in Table 2.
3.1. Verification by the SMC Based ABS Algorithm
The sliding mode control (SMC) is suitable for the systems with uncertainty and disturbance because of its advantages in quick response, insensitivity to parameter change or disturbance, no need of online system identification, and usability of physical implement [28]. Therefore, it is often used for ABS and realizing stable slip ratio control in simulation.
Take the error between the “calculative” slip ratio and the optimal slip ratio and its derivative as the state variables:
The switching function is set as follows: where is a constant, which can be referred to in Table 2.
Adopting the constant speed reaching law of SMC, where and are positive real constants.
With the consideration of the hysteresis of hydraulic brake system, the brake system can be modelled as follows: where is the brake torque applied on the target wheel, is the output from the SMC ABS controller, and is a time constant.
By substituting (3) into (33), we get
Assuming , and putting (34) into (32), we have the output of SMC controller: where is the speed of center of the wheel. By adjusting the braking torque on slipping wheels, the wheel slip ratio could be limit in a suitable range.
During the evaluation, the influence by the sensors noises should be considered for real application. Therefore, sensor noises were treated as white noise and added onto the smooth simulated measurements. Their variances were analyzed based on the experimental data and listed in Table 3. These data are derived from raw signals of sensors used in our test system, which will be introduced in Section 4. Normally, these signals should be prefiltered before being used by controller such as ABS or ESP, and the noises will be reduced significantly. Therefore, if the proposed estimation algorithm performs well on these noises, it will be qualified for the practical usage. Furthermore, it should be noticed that the magnitude of noise of raw wheel speed changes with the value of wheel speed itself: the larger the speed, the higher the noise power. Therefore, a segmented variance of white noise was adopted in this paper.
The simulation results for the SMC based ABS control are shown in Figures 5 to 8. Due to the limited space in this paper, only the results curves of the left front wheel, which are typical in all of the four wheels, are shown and analyzed.
The velocity and wheel pressure without noises are shown in Figure 5. It should be noticed that using the SMC based ABS control, the wheel slip ratio was controlled at the target value accurately, and both of the wheel speed and the brake pressure are smooth and stable. Therefore, it is possible to analyze the estimation performance by means of the step response. Then, the white noises were added onto , , , , and , and the velocity and wheel pressure with noises are shown in Figure 6.
The curves of longitudinal tire forces are shown in Figure 7. The actual curve shows the longitudinal force from the CarSim vehicle model, the estimated (smooth sig) and the estimated (noised sig) show the longitudinal forces estimated by Kalman filter based observer from signals without noises and with noises. It can be seen that the observer showed well-qualified estimation performance. When the road friction coefficient change happened, the estimated tire force based on both smooth signals and noised signals followed the actual value fastly and the lag of the rising time was only about 0.12 s. The errors between the actual value and the observation values were not more than 50 N when the calculations reached the steady state. Thus, it is concluded that the algorithm can respond to the change of road surface conditions effectively, and asymptotically unbiased estimation of the tire forces can be achieved. The Kalman filter also performed well on noised signals, although the estimated based on noised signals wobbled a little bit more than estimated by smooth signals.
The friction coefficients from CarSim vehicle model and RLS based estimator are shown in Figure 8. The estimated friction coefficient based on smooth signals shows satisfied performance with fast response and good accuracy. The rising time of estimated value had a lag of about 0.14 s. As its input is generated by the Kalman filter based observer, the lag of the observer should be considered and the lag of the RLS estimator is pretty small. At steady state, the error between the actual and the estimated value is not larger than 5%, which is precise enough. It is also seen from the estimated curve based on noised signals that the transient performances became a bit worse. More fluctuation, larger steady state error, and extended settling time can be seen. It is concluded that the noise of estimated based on noised signals further affected the RLS estimation negatively. However, the result is still good enough at both response speed and accuracy.
3.2. Verification by the Threshold Based ABS Algorithm
The threshold based algorithm is widely used in practical ABS system. Although the products from various manufactures are different from each other, their principles are the same.(1)Wheel slip ratio thresholds and acceleration thresholds are set as the control targets.(2)The controlled wheel speed will be fluctuating around the target value, and obvious and regular control cycles can be seen.(3)In the first control cycle, large pressure dump will be set to ensure that the wheel pressure decreased to a proper level as soon as possible. In following cycles, there will be pulse pressure increments and decrements.(4)Road surface changes are also identified by preset wheel slip ratio thresholds and acceleration thresholds. Generally, only simple classification of high and low road is provided.
The ABS control cycles of both wheel speed and pressure can be seen clearly in Figure 9. It is seen that when braking on low road, there are 3 or 4 cycles in one second and the amplitudes and periods of these cycles are regular, while, on roads with high friction coefficient, the cycles are less regular. The curves of longitudinal tire forces for the threshold based ABS are shown in Figure 10. It is obvious that the also exhibited “regular cycles” along with the fluctuations of wheel speeds. In spite of this, the Kalman filter based observer also showed quite precise estimation, which provided a good basis for the identification of the road friction coefficient.
The friction coefficient estimation results are shown in Figure 11. It is clear that the errors are more significant comparing with those in Figure 8, which means the fluctuations of ABS control did affect negatively the estimation performance.
The statistic data of the estimation is shown in Table 4. The lag time of the first step is not considered because the estimated value was not updated in the beginning and kept as the initial value 0.42 until the slip ratio exceeded 5%. In the following estimation, the values also kept constant at any time the slip ratio was smaller than 5%.
The lag time in the case of mid- to low--jump was only 0.136 s which is almost the same as that in the SMC case. However, before the threshold based ABS figures out the car driving from a slippery surface to a rough surface, it will apply small pressure increments, and the slip ratio will be tiny, which can be seen in Figure 9. Thus, the increase in estimated value in the case of low-- to high--jump will be slow, whose lag time was 0.372 s in this simulation.
When the estimation values were steady, their means and standard deviations were calculated. The maximum steady state error occurred on the mid- road, which was about 11.42%, and the minimum one occurred on the low- road, which was only 2.4%. Furthermore, all the standard deviations were small enough. It is concluded that the algorithm is accurate even in cases of fluctuations.
4. Verification by Road Tests
The road tests were carried out by the car equipped with the same threshold based ABS. The parameters of the car are listed in Table 2. As shown in Figures 12 and 13, the testing system consists of an ABS, a VBOX system, a dSPACE Autobox system, and a series of sensors. The VBOX captures the brake trigger signal and sends out vehicle speed and acceleration captured by its GPS. The wheel speeds and -jump flag can be achieved from the ABS ECU internal signals. Both of them are configured as nodes on CAN bus and send their information to the Autobox. The Autobox also captures the signals from lateral accelerometer, yaw rate sensor, and the pressure sensors on wheel cylinders. Finally, all of the signals are transferred and saved in a notepad computer. It can be seen in Figure 13 that all of the necessary signals for the estimation, except for the lateral velocity, were captured directly, while the lateral velocity was calculated by the following equation:
Furthermore, the -jump flag is not necessary for the road identification. It is only used as the flags of road changes.
Differing from simulation, the sampling time of the variable devices is different in road test. The lateral acceleration, the yaw rate, and the brake pressures were captured by AutoBox directly at sampling time of 1 ms, the ABS ECU control ran at a loop time of 5 ms, and the sampling time of CAN signals from VBOX was 10 ms. On data processing, linear interpolations were used to achieve 1 ms interval samples for the signals of ABS ECU and VBOX system. By this way, the algorithm also runs at a sampling time of 1 ms.
The tests were carried out in both summer and winter. The summer tests were implied on a road test ground located at the south of China. The car was equipped with all-season tires. The high- road is dry concrete pavement, whose adhesion coefficient is about 0.8 to 0.9, and the low- road is wet basalt tiles road, whose adhesion coefficient is smaller than 0.2. The test road is shown in Figure 14.
The winter tests were implied on a lake located at the north-east of China. The snow tires were equipped on the car. The high- road is dry concrete pavement and the low- road is the ice surface, which is shown in Figure 15. As the concrete pavement is not as rough as the one of summer test ground, its friction coefficient is about 0.7 generally. And the friction coefficient of the icy surface is about 0.2.
(1) The Braking Test from High- to Low- Road. According to the ABS test regulation we used, the car was tested at an initial speed of 80 km/h, and the length of high- road is 5 m. The results of summer and winter tests are shown in Figures 16 and 17, respectively. It is seen that the curves of estimated friction coefficients are quite similar to those in simulation. When braking on the low- roads, the estimation values are very steady. The friction coefficient is about 0.1 for the tiles road and about 0.2 for the icy surface.
However, there are fluctuations for the estimation results on the concrete pavement in Figure 17, and its detailed information is shown in Figure 18. It is seen that the estimated value varied from 0.35 to 0.83. Its valley occurred at 0.522 s which followed the maximum slip ratio of 23%. Furthermore, the absolute value of the car deceleration kept on increasing and reached the peak of 0.78 g at 0.798 s, while the peak of the estimated appeared at 0.794 s. As the brake efficiency cannot reach 100%, it is reasonable to achieve a peak deceleration of 0.78 g on a surface with the peak adhesion coefficient of 0.83. The precision of the estimation is acceptable, but it took the algorithm about 0.25 s to reach the correct level. The fluctuation might be caused by the relaxation characteristics of the snow tires. Furthermore, it was discovered that some snow was taken by the tires from the compacted snowfield in front of the pavement during the test, which might also affect the results.
(2) The Braking Test from Low- to High- Road. In these cases, the car was braked at an initial speed of 60 km/h. The length of the low- road is 10 m in summer test and 2 m in winter test. The results of summer and winter tests are shown in Figures 19 and 20, respectively. Again, the estimation showed perfect performance on the low- roads. And larger lags were shown when the -jump happened, which is similar to simulation. Furthermore, the estimated value on concrete pavement in winter is located between 0.45 and 0.7. Considering the average deceleration of 0.49 g of the car, the estimation is acceptable.
5. Conclusion
In this paper, a road friction estimation algorithm based on discrete Kalman filter and RLS method is proposed. The Kalman filter is used to observe the longitudinal tire forces and the RLS is used to estimate the road friction coefficient. The algorithm was verified by simulation as well as road test. A SMC based ABS and a threshold based ABS were used for the verification. The SMC based ABS can control the wheel slip ratio at the target value smoothly, and the threshold based one generally shows significant fluctuations in pulse pressure control.
Simulations for the SMC based and threshold based ABS showed that the Kalman filter based observer can detect the longitudinal tire forces accurately and quickly, even if there are noises of sensor signals or wheel slip ratio and pressure fluctuations caused by braking control.
As the wheel speed is quite smooth in the case of SMC based ABS control, the algorithm shows perfect performance on road friction coefficient estimation. While the fluctuations caused by the threshold based ABS control will affect negatively its precision as well as response speed, its performance is still satisfactory.
The experiments were carried out in both winter and summer. When braking on more slippery roads, the wheel speed control cycles will be more regular in both amplitude and period. Thus, the estimation performance is better on roads with lower friction coefficient, and estimation errors are larger on roads with higher friction coefficient. Furthermore, when low- to high--jump happens, there will be larger lags than those of high-- to low--jump cases because of the characteristics of the tested ABS algorithm. Still and all, the performance of the proposed algorithm is satisfactory.
In conclusion, the Kalman filter based observer is robust enough and affected slightly by the fluctuations of wheel dynamics, while the RLS based estimator is more likely affected by the pulse pressure control. The smoother and more regular the wheel controlled by the ABS is, the better the estimation performance is.
Notations
| : | Rotation speed of wheel, , , , represent the front left, front right, rear left, and rear right wheel, respectively. |
| : | Longitudinal speed of vehicle |
| : | Lateral speed of vehicle |
| : | Longitudinal acceleration of vehicle |
| : | Lateral acceleration of vehicle |
| : | Yaw rate of vehicle |
| : | Brake moment of wheel |
| : | Brake wheel cylinder pressure |
| : | Tire longitudinal force |
| : | Tire lateral force |
| : | Tire vertical force |
| : | Lateral forces of the front axle |
| : | Lateral forces of the rear axle |
| : | Steer angle of the front wheels |
| : | Mass of the target vehicle |
| : | Moment of inertia of the vehicle |
| : | Distances from the center of mass of the vehicle to the front axle |
| : | Distances from the center of mass of the vehicle to the rear axle |
| : | Wheel base |
| : | Moment of inertia of the wheel |
| : | Tire effective rolling radius |
| : | Wheel vertical load from the mass |
| : | Longitudinal force acts on the wheel from the axle |
| : | System state vector |
| : | System measurement variable |
| : | System process noise |
| : | System measurement noise |
| : | Process noise covariance matrix |
| : | Measurement noise covariance matrix |
| : | “Practical” slip ratio |
| : | Side slip angle |
| : | Gravitational acceleration |
| : | Height of center of mass |
| : | Longitudinal stiffness of tire |
| : | Lateral stiffness of tire |
| : | Tire-road friction coefficient |
| : | Observed values of tire forces estimated by Kalman filter |
| : | Tire brush model parameter vector |
| : | Expression of tire brush model |
| : | Observed noise |
| : | “Calculative” slip ratio |
| : | Optimal slip ratio |
| : | Difference between and |
| : | Switching function |
| : | Output from the SMC ABS controller |
| : | Speed of center of wheel. |
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is partially supported by National Natural Science Foundation (51105169, 51175215, and 51205156), Jilin Province Science and Technology Development Plan Projects (201101028 and 20140204010GX), Science Foundation for Chinese Postdoctoral (2011M500053 and 2012T50292), and Fundamental Research Funds of Jilin University.