Mathematical model under the action of ABS Single-wheel model under ABS In order to accurately reflect the braking characteristics of the vehicle under the action of ABS, the influencing factors including tire elasticity, suspension spring stiffness, and hydraulic shock absorber damping are established. A more complete one-wheeled vehicle model, as shown in the modeling principle.
In the vertical direction, due to the suspension and tire elasticity causing vehicle vibration, the dynamic equation has a vehicle body force balance equation mcZc-Bc(Zc-Zt)+Kc(Zc-Zt)+Fb=0(1) wheel force balance equation is mtZ
t-Bc(Zc-Zt)-Kc(Zc-Zt)+Kt(Zt-Z0)=0(2) Force balance equation between tire and ground Fz=(mc+mt)g+kt(Z0-Zt)-Fb(3) In the horizontal direction, due to the effect of braking torque and ground friction, the vehicle begins to decelerate. The dynamic equations include the vehicle body force balance equation mcv+Ff+Fa=0 (4) the wheel moment balance equation Jwω-FfRw+Tb=0 (5) ground and tires The friction between Ff = μ (s) Fz (6) Where mc is a quarter sprung mass; mt wheel mass; Kc suspension stiffness; Bc shock absorber damping; Kt ​​tire stiffness; Ff tires and road surface The longitudinal friction between the two; Fz tire normal force; Jw wheel inertia, Jw = mtrw2; Rw wheel effective radius, Rw = R0-Z0 + ZC; R0 nominal radius; Fa car horizontal wind resistance, Fa=CAfv2/4; Fb Additional load due to inertial force, Fb=2(mch/L)v; L-axis moment; h-center-of-center-height; Tb braking torque; μ(s) Friction coefficient between tire and road surface v body speed; CA air resistance coefficient; Af car's upwind area.
Tyre Model In the study of ABS, the tire model used has many forms. Among them, the model proposed by Dugoff is easy to calculate and has sufficient accuracy [9]. This model uses equation (7) to describe the friction between the tire and the road surface FfFf. =Css/(1-s), Css/(1-s)<μ(s)Fz/2Fzμ(s)-μ2(s)Fz(1-s)/(4Cs), Css/(1-s) > μ(s) Fz/2 when dry road surface μ(s) = μ0(1-Asvs) (8) μ(s) = μ0exp(-vsvc) on wet road surface; (9) where Cs tire longitudinal stiffness, μ0 The coefficient of friction during pure rolling, the coefficient of adhesion reduction of As, and the rms of vc and the height of the tire maintain a relatively constant characteristic speed.
Hydraulic system model In order to reduce the cost, the current ABS mostly uses high-speed on-off valves to control, limiting the application of continuous control and modern control methods. In this study, a direct acting electro-hydraulic proportional directional valve was used in place of the on-off solenoid valve to achieve continuous control of the braking process and solve the problem of large pressure fluctuations in the wheel cylinder. The principle of the hydraulic control system is shown.
Set the master cylinder and wheel cylinder pressure respectively to pm, ps, in the pressurized state, the brake fluid in the master cylinder flows through the proportional valve to the wheel cylinder, the flow into the wheel cylinder q1 = CdA (xv) 2Ïpm-pssign ( Pm-ps) (10) Under reduced pressure, the brake fluid in the wheel cylinder enters the low-pressure accumulator via the left position of the proportional valve. The flow rate is q2 == CdA(xv)2Ï(ps-pa)(11) Because the spring stiffness of the brake is very large, the speed of the piston of the wheel cylinder is very small. Omitting this part of the flow rate can simplify the pressure change rate of the rear wheel cylinder dpsdt=βeν(CdA(xv)2Ïpm-pssign(pm-ps)-CdA (xv)2Ï(ps-pa) (12) where the flow coefficient of the Cd valve; xv proportional valve spool displacement; A valve flow area; Ï oil density; βe fluid equivalent elastic modulus.
Basic Control Principles Principles and Structures of Controllers Because of the strong nonlinearity and uncertainty in the automotive braking process, modern model-based control theory methods are difficult to implement in ABS, while fuzzy control methods have independent objects. The characteristics of the mathematical model are applicable to the control of complex and non-linear systems. Therefore, this method uses this method to control the ABS. The total control loop includes three parts: the calculation of the optimal slip ratio, the recognition of the road surface status, and the fuzzy controller. Pavement condition recognition is achieved by vehicle body deceleration and wheel angular deceleration and their rate of change. The optimal slip rate occurs when the tire and road friction are the greatest, and is used as the reference input of the fuzzy controller. The fuzzy controller controls the slip rate to track the reference input so that the friction between the tire and the road is always near the maximum value. The control principle is as shown.
In order to make the output of the system have more ideal characteristics, based on the basic fuzzy controller, the reference model and the fuzzy inverse model method are adopted, and the auxiliary signal is used to improve the self-adaptive ability of the system to external environment and road surface change. The principle is shown as .
According to the reference model of [6], take dsm(t)dt=-10.0sm(t)+10.0s*(13) where s* is the input value of the whole system, ie the best slip rate; the sm(t) reference model Output.
The calculation of the best slip rate When the car is in the state of the best slip ratio, the maximum friction between the tire and the road, that is Ffss = s* = 0 (14) for the equation (7) derivative, into the dry The formula for the friction coefficient of the road surface (Eq. (8)) is simplified and the third power term of s is omitted. The formula for calculating the best slip ratio for dry road surface is s*μ0Asv (4CsFz+2μ0+μ0Asv).
(15) On the wet road surface, take the first two terms of the Taylor expansion of equation (9) to obtain μ(s)≈μ0(1-vcs)(16) comparison equation (8) and the above equation, and obtain As1vc ( 17) Therefore, Formula (15) is suitable for dry roads as well as wet roads. When wet roads, 1/vc is used instead of As in the formula. Detailed derivation can refer to [7].
Road surface recognition principle According to the literature [8], the condition of the road surface is judged by the deceleration of the vehicle body, the angular deceleration of the wheels and their rate of change, because on the dry road, the deceleration of the vehicle body is greater than the wet road surface, if the deceleration is measured. If the value is greater than the preset value, it can be determined that the current road surface is dry road surface, otherwise it is a wet road surface. When the vehicle is braking on a certain road surface, if the rate of change of the wheel angle deceleration and the vehicle body deceleration is greater than or less than the set value, it can be determined that the road surface condition has changed. The research uses the stateflow module based on finite state machine theory in Matlab software to achieve this function. Set road=0.2 for dry roads and road=0.1 for ice roads.
Analysis of ABS characteristics based on fuzzy control Based on the established mathematical model of the controlled object, the characteristics of ABS under the condition of measurable vehicle speed, using the fuzzy control toolbox of the fuzzy controller principle software adopted by Matlab, the mathematical model of the aforementioned fuzzy control system Convert to a computer simulation model. A simulation study of braking on a dry road with an initial speed of 22m/s was carried out. The results are shown. From the simulation results, it can be seen that the slip ratio of the wheel is maintained at an optimum value of 0.2 in the entire braking process, and the adhesion coefficient is also kept substantially near the maximum value, indicating that the wheel has always been in an ideal rotation state, ensuring The car has good steering maneuverability, stability and optimal braking effect when braking. In the same way, this method can achieve a very good braking effect even when the road surface mutates.
The characteristics of the ABS system using the reference speed The previous studies have shown that the ABS braking effect is ideal in situations where the vehicle speed can be accurately measured. However, so far, there is no practical speed detection method and sensor in the world. Therefore, in the ABS control, the estimated “reference speed†is used instead of the true speed of the vehicle. The literature [5] proposes a numerical analytical method for directly obtaining the reference vehicle speed of each wheel based on the wheel braking force. The calculation formula of the reference vehicle speed is v=-agμ(v)a-μ(v)h(18) when the wheel speed ω is already When knowing, the slip rate s becomes a function of the vehicle speed v, ie, μ(s) = μ(v). Equation (18) is a first-order differential equation for v. It can be solved numerically by using the fourth-order Runge-Kutta method. The result obtained is the reference speed in the braking process of the car. c. For the braking process, the actual vehicle speed and the reference vehicle speed estimated from equation (16), it can be seen that the calculated reference speed is slightly less than the actual vehicle speed.
Acceleration signal real-time correction reference vehicle speed ABS characteristics As the ABS system itself has uncertainty, the estimated vehicle speed is always biased and affects the braking effect of ABS. In order to solve this problem, a new acceleration sensor is used to measure the vehicle body acceleration. As a reference value of the acceleration, the acceleration obtained by the numerical solution method is used to approximate this reference value, and the method of correcting the reference speed and improving the estimation accuracy is performed.
Conclusion The fuzzy control method was used to simulate the characteristics of the anti-lock braking system of the car under the conditions of measurable and unmeasurable vehicle speed. The simulation results of the braking process of ABS on the dry road surface at the time of vehicle speed can be measured. The special results show that when the vehicle speed can be accurately measured, the effect of ABS will be very satisfactory. If the reference slip speed is used to calculate the slip ratio, the initial braking effect is better, but as the vehicle speed decreases, the slip rate will deviate from the optimal value, and the wheel locking process will still occur at low speeds. Using the newly proposed method of real-time correction of the reference vehicle speed using the acceleration signal, the slip ratio is always maintained at the optimum value during the entire braking process, the wheel is not locked, and the braking effect is very close to the vehicle speed. Ideally, in the absence of practical speed detection methods and sensors, the new method has the advantages of good effect and practical feasibility.
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