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Fuzzy Logic Anti-Lock Brake System for a Limited Range Coefficient of Friction SurfaceD.P. Madau, F. Yuan, L. I. Davis, Jr., and L. A. FeldkampResearch Laboratory, Ford Motor Company Suite 1100, Village Plaza, 23400 Michigan Avenue Dearborn, Michigan 48124 davis@A b s t~ u c t - The use of fuzzy logic has recently gained recognition as an approach for quickly developing effective controllers for higher-order, nonlinear time-variant systems. This paper describes the preliminary research and implementation of a fuzzy logic controller to co’ntrol wheel slip for an anti-lock brake system. The dynamics of braking systems are highly nonlinetlr and timevariant. Simulation was used to derive an initial rule base which was then tested on an experimental brake system. The rules were further refined by analysis of the data acquired from vehicle braking maneuvers on a surface with high coefficient of friction. The robustness of the fuz:zy logic slip regulator was further tested by varying operating conditions and external environmental variables.
I. INTRODUCTIONWhen braking force is applied to a rolling wheel, it will begin to slip; that is, the wheel circumferential velocity Vwhl will be less than the vehicle velocity h e h . Slip is defined as the difference between vehicle velocity and wheel circumferential velocity, normalized to vehicle velocity:
If sufficient braking force is applied, the wheel will“lock up,” that is, slide without turning at all. A locked wheel has no lateral stability and usually slides with much less friction than a wheel with X< 1.0. The relationship between slip, vehicle velocity, and the coefficimt of friction p is complicated and changes with different 13urfaces. Figure 1 is a plot of typical p-slip functions, both lateral and longitudinal. The lateral coefficient of friction is greatest at zero slip. Lateral friction provides lateral stability, the ability to steer and control the direction of the vehicle. The longitudinal coefficient of friction is zero at zero slip and typically exhibits a peak in p at some intermediateO-7803-0614-7/93$03.00Ql993EEE
value of slip. For most surfaces, as braking force is increased longitudinal p increases with slip until a point is reached where p decreases for increasing slip. If braking force is not quickly reduced at this point, the reduction in road force leads to a rapid increase in slip and eventual lockup. Anti-Lock Brake Systems (ABS) sense when this point has been reached and reduce braking force so that lockup is avoided. The two curves shown in Figure 1 for longitudinal coefficient of friction are typical of hard surfaces. It would appear that maintaining slip at the value of X which gives the peak value of p would be ideal. Unfortunately, the position of the peak varies for digerent surfaces and speeds (for example, the curve may not have an extremum at all until A= 1.0). Most control strategies define their performance goal as maintaining slip ne
ar a value of 0.2 throughout the braking trajectory. This represents a compromise between lateral stability, which is best at zero slip, and maximum deceleration, which usually peaks for some value of slip between 0.1 and 0.3. The goal of ABS control then becomes the regulation of slip to a known and desired level. In terms of practical ABS design, there are many complicating factors. For example, the amount of brake torque at the wheel is nonlinearly related to the temperature of the brake linings. The viscosity of the brake fluid is affected by the temperature of the brake fluid and affects the rate at which the brake pressure can be increased and decreased. Also, the anti-lock brake system must handle external disturbances such as variations in the adhesive force between the road and tire due to changes in road surfaces, loading, and steering, and variations in the frictional force due to irregularities in the road surface. External variables must be considered, including anomalies in the wear patterns of the braking components which may increase noise in the wheel velocity sensor, variations in the adhesion of the brake linings, and brake hysteresis. The control algorithm must accommodate variations in tire inflation pressure and tire wear patterns (as well as possible use of a mini-spare tire). Tolerances on the sensor wheel and tooth errors can significantly affect the accel-
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Figure 1: Coefficient of Friction mu,as a function of wheel slip. eration calculations. In summary, the internal ABS plant dynamics are difficult to model accurately and tend to be nonlinear, time-varying, and complicated by the inclusion of higher order terms. In this paper, we describe the development of a fuzzy logic controller for anti-lock braking. Fuzzy logic is being recognized as a useful tool in developing robust controllers for higher-order, nonlinear, time-varying systems. We begin with a description of the initial controller design via simulation and proceed to follow the design process through practical system considerations and development issues, test results, and adjustment of the control strategy. Although the development has not been carried through as far as a production system, the resulting controller shows great promise.
Figure 2: Initial ABS Rule Set. Input membership functions are shown on the right, with corresponding output functions on the left. conditions and then the dynamics were simulated according to what brake output the controller produced at each step. The model maintained separate state variables for the“actual” system and an estimate of the state variables for deriving sensor readings to be provided to the controller. Simulation time steps for the model were te
n times finer than the controller cycle times. At each controller cycle, the model made available to the controller three values: estimates of vehicle speed, and estimates of wheel speed and acceleration. These estimates included noise and integration error accumulated during the simulated trajectory. The resulting initial set of rules is pictured in Figure 2. The input variables used were slip A, as calculated from estimated vehicle and wheel speed, and estimated wheel acceleration a, h l . There was only one antecedent for each rule. The output 6u, was chosen to be change in braking force in anticipation of the use of hydraulic actuation. These rules resulted from simulating many braking trajectories and adjusting for minimum stopping distance. The rules and the control they represent was presented as a generic starting point, with important details such as scaling of the inputs and output to the parameters of the vehicle and braking system remaining to be established.
FUZZY LOQICABS CONTROLLER 11. INITIAL DEVELOPMENTAn initial fuzzy logic controller was developed using a simple simulation model for the braking process. By observing simulated trajectories we were able to propose, test, and evaluate rules corresponding to particular strategies. Once a reasonable set of rules had been established, their corresponding membership functions were adjusted to improve results. Evaluations were made on the basis of statistical comparisons of tests performed while varying road surface p, sensor noise, initial conditions, etc. The simulations of ABS control were performed using a simple, one-wheel, straight-line model as described in[l]. The system model was initialized to a standard set of884
111. PRACTICAL CONTROLLER ABS DEVELOPMENTThe inputs necessary to implement the rules in Figure 2 are, for each wheel, wheel velocity, wheel acceleration, and slip. The only available inputs to the ABS controller that are actually measured are the wheel velocities. All
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-Figure 4: ABS Hydraulic System The following five rules were used to drive the fuzzy logic ker ne1:1. If X is Pos. Small then 6u is Pos. Small 2. If crwhl is Neg. Large then 6u is Neg. Med. 3. If a w h 1 is Pos. Large then 6u is Pos. Large 4. If X is Pos. Large then 6u is Neg. Large 5 . If A is Pos. Medium then 6u is Neg. Small
Figure 3: Control Block Diagram other potential inputs are derived from these measurements. The wheel velocities are provided by a wheel speed sensor at each of the four wheels. A wheel speed sensor generates a signal whose frequency is directly proportional to the velocity of the wheel V w h l . The computed derivative of the wheel speed is used to estimate the angular acceleration of the wheel crwhl. The velocity of the vehicle%eh is necessary in calculating the individual wheel slip X according to Eq. 1. How
ever, the determination of the vehicle velocity can become a very uncertain task if all four wheels are slipping at once. Generally an estimate of% e h, called the reference velocity Vrej, is calculated using certain assumptions. Under normal (non-ABS) conditions, for each wheel is the average of the non-driven wheel velocities. As ABS operation begins fix a wheel, its Vre,starts at the prevailing average and thten is decelerated at a fixed rate. Any time the wheel’s velocity exceeds its V r e f, Vref is allowed to track the wheel velocity. The value of V r e j continues to be updated in this manner until the wheel leaves ABS mode. The amount of slip for each wheel is calculated using Eq. 1, but with V,,J instead ofKeh.
The fuzzy logic controller consists of three blocks (Figure 3): the preprocessing block, the fuzzy logic control block, and the postprocessing block. The preprocessing block handles the hardware interface and calculates the necessary variables for the fuzzy logic control block. The fuzzy logic control block fuzzifies the inputs and maps them to a rule base to determine the control output 6u.885
The fuzzy logic kernel utilizes standard max-min fuzzy inference. The center of area method was used to generate the control output 6u. The ABS hydraulic system components include an inlet valve and an outlet valve located at each wheel used to regulate the pressure at the caliper (Figure 4). For the simulation, there was just one output, change in brake force; for the real brake system, we must derive two output signal rates, one for each valve. In the build pressure state, the inlet valve is open and the outlet valve is closed, allowing for pressure to build at the caliper. The dump pressure state opens the outlet valve while closing the inlet valve, which allows for pressure reduction. The hold pressure constant state closes both valves, maintaining a constant pressure. The outlet valve orifice size is twice that of the inlet valve to accommodate the lower pressures at the outlet valve. Note that the inlet pressure at the build valve is unknown and varies as a function of the driver pedal effort. This leads t o some uncertainty as to the rate of increase of brake pressure. It is possible for the
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6uFigure 5: Brake Output Scaling: Build, hold, and dump rates outlet pressure to be estimated if,U can be estimated, under the assumption that the control algorithm is operating in the optimal braking slip range. We did not, however, include,U estimation in our strategy. To accommodate integer math used by the microcontroller, the output was scaled to a domain from 0 to 40, such that 20 is equivalent to no change in brake pressure (see Figure 5). When the outp
ut is negative (i.e., less then 20) the controller commands a specific dump pressure whose rate increases as the output approaches zero. A positive output specifies a build pressure rate whose rate increases as the output approaches 40. These rates are not interpolated linearly from the output but are derived from a lookup table in the postprocessing block. The postprocessing lookup table maps the output from the fuzzy logic control block 6u to a specific build or dump rate as indicated in Figure 5 . Each rate specifies an on time and a period for the valve. Normally, the adjustment of the table of these rates would be determined from bench testing of the various valve flow rate capabilities in conjunct4ionwith the hydraulic system. Instead, we developed the postprocessing block of the fuzzy logic ABS from in-vehicle testing. algorithm took far in excess of the 5 ms loop time. By reducing resolution in the defuzzification portion of the max-min inference and in-lining the code, we were eventually able to squeeze the algorithm time to within the necessary 5 ms constraint. In order to record the values of controller variables during vehicle braking tests, an internal serial data link was used to communicate to external data acquisition equipment. The link could read four data words located in memory at a maximum rate of 9600 baud within the 5 ms loop time. As a result of this time constraint, the data that could be collected largely consisted of the reference velocity, one of the individual wheel speeds, the wheel valve signal, and the output of the fuzzy controller signal621.
ISSUES IV. DEVELOPMENTThe implementation of the fuzzy logic algorithm in software is limited by constraints that are embedded in the base architecture design for the ABS system. The loop time is constant and fixed at 5 ms. During the loop time, the software must preprocess the inputs, determine the control action, drive the outputs for all four wheels, check for hardware fail-safe, and communicate to the serial data acquisition system. Initially, processing the fuzzy logic886
The performance of the initial fuzzy controller was far from optimal. The system did not attempt to take control of the wheel on the first cycle until it was deep into slip. In order to recover, the decrease in brake pressure necessary to reaccelerate the wheel resulted in an excessive loss of brake fluid. In a closed loop hydraulic configuration there exists the potential for depleting the total energy of the system more rapidly than the internal ABS pump motor can compensate. As a result, the inlet brake pressure may not be able to provide enough energy to allow the wheel to climb up the stable side of the 1.1 - A curve and reach the optimal braking force. This results in the cycling of the build and dump valves at a greater frequency at the beginning of the stop than later in the stopping maneuver. We observed this to be the case for the initial rule set provided from the simulation studies. Sinc
e most of the energy in the system was used in recovering the initial wheel’s velocity, the length of time spent in build pressure
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Figure 6: Fuzzy Logic ABS: High p Results state near the end of the stop is significantly greater than at the beginning. The total deceleration rate for the stop on a high p surface was 8.36 m/ s 2 (= 0.853 G). The theoretical optimal deceleration rate for a vehicle on high p is 9.8 m/ s 2 (= 1.0 G). Although we do not expect to achieve the theoretical result (for high p, zero degrees pitch, straight line stop), we expect to see deceleration rates greater than 8.82 m/ s 2 (0.9 G ) . Since dumping brake pressure quickly over a short time interval severely limits the ability of achieving optimal deceleration, the output scaling was modified to decrease the dumping of brake pressure. After this modification, it was still evident that the system did not try to control the initial slip of the wheel until it was deep within slip. On the other hand, the build and dump frequencies were similar during the entire stop, leading to the conclusion that the system energy was better preserved during the stop. Initial vehicle stability was still an issue due to the deep first cycle of the wheels. The next stage of development was to ad,just the trigger points that initiate ABS control t o enable the control algorithm to react more quickly t o a pending lockup situation. With this modification, wheel dip was better maintained within desired limits throughout the stop. Although this did not improve the deceleration rate very much (8.62 m/ s 2= 0.88 G), the stability of the vehicle was increased. Another consideration during the testing was the temperature of the brake linings. During repeated stops, the temperature of the brake linings will increase, thus affecting the ability for the brake to displace any additional energy and thereby reducing braking potential. Of concern is that at a certain point the brake pads, would begin to burnish which would further decrease the maximum amount of brake torque available. In observing the performance of the control from initial stop to several stops887
Figure 7: Conventional ABS: High p Results later there was evidence that the initial brake stop displayed a greater force on the wheels than the final stops. On the other hand, we noted that the control of the hot brakes was actually better: with less force able to be applied t o the wheel, stabilizing the wheel required less effort. This was an indication that the algorithm tended to allow pressure to build too fast. With hot brakes, the algorithm regulated the slip of the wheel to a desired level without going deep into the unstable region. Finally, after slight modification to the postprocessing valve
timing, the following results were achieved on a high p surface. The average deceleration rate for the entire stop was 8.91 m/ s 2 (= 0.91 G). The wheel slip cycled around 0.9 of VTej. As the vehicle velocity decreased the wheel was allowed to cycle deeper (around 0.8). The algorithm appeared to gain better control of the wheel as time increased. The wheel oscillations were kept to a minimum near the end of the stop, which increased the stability of the vehicle.
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The main design intent for this first set of experiments was preventing the wheels from locking and decreasing the stopping distance. The results as recorded with an external data acquisition system were very promising, as shown in Figure 6 . Results for a conventional ABS are shown in Figure 7. On the high p surface the performance of the fuzzy ABS system was quite comparable to that of the production ABS system. The wheels spend most of the time around the reference velocity. In a straight-line stop, the fuzzy system decelerated the vehicle at a slightly higher rate than that of a production system. A characteristic of this fuzzy logic approach to ABS is that no attempt is made to estimate the overall coefficient of friction. None of the rules directly utilize the
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Figure 8: Fuzzy Logic ABS: Low p Results value for p; therefore it was interesting to test the robustness of the fuzzy logic controller with a lower coefficient of friction test surface. The initial results for a low p surface are shown in Figure 8. Although the controller was surprisingly functional, the performance naturally lacked features that further tuning undoubtedly would provide. The control algorithm cycled the wheels over a greater slip range than for the high p case. The wheels spent too much time deep in slip which limited the steering ability of the vehicle. Further analysis of the controller output indicated that the difficulty of slip regulation was not necessarily the result of the fuzzy control but rather the limited resolution of the postprocessing block. Since the build valve on time was limited, the controller could not request a slow enough pressure build for the valve to keep the wheel from locking while maximizing the available frictional forces and maintaining wheel stability. Further work with a better valve resolution is warranted.
Figure 9: Final Rule Set for ABS performance was found to be surprisingly functional. The controller cycled the wheels over a wider slip range and consequently the steering ability was not very good; but it was not clear whether this was due to any deficiency in the strategy or merely a result of a limitation in the valve response. Further simulation suggests that low p performance can be improved without degrading performance at h
igh p by adding additional rules. One observation is clear from our experience: fuzzy logic supplies a framework for quick development of prototype controllers without detailed understanding of the internal plant dynamics. This is apparent both in the process of refining the controller and in the initial development of a rule set. Comparison of the final rule set (Figure 9) with the initial rule set shows how close one may come to a final controller through simulation. F'urther work will include addition of rules to handle various braking maneuvers, road surfaces, and refinement of the valve resolution time.
VI. DISCUSSIONThe intent of this work was to establish an initial practical understanding of the ability of fuzzy logic to control a system with nonlinear dynamics and variable parameters, specifically, an anti-lock brake system. Simulation was used to establish an initial rule base for Use in the fuzzy logic controller. With minor tuning of the rules and postprocessing block for the high p surface, results achieved were comparable to that of a production ABS controller. When tested on a low p surface, the untuned888
REFERENCES[l] L. I. Davis, Jr., G. V. Puskorius, F. Yuan and L.A. Feldkamp (1991). Neural Network Modeling and Control of an Anti-Lock Brake System. Proceedings of the Intelligent Vehicle '92 Conference, June 29 July 1, 1992, Ypsilanti, Michigan.
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