ABS Prof. R.G. Longoria Spring 2002 v. 1
Anti-lock Braking Systems These systems monitor operating conditions and modify the applied braking torque by modulating the brake pressure. The systems try to keep tire operating within a desired range of skid, and by preventing wheel lock-up during braking they can help retain steerability and stability Anti-lock braking systems are closed loop control systems within the braking system.
Desired Range of Slip or Skid Bosch (1999) Trying to control slip in a desirable range is complicated by changing road conditions. Wong (1993)
Background Concept dates back to early 1900s, and first patent went to Bosch in 1936. Concept is now well established. Figures from Gillespie (1992)
Some ABS Requirements Adapted from Bosch (1999) Maintain steering response at all times, regardless of road conditions Adapt to and exploit available friction to maximum effect, but put emphasis on stability and steering rather than stopping distance, independent of how driver applies pedal force Effective over large speed range (walking speed) Control yaw effect in split-µ conditions Self-diagnostics
ABS Closed-Loop Concept Controlled Variables: wheel-speed and data measured from wheel (3) Manipulated Variable: brake pressure (1-3) Disturbance: roadsurface conditions, brake condition, vehicle load and tire characteristics Bosch (1999) Controller: wheelspeed sensors and ABS unit (4) Reference input: pressure applied to brake pedal (2) Controlled system: vehicle with wheel brakes, wheels and friction between tires and road surface
Ideal Braking Concept Ref. Bosch Idealized concept: nondriven wheel, 1/4 mass of vehicle, stable and unstable regions. Applied braking torque builds up linearly over time. Road surface torque lags slightly, and reaches maximum (saturates). Note that the torque differential can provide a measure of wheel acceleration. The opposing response is a good indicator of conditions at tire. 200 ms
Control Issues Wheel-speed sensors must be used to find wheel peripheral deceleration/acceleration brake slip reference speed and vehicle deceleration It is not practical to use wheel acceleration or deceleration or the slip as the controlled variable. How can this information be used?
Estimation of Reference Speed Since vehicle speed can not be measured directly, the ECU must estimate an appropriate value. The Bosch ABS system, for example, uses information from diagonal wheels, and bases an estimate on information obtained during non-abs usage. Under moderate braking, the ECU will estimate reference speed based on the diagonal wheel that is turning the fastest. During panic stops, a ramp-shaped extrapolation of the speed collected at the start of the cycle is used to calculate the reference speed.
Some Example Ways for Predicting Lock-up Wong (1993) Anti-locking may be initiated when the product of angular acceleration of the tire and its rolling radius exceeds a predetermined value (e.g., 1 to 1.6 g). Passenger car ABS may have a track-and-hold circuit that stores any wheel acceleration values of 1.6 g and higher. If the measured angular speed drops 5% during a predetermined time (e.g., 140 ms), and if the vehicle acceleration as measured by an accelerometer is not higher than, say, 0.5 g, then the tire is considered to be at the point of locking, and the brake is released when angular speed decreases by 15% of stored value. If the vehicle deceleration is greater than 0.5 g, locking is predicted and brake is released when angular speed decreases by 15% of stored value.
Some Example Ways for Reapplying Pressure Wong (1993) As soon as none of the conditions discussed for decreasing pressure at the brake are met, reapply the brake. Use some hysteresis; that is, wait a fixed amount of time before reapplying pressure. In some systems, the brake is reapplied as soon as the product of the angular acceleration of the tire and the rolling radius exceeds a predetermined value. Some typical values are 2.2 to 3 g, and sometimes the build-up rate may depend on actual value of acceleration.
Two-Position or On-Off Controllers In a two-position control system, the actuating element can take on only two positions, and often this is either on or off. This is a very common and inexpensive way to control systems. For example, a level controller can be built this way, as shown below. A simple two-position controller could follow the basic rule, M1 e() t > 0 mt () = M2 e() t < 0 Ogata (1978) Ogata (1978) While the controller is in a given position, the system may behave linearly, however on-off controllers are classified as nonlinear because they are not amenable to classical linear control design methods.
On-Off Controller Issues Sometimes, an on-off controller may have some hysteresis, and the range that the error signal must go through before actuating either way is called the differential gap. This hysteresis may be unintentional (caused by friction or gap in the mechanism), or it may be intended. One reason to purposefully include hysteresis in an on-off controller is to slow down the switching between the two-states. Switching too often can lead to reduced life in the control actuating element. A differential gap, however, will cause the output to have some oscillations, the amplitude of which can be reduced by decreasing the gap. Ogata (1978) Typical oscillations, as induced by on-off control of tank level.
Bang-Bang Control The bang-bang principle of control says that a system being operated under limited power can be moved from one state to another in the shorted time possible by at all time utilizing all available power. This was hypothesized and proven experimentally and theoretically long ago. Not all bang-bang implementation guarantee time-optimal control, of course, but for certain systems this is the case. For, With the input, Gs u u u o Y () = = n n 1 U as n + an 1s + + a0 o It can be proved that bang-bang gives time-optimal control, and you can reach a desired state in at most n-1 switches. b
Example Operation Heavy Vehicle with Pneumatic Brakes (Wong, 1993) This figure shows results for a heavy vehicle braking on wet pavement. The cycle of reducing and restoring pressure can be repeated from 5 to 16 times per second. The ABS operation is usually deactivated once the vehicle slows to about 2 or 3 mph.
Wheel-speed Sensors May generate 90 to 100 pulsed per wheel revolution. Bosch (1999) The wheel-speed sensor's pole pin with its external winding is located directly above the sensor ring, which is a type of pulse rotor joined directly to the wheel hub (the wheel-speed sensor is also installed in the differential in some applications). The pole pin is connected to a permanent magnet producing an electrical field that extends outward to the sensor ring. As the ring turns, the pole pin is exposed to an alternating progression of teeth and gaps. This results in the magnetic field changing continuously so that a voltage is generated in the sensor's winding, the frequency of which provides a precise index of the current wheel speed.
Pressure Modulator The hydraulic pressure modulator contains an accumulator, return pump, and solenoid valve.
Brake-Pressure Modulation Bosch (1999) Pressure buildup Hold pressure Reduce pressure Pressure decrease at appropriate wheel Pressure hold Pressure increase
Typical ABS Control Cycle Not your basic on-off control! This figure shows cycling on a high traction surface. Monitoring peripheral acceleration. The reference speed is used to determine the slip switching threshold. Pressure is dropped as long as peripheral acceleration is below threshold. Increases in acceleration will lead to pressure build-up, but there are gaps where system waits. Maintain pressure hold Bosch (1999)
Modeling the ABS System Vehicle dynamics and base brake system Wheel speed sensors Hydraulic modulator and valves Control Module (all logic, diagnostics) Other: pedal travel and switch, accelerometers, accumulator and electric priming
ABS Block Diagram DRAFT T b T t T d + + T wheel G w ω w Tire & vehicle Loads v x G b ABS and brake actuation or s r λ r ECU 1 Sensor Vehicle speed estimation
MathWorks ABS Simulation ABS Braking Model 0.2 Desired relative slip ctrl Bang-bang controller 100 TB.s+1 Hydraulic Lag 1 s Brake pressure Kf Force & torque brake torque 1/I tire torque 1 s Wheel Speed mu-slip friction curve slp -K- Weight Ff -1/m Explain Rr 1 s Vehicle speed STOP 1/Rr Vehicle speed (angular) 1 s Stopping distance Mux yout Double click to run model and plot the results 1.0 - u(1)/(u(2) + (u(2)==0)*eps) Relative Slip ωw 1 Developed by Larry Michaels Open function: runabs ωv The MathWorks, Inc PreLoadFcn = absdata (in a Simulink model, you set this using set_param( ) on MATLAB command line.
ABS Simulation Results 80 Vehicle speed and wheel speed Speed(rad/sec) 60 40 20 Wheel speed (ω w ) Vehicle speed (ω v ) 1 0.9 0.8 0.7 mu-slip curve Normalized Relative Slip 0 0 5 10 15 1 0.8 0.6 0.4 0.2 Time(secs) Slip 0 0 5 10 15 Time(secs) µ 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 slip Note: should also show stopping distance plot!
ABS Simulation Results 80 Vehicle speed and wheel speed Lower µ surface Speed(rad/sec) 60 40 20 Wheel speed (ω w ) Vehicle speed (ω v ) 1 0.9 0.8 0.7 mu-slip curve Normalized Relative Slip 0 0 5 10 15 20 25 1 0.8 0.6 0.4 0.2 Time(secs) Slip µ 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 slip 0 0 5 10 15 20 25 Time(secs)
ABS Simulation Results Use the simulation model to examine 3 different cases, with the last one the lower m case with no ABS. e c s m e, T i 1400 1200 1000 800 600 Stopping Distance Higher mu with ABS No ABS lower mu Lower mu with ABS This demonstrates the basic advantage of ABS. 400 200 0 0 5 10 15 20 25 Distance, m
Other Examples Ulsoy and Peng (1997) Many articles in the literature Linear ABS controller (example 8.4) using linearized model of basic braking model (no actuator) integral (error in slip) plus state feedback (both states, proportional) poor performance, even for linearized system not practical to assume you will know µ-slip curve, and there will be many other uncertainties in parameters Nonlinear controller (example 8.5) rule-based, or look-up table to give torque change needed to achieve desired result very hard to tune, calibrate, etc difficult to give any guarantee on performance or stability
Driven vs. Non-Driven Wheels There is a significant difference in applying ABS on non-driven versus driven wheels. The deceleration/acceleration rates for the nondriven wheel are generally a good indication that you may be inducing wheel lock. The engine will act on the driven wheels, resulting in extra moment of inertia coupled into the wheels, and they will react as if they are heavier. This means that the peripheral deceleration rates may not exhibit the same sensitivity to entering the unstable region, an event that precedes lockup. Use bond graph to explain inertia coupling in driven wheel
Summary How can you model this in a simple way? In the end, how good are model predictions? Will you always have to test and tune? There exist very good solutions, there are many open questions, and what goes into production is usually proprietary. How well can you quantify reliable operation?