algorithm to generate target for anti-lock braking system
TRANSCRIPT
2019-XX-XXXX
Algorithm to Generate Target for Anti-Lock Braking System using Wheel PowerEhsan Arasteh, Francis Assadian, Louis Filipozzi
Department of Mechanical and Aerospace Engineering, University of California, Davis, CA
Abstract
This paper discusses creating a suitable reference for the Anti-LockBraking (ABS) control system that maximizes power recovery ordissipation of the brake system. Current anti-lock brake systemsimplement a finite-state method algorithm to respond to the wheellongitudinal slip by keeping the slip at the given target to maximizebraking force. Variables in the logic can include wheel-speedmeasured from sensors, brake target slip which is estimated by theElectronic Control Unit (ECU), reference speed, which is alsoestimated, and vehicle acceleration/deceleration. Angular accelerationand calculated actual wheel slip are used as indicators of the wheel’sstate of motion. To find the maximum braking force, a typical ABSover-brakes such that the tire longitudinal force exceeds its maximumvalue and therefore, the wheel starts locking up. The ABS then adjustsby under-braking, and repeats this cycle to keep the tire longitudinalforce around its maximum. This paper uses the power absorbed by thewheel to find the maximum power dissipated. Using this strategy, theABS utilizes a continuous control strategy and can better approximatethe maximum brake force without the cyclic method that has beenused in the conventional ABS. In addition to maximizing the brakeforce as a result of this approach, driver comfort also increases due tothe continuous control vs. on/off (cyclic) control during an ABSevent. The paper discusses theoretical aspects of the problem, andsoftware simulation using MATLAB and Simulink.
Introduction
Current anti-lock brake systems implement a finite-state methodalgorithm to respond to the wheel longitudinal slip by keeping the slipat the given target to maximize braking force. Variables in the logiccan include wheel-speed measured from sensors, brake target slipwhich is estimated by the Electronic Control Unit (ECU), referencespeed, which is also estimated, and vehicle acceleration/deceleration.Angular acceleration and calculated actual wheel slip are used asindicators of the wheel’s state of motion. Current continuous controlmethods focus on nonlinear control techniques such as sliding modeto keep the vehicle’s slip at a desired value [1] [2] [3]. Suchtechniques are often not intuitive, hard to tune and require a lot ofcomputational power to be implemented on the production vehicle.
To find the maximum braking force, a typical ABS over-brakes suchthat the tire longitudinal force exceeds its maximum value andtherefore, the wheel starts locking up. The ABS then adjusts byunder-braking, and repeats this cycle to keep the tire longitudinalforce around its maximum.
In this paper, we use the one-wheel vehicle model to derive a unique
control strategy for the control of a vehicle ABS, using the powerabsorbed by the wheel to find the maximum power dissipated. Usingthis strategy, the ABS utilizes continuous control and can betterapproximate the maximum brake force without the aforementionedcyclic method that has been used in the conventional ABS. In additionto maximizing the brake force as a result of this approach, drivercomfort also increases due to the continuous control vs. on/off(cyclic) control during an ABS event.
The first part of the paper consist of modeling the system. Then, usingthe dynamics equations of the model, we will lay the groundwork ofthe use of dissipated power in the context of ABS continuous control.The paper then continues with the simulation results (done inMATLAB and Simulink) and follows with the discussion of theresults.
Vehicle and Brake model
Figure 1 shows a single wheel tire and all the forces, moments andvelocities while braking. This simple model is chosen for the purposeof initial results on the ABS algorithm proposed in this paper. [4].
Figure 1: Schematic of forces and torques on the wheel
Equations 1 and 2 describe the dynamic equations for this singlewheel tire model.
Jwωw = FbRw − τb (1)
Mu = −Fb (2)
Where M is the vehicle’s mass, m is wheel’s mass, Fb is the brakingforce, N is normal force from the ground, u is vehicle’s forwardvelocity, ωw is angular velocity of the wheel, Jw is wheel’s momentof inertia, Rw is wheel’s radius, τb is the brake torque. For the Fb,longitudinal tire force, we can write
Fb = µ ·N (3)
Where µ represents the longitudinal friction coefficient between thetire and the ground. There are different models for the tire forces such
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as Pacejka magic tire formula [5], Dugoff [6], LuGre [7], andBurckhardt [8]. In this paper, we choose a simplified Burckhart modelto represent the longitudinal tire model, for the sake of its simplicityand being able to capture longitudinal tire saturation well. Thelongitudinal friction coefficient is defined as
µ = c1 · (1− e−c2λ)− c3λ (4)
where c1, c2 and, c3 are tire constants that depend on the road surfacetype (dry asphalt, wet asphalt, snow, ice) and λ is wheel slip andduring the brake it is defined as
λ =u−Rwωw
u(5)
This relationship (Longitudinal friction Coefficient vs. Slip) is shownfor a few of the road surfaces in Figure 2.
Figure 2: Longitudinal Friction Coefficient vs. Slip for different road surfacesusing Burckhardt tire model
A brake actuator was also implemented similar to the one discussed in[9]. This actuator resembles the dynamics of an Electro-Hydraulicbrake. The input to the system is the pressure input on the brake fluidlines and the output is the caliper position which then translates to thebrake torque τb with the following equation
τb =
{2µcalreffkcal (xcal − x0) , if xcal ≥ x00, otherwise (6)
Where µcal, kcal, reff , xcal, and x0 are the friction coefficientbetween the brake disk and caliper, brake pad stiffness, brake padeffective radius, caliper position, and the spacing between the brakepad and brake disk when there is no braking, respectively.
Power Method
The overall control architecture of new continuous ABS algorithmmethod is shown in Figure 4. The main idea in the this architecturelies under the ”Power Method” block. It provides the brake torquereference to the control loop. Additionally, a low-level controller wasdesigned for the brake system using Youla-Kucera robust controltechnique [4][10].
An overall schematic of the power method reference torque generatoris given in Figure 3. The inputs to the Power Method referencegenerator are wheel angular velocity, the brake torque, and braketorque rate of change. The wheel angular velocity can be obtainedthrough the wheel rate sensor. However, the brake torque is notreadily available with the current sensors in a commercial vehicle.Therefore, estimation techniques can be utilized to obtain theestimated brake torque. The output of the reference generator is
torque reference (τref ). We will discuss about dτbdt
later in this sectionand also in the Mathematical Background section. Writing a simplepower balance for the wheel yields to Equation 7.
Power = Pωω + τbω + Fb (u−Reffω) = Fbu (7)
Figure 3: Power Method Reference Torque Generator for ABS
Where Pω, ω, τb, Fb, u and Reff are the derivative of angularmomentum (Pω = Jyω, where Jy is wheel moment of inertia), wheelangular velocity, brake torque, brake force, vehicle speed andeffective wheel radius, respectively. A steady state case is consideredwhere there is no power exchange due to the tire inertia. This is agood approximation of operating in the linear region of thefriction-slip curve where saturation momentum has less of an effect.Therefore, the power balance would result in Equation 8.
Power dissipated = τbω = FbReffω (8)
Equation 8 shows that braking force of the vehicle is associated withthe power dissipated by the brakes on wheel. If the power dissipatedby the brakes is maximized, the braking force is also maximized. Thishappens near the peak of the friction-slip curve. If the slip becomesgreater than the maximum slip, this would result in the wheel lock up(and therefore zero angular velocity of the wheel) and the powerwould immediately go to zero. To search for maximizing thedissipation power, the algorithm shown in Figure 3 is used. Itcompares the previous power dissipation with the current powerdissipation and if that power is increasing, then the dissipated powerstill hasn’t reached its maximum. Therefore, the wheel torque brakecan increase. Once the previous power dissipation is more than thecurrent power dissipation, the maximum power dissipation has beenreached and therefore, the previous brake torque is the maximumtorque brake that can be exerted by the brake. Brake torqueincrease/decrease in each step is called dTb
dt(also referred as torque
rate in this paper) . This variable plays a very important rule in thisheuristic search as we will discuss in the following sections. We willdiscuss different ways to find a solution for an optimal dTb
dtas it is a
very important variable in the power method algorithm. Loyola et. alhas suggested a 2-D lookup table based on acceleration and velocityof the vehicle for this variable. [4] In this paper, for moreeffectiveness and robustness of the method, we use an adaptive torquerate ( dτb
dt) during the ABS event.
Power Method
ω
τbrake
dτbdt
Low-LevelController
Brake Vehicleτref ∆τ Pb τbrake ω
Figure 4: Overall control architecture of the new ABS method
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Rate of Torque Calculation Using Coefficient ofFriction
Linearizing the relationship between longitudinal force andlongitudinal slip velocity requires,
Fb =∂Fb∂us
us + Fb0 (9)
where us is the longitudinal slip velocity and is equal to
us = u−Rwωw (10)
And in the linear portion of the curve Fb0 = 0. If we differentiate thewheel spin dynamics of Equation 1 with respect to time and substituteEquation 9; then we can write,
Jwdωwdt
=d
dt
(∂Fb∂us
usRw
)− dτb
dt(11)
Hence, we can simplify Equation 11 and approximate dτbdt
by thefollowing equation,
dτbdt
= NRw∂µ (us)
∂us
dusdt
(12)
where in approximating Equation 11, we assumed that the angularjerk is relatively small compare to the other terms of Equation 11, thetire radius and ∂Fb
∂usare approximately constant and do not change
with time, and Fb = µN , where N is the tire normal force and it isassumed that the normal force does not change with time (it has aslower dynamics than the wheel dynamics). Now, if we substituteEquation 5 into Equation 12 and replace ωw by its equivalentexpression in 1, we can write,
1∂µ(us)∂us
dTbdt
− NR2w
JwTb = NRw
du
dt− NR3
w
JwFb (13)
Equation 13 describes an approximate transient behavior of the tirewith the longitudinal brake force as input and the brake torque asoutput. Furthermore, using Equation 2, we have
du
dt= −Fb
M(14)
In addition, our search is for Fbmax where,
Fbmax = Nµmax (15)
We substitute Equation 14 and Equation 15 into the Equation 13, andwe divide Equation 13 by NRw, then we can write,
1∂µ(uS)∂uS
NRw
dτbdt
=NR2
w
Jw
τbNRw
− NR2w
Jwµmax − N
Mµmax (16)
When working in the linear region of the longitudinal force vs. slipcurve the term τb
NRwis equal to under the tire coefficient of friction µ.
In addition, in Equation 16, the term NM
� N R2w
Jw; hence, we can
ignore the term NMµmax, and rewrite the Equation 16 as,
dτbdt
= α (µmax − µ) (17)
Where the proportionality constant is
α = −∂µ (us)∂us
R3WN
2
JW(18)
Therefore, Equation 17 can be used for estimation of dτbdt
, provided anestimation of the road surface maximum coefficient of friction (µmax)
is available. As stated earlier, it is important to note by measuring thebrake torque τb, and either computing or measuring the normal forceN, it is then possible to compute the under tire coefficient of friction.Another important fact is that by having access to the road surfacemaximum coefficient of friction and the under tire coefficient offriction, it is possible to directly implement a force control strategy.This strategy simply minimizes the error between the under tirecoefficient of friction and the road surface maximum coefficient offriction. Therefore, having the knowledge of µmax and µ, an ABScontroller would not be needed and can be replace with a force controlstrategy. In the next subsection, we discuss a simpler method thatdoes not require a knowledge of µmax and and lends itself very wellin the power method brake torque generator for a continuous ABS.
Rate of Torque Calculation Using Longitudinal TireForce
Assuming that ωw is zero during the steady-state parts of the brake,from Equation 1, we can write
τb = Rw · Fb (19)
Then dτbdt
would become
dτbdt
= Rw · dFbdt
(20)
Which means that dτbdt
can be calculated from the derivative of the tirelongitudinal force. This would require an estimation of tire’slongitudinal force which has been research by many researchers suchas [11],[12],[13],[14], and [15]. This force estimation along with thederivative would inherently add delays to this signal. Therefore, wehave used a 10 millisecond delay for this signal.
In the next section, we present a few simulation results due toimplementation of the adaptive- dτb
dtpower method using a rate of
torque being calculated by the derivative of longitudinal tire force andits comparison with a constant rate of torque.
Results and Discussion
In this section, we discuss the results of the power method ABSalgorithm which is proposed in the previous sections. The initialconditions for the simulations are different initial velocity of thevehicle, different road surfaces, and an initial slip which accounts forthe time that ABS starts its process which is usually after moments ofhard braking by the user. The initial slip is tested on different values(0.3 and 0.7) to ensure that the ABS power method algorithm worksfor initial slips near wheel lock (λ = 1) and smaller values which arestill far from wheel lock.
Figures 5, 6, 7, 8 and 9 show the results of the adaptive dτbdt
using thederivative of longitudinal force. As shown in the Figures, this methodperforms well in different road surfaces, initial velocities, and initialslips. Figures 6 and 5 show the coefficient of friction (Fb
N) in different
conditions. The maximum coefficient of friction is also plotted for thecomparison. As shown in these figures, the vehicle is really close tothe maximum friction coefficient without even getting near thelocking of the wheel at any point (Figures 8 and 7). Figure 9 showsthe reference brake torque generated using the power method atdifferent speeds on a dry asphalt with the initial slip of 0.7. It takeslonger for the algorithm to find the steady-state solution since at thehigher velocities, the power dissipated would be larger; and therefore,the peak of the power happen at a later time.
As seen in Figures 11 and 12, a constant dτbdt
can become sensitiveover different conditions and will lock-up the wheel if this constantvalue is too high under certain road surfaces, initial velocities, andinitial slips. And if the value of rate of torque is set low, it will result
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Figure 5: Coefficient of Friction vs. Time for various vehicle velocities (U0)and road surfaces with the intial slip of λ = 0.7
Figure 6: Coefficient of Friction vs. Time for various vehicle velocities (U0)and road surfaces with the intial slip of λ = 0.3
in a sub-optimal friction which would result in a longer stoppingdistance. Two different values of rate of torque has been tested andcompared with the adaptive rate of torque which uses the derivative oflongitudinal force.
As show in the Figures, this constant value is dependent on the roadsurface, initial velocity and initial slip that the ABS algorithminitiated which would require a look-up table and a lot of calibrationand tuning during the implementation. In all the cases, the adaptiverate of torque shows a better performance (higher friction coefficient,µ) and does not lock the wheel. Figure 3 compares the dissipatedpower for to cases of constant rate of power and the adaptive rate ofpower for an initial velocity of 40 mph on a wet asphalt with theinitial slip of 0.7. As we can see from Figure 11, when dτ
dt= 200, the
Figure 7: Slip vs. Time for various vehicle velocities (U0) and road surfaceswith the intial slip of λ = 0.7
Figure 8: Slip vs. Time for various vehicle velocities (U0) and road surfaceswith the intial slip of λ = 0.3
wheel locks up. This can be seen in Figure 3 as well since thedissipated power doesn’t reach its peak and goes to zero before theother two cases. When dτ
dt= 200, although it doesn’t lock the wheel,
and it looks like an optimal solution from a power point of view, thesteady-state value of coefficient of friction doesn’t stay close to theoptimal solution and decreases. The only explanation for this wouldbe that because of the nature of constant torque rate, once thealgorithm reached one step before the peak of power (which wouldtranslate to the optimal brake torque), it then added one more constantstep to the near optimal brake torque and it caused it to go over thepeak of power; therefore, the coefficient of friction immediatelydropped. This just proves what we discussed earlier regarding theconstant rate of torque being hard to tune.
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Figure 9: Reference Torque from Power Method vs. Time for various vehiclevelocities (u0) on a Dry Asphalt with the intial slip of λ = 0.7
Figure 10: Comparison of constant dτbdt
vs. a variable dτbdt
- Dissipated Powervs. Time for initial velocity of 40 mph on a dry Asphalt and an initial slip of0.7)
Conclusion
We presented am ABS target generator using the dissipating power ofthe wheel for a continuous ABS control scheme. Different methodswere used to find the a solution for a rate of torque that gives the bestresults. An adaptive rate of torque using the derivative of longitudinalforce was used and showed promising results.
Figure 11: Comparison of constant dτbdt
vs. a variable dτbdt
- Slip vs. Time forvarious vehicle velocities (U0), road surfaces and intial slips (λ0)
Figure 12: Comparison of constant dτbdt
vs. a variable dτbdt
- Coefficient ofFriction vs. Time for various vehicle velocities (U0), road surfaces and initialslips (λ0)
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Contact Information
Ehsan Arasteh,University of California, Davis, USA,[email protected]
Prof. Francis Assaidan, University of California, Davis, USA,[email protected]
Louis Filipozzi, University of California, Davis, USA,[email protected]
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