Brake and steer by wire

At the most fundamental level, the control algorithm consists of three functional blocks: 1 means of determining the desired response of the vehicle in the yaw plane; 2 vehicle model sto

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SAE TECHNICAL

Unified Control of Brake- and Steer-by-Wire

Systems Using Optimal Control

Allocation Methods

Aleksander Hac

Delphi Corporation

David Doman and Michael Oppenheimer

Air Force Research Laboratories

Reprinted From: Brake Technology 2006

(SP-2017)

2006 SAE World Congress

Detroit, Michigan April 3-6, 2006

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Printed in USA

A new optimal control strategy for dealing with braking

actuator failures in a vehicle equipped with a

brake-by-wire and steer-by- brake-by-wire system is described The main

objective of the control algorithm during the failure mode

is to redistribute the control tasks to the functioning

actuators, so that the vehicle performance remains as

close as possible to the desired performance in spite of

a failure The desired motion of the vehicle in the yaw

plane is determined using driver steering and braking

inputs along with vehicle speed For the purpose of

synthesizing the control algorithm, a non-linear vehicle

model is developed, which describes the vehicle

dynamics in the yaw plane in both linear and non-linear

ranges of handling A control allocation algorithm

determines the control inputs that minimize the

difference between the desired and actual vehicle

motions, while satisfying all actuator constraints The

algorithm can be applied to either vehicles with a

brake-by-wire system only or to vehicles with both brake and

steer-by-wire systems and can be adapted to specific

conditions The results of simulations using a high

fidelity vehicle model demonstrate the benefits of the

proposed control method

INTRODUCTION

Brake-by-wire (BBW) [1] and Steer-by-wire (SBW) [2]

systems for motor vehicles have reached at least the

prototype stage of development with some systems

being implemented in production vehicles In BBW

systems, braking of each wheel is usually controlled by

independently operating mechanical or

electro-hydraulic, actuators Consequently, failure modes that

may occur in these systems are different from those

experienced in conventional (e.g hydraulic or

pneumatic) brake systems Since BBW systems

generally include some level of redundancy in

measurements, algorithms have been developed for

detection and identification of failure modes [3] These

algorithms either rely on sensor redundancy or use model-based techniques to detect and specify failure modes It is therefore assumed here that when a failure occurs, it is sensed and reported to the control system Furthermore, the brake system operates in “fail silent” mode; that is, in the event of a failure, the actuator does not produce braking torque As BBW actuators can operate independently of each other, the functioning actuators can be used during failure modes to compensate, at least in part, for the braking lost due to the failed actuator

In SBW systems, the steering angle of front (or rear) wheels is controlled by actuators In the true SBW systems or Active Rear Steer (ARS) systems, this is achieved without a direct mechanical link to the steering-wheel, while in an Active Front Steer (AFS) system, a steering correction can be applied to the front wheels in addition to the driver steering input One advantage of these systems is that a corrective yaw moment can be applied to the vehicle, independently of the driver, by changing the front (or rear) steering angle

This paper describes a new optimal control strategy for dealing with failures of brake actuators in a vehicle equipped with a BBW system and possibly with a SBW system From the point of view of vehicle level control, failure of a brake actuator results in two undesirable effects: 1) vehicle deceleration is less then desired since the total braking force acting on vehicle is reduced; 2) brake force distribution becomes asymmetric, creating

an unbalanced yaw moment, which pulls the vehicle to the side In order to minimize these effects, the control strategy must be modified during failure, by redistributing the braking forces to the remaining three actuators and possibly by introducing a steering correction to balance

at least a part of the yaw moment generated by asymmetric braking, if the vehicle is equipped with a SBW system The main objective of the control algorithm during the failure mode is thus to redistribute the control tasks to the functioning actuators, so that the vehicle

2006-01-0924

Unified Control of Brake- and Steer-by-Wire Systems Using

Optimal Control Allocation Methods

Aleksander Hac

Delphi Corporation

David Doman and Michael Oppenheimer

Air Force Research Laboratories

Copyright © 2006 SAE International

performance remains as close as possible to the desired

performance, in spite of a failure This type of control

problem, usually referred to as a control allocation

problem, has been extensively studied in the aerospace

industry and a number of control methods have been

developed [4] The solution to a control allocation

problem generally depends on the desired motion, on

the operating point of vehicle, and on actuator

constraints One control method, which has been

successfully applied to aircraft control problems, and is

particularly well suited to the problem considered here,

is the optimal control allocation method, which uses

linear programming to determine the optimal control

input in real time [5] This optimal control allocation

method provides means of determining the optimal

solution under all operating conditions, while satisfying

all actuator constraints It also permits a unified control

approach to vehicles equipped with BBW systems only

or vehicles with both BBW and SBW systems In this

paper, this methodology is applied to the reconfigurable

control of a vehicle during brake actuator failure

This paper is organized as follows In the next section,

the control algorithm development is described It

includes a brief overview of the control allocation

problem formulation and a description of the reference

model, which generates the desired response of vehicle

Also, the vehicle model used for development of control,

actuator constraints, and a summary of the control

algorithm are provided Results of simulations using a

validated high fidelity dynamic vehicle model are then

presented, followed by concluding remarks

CONTROL ALGORITHM DEVELOPMENT

This section describes, the control algorithm that was

developed for the purpose of controlling a vehicle in the

presence of a braking actuator failure As discussed in

the Introduction, it is assumed that the failure is detected

and reported to the Central Processing Unit (CPU) and

that the failed actuator does not produce any braking

torque The control algorithm should be capable of

reconfiguring the control tasks, using the remaining

brake actuators and possibly the steering actuator, so

that the vehicle response remains as close as possible

to the desired response The control problem is

formulated as a control allocation problem, in which an

optimal solution is determined in real time using efficient

and reliable algorithms based on linear programming

techniques At the most fundamental level, the control

algorithm consists of three functional blocks: 1) means

of determining the desired response of the vehicle in the

yaw plane; 2) vehicle model (stored in an on-board

microprocessor), which describes the vehicle’s behavior

with sufficient accuracy and is used for determination of

the control inputs and actuator constraints in real time;

3) a numerical optimization algorithm to calculate the

control inputs

The desired motion of the vehicle in the yaw plane is

determined using driver steering and braking inputs

along with vehicle speed For calculation of control inputs, a non-linear vehicle model is used, which describes the vehicle’s dynamics in the yaw plane in both linear and non-linear ranges of handling The structure of the model equations is selected to facilitate the use of dynamic inversion A control allocation algorithm determines the control inputs, which minimize the difference between the desired and actual vehicle motions, while satisfying all actuator constraints The algorithm provides means of varying the weightings among different control objectives and control inputs, which can be used to adapt the control algorithm to specific situations The braking and steering commands are realized by the brake and steer actuators using actuator-level control system; this aspect of design remains the same as during normal operation and is not discussed in this paper

Since the reader may not be familiar with the control techniques used here, a brief overview of the control allocation problem for dynamical systems is first given This is followed by a description of each one of the three functional blocks in control design defined above

DYNAMIC INVERSION AND CONTROL ALLOCATION OVERVIEW

Consider a dynamic system described by the following state equation

u) f(x,

x  (1)

where x and u are the state and control vectors,

respectively, and a dot denotes a derivative with respect

to time A dynamic inversion problem is the problem of

finding a control vector, u, which yields the state vector equal to a desired state vector, x des It is often convenient to define a set of pseudo controls, which quantify the total effect of all actuators acting on the system These pseudo-controls can correspond to any combination of desired forces, moments or accelerations, which result from the influence of the actuators on the vehicle A control allocation or mixing problem results when one attempts to find an optimum combination of actuator inputs that deliver a desired set

of pseudo-controls subject to a set of inequality constraints on the control input vector:

u min dudu max (2)

Here u min and u max are constant, known vectors that represent force or position limits of the actuators In some cases, limitations on the rates of change of control inputs may be imposed which can be expressed using (2) when implemented on a digital computer Depending

on the dimensions of the state and input vectors and the particular form of the state equation (1) and the constraints (2), the problem can have an infinite number

of solutions, a unique solution, or no solution If an exact solution cannot be found without violating constraints, then the control input is determined that minimizes the

difference between the desired and actual values of the

pseudo-controls In this case, some of the control inputs

usually reach their limiting values When using control

allocation combined with dynamic inversion, the

structure of the model equation (1) is of great

importance In order to perform dynamic inversion

efficiently, it is desirable that the equation of motion be

linear in the control vector u Specifically, if the state

equation can be expressed as:

g(z)u z) f(x,

x  (3)

where z is a vector of parameters or variables, which

can be measured, then given the desired state vector,

x des, the control input can be determined by application

of linear programming techniques Linear programming

techniques can be used since both the state equation

and the constraints are linear in the input vector Linear

programming techniques in essence reduce the

optimization problem to solving sets of linear algebraic

equations iteratively, thus they are numerically more

efficient and reliable than non-linear optimization

algorithms

In many control problems, including those of controlling

flight dynamics, the last term at the right hand side of

equation (3) has a more general form, namely, g(z,u)

[5] In order to obtain the dynamic inversion control law,

the desired value of the derivative of the state vector,

dx des/dt, is specified Then, solving the state equation for

g(z,u), one obtains:

[ xdesf(x, z)] g(z, u) (4)

The left hand side of this equation defines the desired

pseudo-controls and solving the above equation for the

control input vector, u, poses a control allocation

problem as a nonlinear root finding problem with

constraints on u In order to solve this more general

problem using efficient algorithms based on linear

programming techniques, Doman and Oppenheimer [6]

proposed to approximate the term g(z,u) as follows:

g(z, u) B(z)uİ(z, u) (5)

Here the intercept term H(z,u) represents a correction for

any additional terms in g(z,u), not represented in the

linear term, B(z)u Because the control allocation

algorithm operates in discrete-time, the problem can be

posed as follows: Find the control vector, uk+1, such that

1 k k k

k k k des f(x , z ) İ(z , u )] B(z )u

x

subject to u min dudu max Here the subscript k refers

to the kth discrete time instant and H(z k ,u k ) represents an

intercept correction term, which compensates for

nonlinearities in the steering and braking force and

torque vs control deflection maps A solution to

equation (6) subject to the constraints (2) can be

obtained using linear programming techniques Since

dynamic inversion and control allocation has to be performed on line, efficiency of the algorithm is of primary importance

Application of the control allocation algorithm to control the dynamics of the vehicle in the yaw plane requires determination of the desired values of vehicle state variables and their time derivatives Also required is the development of a dynamic model of the vehicle that describes the actual vehicle with sufficient accuracy and possesses a structure that makes it suitable for the application of an efficient control allocation method These two elements are described in the next two sections

REFERENCE MODEL

The reference model is used to determine the desired values of the vehicle state variables and their derivatives using driver inputs, specifically hand wheel angle, brake pedal force, and vehicle speed Vehicle motion in the yaw plane is uniquely determined by three state variables: longitudinal velocity, vx, lateral velocity, vy, and yaw rate, : The desired values, vxdes, vydes, :des and their time derivatives are determined as follows First, the steady-state desired longitudinal acceleration is computed from the measured brake pedal force:

a xdesss KF pedal (7)

where K is a constant and F pedal is the brake pedal force The minus sign indicates that acceleration during

braking is negative The magnitude of a xdesss is subsequently limited to a reasonable value, for example

10 m/s2, yielding a xdessslim The desired value of

longitudinal acceleration, a xdes, is obtained by passing

a xdessslim through a low-pass filter, representing the dynamics of the brake system:

) ( 1

1 )

s T s

a xdes  (8)

where s is the Laplace operand and T a is a constant parameter The desired value of the derivative of longitudinal velocity and the longitudinal velocity itself are then determined as follows:

des ydes xdes

v  : (9a)

t v v

v xdes(k1) xdes(k) xdes' (9b)

Here v ydes and :des are the desired values of lateral velocity and yaw rate, which are calculated below, the

subscript k refers to the k-th discrete time instant, and 't

is the sampling time

The desired yaw rate, :des, is calculated using the measured steering angle (at the hand wheel) and the estimated vehicle speed in a similar manner as is done

in Electronic Stability Control (ESC) systems In this study, the desired steady-state value of yaw rate, : ,

was first determined from a look-up table, with data

which was dependent on steering angle and speed In

the linear handling range, the values in the look-up table

closely approximate those derived from the linear bicycle

model, that is

2

x u

d x desss

v K L

v



(10)

Here v x is the estimated vehicle speed, Gd is the steering

angle at the front wheels corresponding to the steering

wheel angle commanded by the driver, L denotes

vehicle wheelbase, and K u is the understeer gradient

Outside the linear handling range, the values of desired

yaw rate at steady state were determined empirically

The magnitude of steady-state yaw rate is then limited

by g/vx, where g is acceleration due to gravity This

yields the desired and limited yaw rate at steady state,

:dessslim This value is then passed through a low pass

filter to yield the desired yaw rate:

) ( 1

1 )

s T



:

:

(11)

where T: is an appropriate filter constant, which may

depend on vehicle speed In the event of combined

braking and steering maneuvers, when the resultant

desired acceleration of the vehicle exceeds surface

limits, desired yaw rate and longitudinal deceleration

undergo further limitation Specifically, if

a xdes2 v x:des2 !g (12)

then both the desired values of yaw rate and lateral

acceleration are multiplied by the factor

2

assures that the resultant desired acceleration of the

vehicle (which is a vector sum of the desired longitudinal

and lateral accelerations) does not exceed the surface

limit The desired lateral velocity, v ydes, is then

determined using the following relationship:

0

2

L C

Mav b v

r

x des

ydes :  (13)

Here C r0 is the constant cornering stiffness of the rear

axle in the linear range of handling, a and b are

distances from the vehicle center of gravity to the front

and rear axles, and M is vehicle mass Relationship (13)

between the yaw rate and lateral velocity holds for the

steady-state values derived from a linear bicycle model

It does not reflect different transient dynamics in the

lateral velocity response versus yaw response in highly

dynamic maneuvers, but it constitutes a sufficiently

accurate approximation for the control algorithm used in

this study Derivatives of yaw rate and lateral velocity

are computed by passing the signals through high pass

filters, which approximate time derivatives Specifically, they are determined as follows:

1 )

s T

s

c

 : (14)

1 )

s T

s s

c ydes



 (15)

where T c is the filter constant

VEHICLE MODEL FOR CONTROL DEVELOPMENT

In this section, a mathematical model of the vehicle, developed for the purpose of synthesizing an optimal control allocation algorithm, is described A good model

is instrumental in developing any optimal control algorithm, since on-line optimization is performed under the assumption that the model correctly describes the actual system Thus the model should accurately describe the vehicle’s dynamics in both linear and non-linear ranges of handling At the same time, the model complexity must be held to a level in order to make on-line computation possible In the case of control allocation algorithms, the structure of the model equation

is of great importance Since, dynamic inversion is used

in this work, it is desirable that the equations of motion

be linear in the control input vector Parameters of the model should be either constant or depend on directly measured variables in order to facilitate on line calculations

The vehicle model for reconfigurable control development should include the effects of steering and braking inputs on vehicle behavior in the yaw plane In normal driving conditions, when vehicle tires are in the linear range of operation, the effects of steering and braking are largely independent of each other and the forces generated are approximately proportional to the driver inputs However, when vehicle tires approach the limit of adhesion, the magnitudes of resultant tire forces are limited by the product of the surface coefficient of adhesion and the tire normal force; consequently tire longitudinal forces must be traded off against lateral forces and vice versa The model should describe both the linear and non-linear ranges of vehicle operation, thus it must include a non-linear tire model

Selection of control variables is important in formulating the reconfigurable control problem, since it affects the form of the state equation and constraints In this paper, the primary focus is the development of a control algorithm at the vehicle level Therefore, we select as control variables those that directly affect vehicle motion

in the yaw plane Specifically, the brake forces on the tire-road interface are chosen as control variables for BBW system and the angular position of the front wheels relative to the vehicle centerline is selected as a control variable for the steering system These variables are not controlled directly, but due to the fast speed of response

of the actuators relative to the rigid body vehicle dynamics, they can be considered as acting

instantaneously to a command as long at that command

does not violate actuator rate or position In reality, a

BBW system controls brake actuating force via its own

(actuator level) feedback control algorithm using input

voltage as a manipulated variable The brake force on

the tire-road interface is directly related to the actuating

force Similarly, the SBW system controls the steering

angle of the front wheels by controlling the steering rack

position through the voltage commanded to an electric

motor using a feedback control algorithm

It is noted that the brake forces at the tires have a

directly proportional effect on the vehicle response,

which simplifies the equations of motion The front

steering angle, however, does not provide this

advantage Since by controlling front wheel steering

angle, one also controls the tire side-slip angle and

therefore the lateral force per axle, it may be tempting to

select the axle lateral force as a control variable This

would have an advantage of simplifying the equations of

motion However, steering the front wheels affects not

only the lateral forces of the front axle, but also the rear

axle through vehicle dynamics Commanding a front

steering angle initially causes a slip angle of the front

axle, which results in a corresponding lateral force The

vehicle begins to rotate (yaw) and develop the slip angle

on the tires of the rear axle and consequently a lateral

force at the rear In order to reflect this dynamic process

in the equations of motion, one has to incorporate tire

dynamic properties, which reduces the potential for

simplification of the equations of motion

FxRF

FxLF

FyRF

FyLF

tw

b

a :

vy

vx

y

x

Fdrag

Gf

Figure 1 Vehicle Model in the Yaw Plane

A planar view of a vehicle model in the yaw plane is

shown in Figure 1 The total mass of vehicle is M and

the moment of inertia with respect to the yaw (vertical)

axis passing through the center of gravity is I zz Symbols

a and b denote the distances from the front and rear

axle to the center of gravity, respectively, and d is the

half-track width of the vehicle (assumed the same, front

and rear) The vehicle is subjected to longitudinal and

lateral tire forces at each corner, as well as a drag force,

F drag Other forces acting on vehicle, such as those due

to road inclinations, side winds, etc., are considered to

be disturbances and are not included in the model The steering angle of the front wheels is Gf As discussed earlier, the control variables are the brake forces at all

four wheels, F xLF , F xRF , F xLR , F xRR, and the front steering angle,Gf

The lateral and longitudinal accelerations of the vehicle center of gravity expressed in the vehicle reference frame are:

a x vxv y: (16a)

a y vyv x: (16b)

where a x and a y are longitudinal and lateral

accelerations, respectively, v x and v y are longitudinal and lateral velocities of the center of gravity, and : is the vehicle yaw rate The following variables are assumed to

be measured or estimated: longitudinal acceleration, ax,

lateral acceleration, a y, yaw rate, :, front wheel steer angle,Gf , brake forces, F xLF , F xRF , F xLR , F xRR, and vehicle

speed, v x With the exception of maneuvers performed

at low speeds, the front steering angle in a SBW vehicle will be small Using a small steering angle assumption (sinGf # Gf and cosGf# 1), the equations of longitudinal, lateral, and yaw motions of the vehicle are

M

F F F F M

v C v

v x y d x xLF  xRF xLR xRR



 :

2



f yRF yLF

M

F F

G



 (17a)

M

F F F F v

v y x yLF  yRF  yLR yRR

 :





f xRF xLF

M

F F

G



 (17b)

zz xRR xLR xRF xLF zz

F F I

a F

F F F I

d











 :

zz yRR yLR zz

F F I

a F

F I

b

G









zz

F F I

d

G



 (17c)

In the above equations, F x and F y denote tire longitudinal and lateral forces (LF – left front, RF – right front, LR – left rear, RR – right rear) Braking forces are assumed positive (and tractive forces negative) The aerodynamic drag force is F drag C xUAv2x /2, where C x is the drag coefficient, U is air density, and A is the frontal area of

vehicle (1/2Uv x is dynamic pressure of the air) The

product C x UA/2 = C d can be considered a constant in most realistic situations

Note that the products of the brake forces, F xLF , F xRF,

F xLR , and F xRR, and the steering angle, Gf, appear in equations (17b) and (17c), making the equations bilinear, rather than linear in the control inputs Furthermore, the products of lateral forces and the front

steering angle appear in equations (17a) and (17c) The

lateral forces depend on the steering angle and the state

variables, in particular, tire slip angles In order to avoid

undue complication of the equations of motion, the

lateral forces appearing in the products with the steering

angle will be expressed as functions of directly

measured variables These products are generally

smaller than other terms, thus the above simplification is

justified in application to these terms First, the total

lateral force of the front axle, F yLF + F yRF can be

approximated as follows:

L

Mb a

M F

F yLF yRF f yf y (18)

Here M f = Mb/L is the mass of the vehicle associated

with the front axle and a yf a ya: is the lateral

acceleration at the front axle location, which can be

determined from the measured lateral acceleration, a y,

and the yaw rate, : Let us assume that during

cornering the lateral tire forces are proportional in

magnitude to the normal loads, that is,

y

y

RF

LF yRF

yLF

a d

h g

a d

h g N

N F

F





(19)

Here N LF and N RF are the normal forces of the left front

and right front tires, respectively, g is acceleration due to

gravity, and h is the height of the vehicle center of

gravity above the ground In calculating the normal loads

in equation (19), quasi-static equations were used and

the roll moment distribution between the front and rear

axle was assumed to be proportional to the mass

distribution Combining equations (18) and (19) yields

yf y

f yRF

dg

h M F

Here, it is seen that the difference in lateral forces is

roughly proportional to the lateral acceleration squared,

with the sign determined by the sign of the lateral

acceleration of the body at the front axle, ayf The above

equation is substituted into the last term of equation

(17c), while equation (18) is used in the last term of

equation (17a)

The axle lateral forces appearing as linear terms in

equations (17b) and (17c) can be expressed as products

of cornering stiffness (per axle) and tire slip angles, as

follows:

¸

¸

¹

·

¨

¨

©







x

y f f f f yRF

yLF

v

a v C

C F

x

y r r r yRR

yLR

v

b v C C

F





The cornering stiffness values of the front and rear axle are functions of tire normal loads, which depend on longitudinal and lateral accelerations and on braking forces Thus

f

where all variables a x , a y , and F xij are either directly

measured (a x , a y ) or estimated (F xij) More explicitly, cornering stiffness of the front axle is the sum of cornering stiffness values of both front tires:

C f C yLF C yRF (23)

where C yLF and C yRF are the cornering stiffness values of left front and right front tires, which depend on the operating point of the tire, primarily on the normal load and braking force Specifically,

¸¸

¸

¹

·

¨¨

¨

©

§



¸

·

¨



2 2 0

0 0

LF

xLF LF

LF y yLF

N

F N

N N k N

N C

and similarly for all other tires In the above, C y0 denotes

tire cornering stiffness at nominal normal load, N 0,

(which is usually close to the static load), N LF is the

normal load, and k is the tire stiffness sensitivity

coefficient to a normal load The above model is perhaps the simplest tire model reflecting the effects of normal load transfer and braking on tire lateral forces [7, 8] It comprehends the most important effects of braking on steerability For example, during light braking, the lateral stiffness of the front tires increases almost in proportion

to the normal load, improving steerability During heavy braking, however, steerability is generally reduced, which is captured by the term (1  F xLF

2

/N LF 2), which

decreases rapidly as braking force, F xLF, approaches the

normal force N LF (e.g the limit of adhesion) The normal tire forces are given by:

y f x

d

Mh a

L

Mh g L

Mb N

2 2

N ' ' (25a)

y f x

d

Mh a

L

Mh g L

Mb N

2 2

N ' ' (25b)

y r x

d

Mh a

L

Mh g L

Ma N

2 2

N ' ' (25c)

y r x

d

Mh a

L

Mh g L

Ma N

2 2

N ' ' (25d)

For each tire, the normal load is the sum of the static

load (front, N fst , or rear, N rst), the normal load transfer due to braking, 'N(ax), and the normal load transfer due

to cornering, 'N f (a y) and 'N r (a y) Symbols Nf and Nr

denote the fraction of the total roll stiffness of

suspension contributed by the front and rear

suspension, respectively, with Nf + Nr = 1 Here, it can be

seen that the normal tire load depends on the known

vehicle parameters and directly measured longitudinal

and lateral accelerations Acceleration due to braking

(e.g deceleration) is assumed negative and lateral

acceleration is positive in a right turn The axle cornering

stiffness is therefore a function of directly measured and

estimated variables: longitudinal and lateral

accelerations and braking forces of a given axle, as

indicated by equation (22)

Substituting equations (18) and (20) into (17a) and

(17c), respectively, then substituting equation (21) into

(17b) and (17c) yields the following equations of motion:

M

F F F F M

v

C

v

v x y d x xLF  xRF xLR xRR





:

2



f yf f

M

a M

G

 (26a)

:

¸

¸

¹

·

¨

¨

©









x

r f x y x

r

f

y

Mv

b C a C v v Mv

C

C

v

f xRF xLF f

M

F F C

G





:









:

x zz

r f

y x

zz

r

f

v I

b C a C v v

I

b C

a



zz

F F F F

I

d











f zz

yf y f xRF xLF

f

I

g

a a h M F

F

C

a

G







(26c)

The cornering stiffness values, C f and C r, are not

constant, but they depend on directly measured

variables: a x , a y, and braking forces, according to

equations (22) through (25) Equation (26) represents a

set of nonlinear differential equations They still involve

products of control inputs (bilinear terms) However, in

most operating conditions, the bilinear terms will be

smaller than the terms which are linear in control inputs

Also, even though the cornering stiffness values depend

on the braking forces and lateral and longitudinal

accelerations, this dependency is rather weak, unless

the vehicle is close to the limit of adhesion Equation

(26) can be written in the following general form:

u) g(z, z) f(x,

x  (27)

where x is a vector of state variables, u a vector of

control inputs, and z a vector of directly measured

variables, which also include some control inputs These

vectors are as follows:

>v x,v y,:@T, u >F xLF,F xRF,F xLR,F xRR,Gf @T,

x

xRR xLR xRF xLF yf y

In the above, the superscript T denotes a transpose The

non-linear functions f(x,z) and g(z,u) are as follows:

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x zz

r f

y x zz

r f

x

r f x y x

r f

x d y

v I

b C a C v v I

b C a C

Mv

b C a C v v Mv

C C

M

v C v

2 2

2

z x,

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f xRR xLR xRF xLF

zz zz zz

zz

yf f

F F F F

g I

d I

d I

d I d

g M

a M M M M M

G

35

25

0 0 0 0

1 1 1 1

u) g(z,

M

F F C

g f  xLF  xRF

25

zz

yf y f xRF xLF f

I

g a a h M F

F C a g

/

35







(30)

Equation (27) has a more general form than the desired form of equation (3), which is linear in the control input

However, the matrix g(z,u) is almost linear in the control vector, u, with the control influence coefficient matrix in

equation (30) being dependent only on directly measured variables (which, however, include the control

inputs F xLF , F xRF , F xLR , F xRR) In most conditions, the linear terms reflect the major control influence, with the bilinear terms being generally smaller Thus, it is a case

of mild nonlinearities, which can be handled by using an intercept term to compensate for nonlinearities

Another important observation from the state equations (26), which is quite intuitive, is that braking forces have a direct and strong effect on vehicle deceleration, while steering input, Gf, has very little effect on vehicle deceleration The latter effect manifests itself through the last term in equation (26a) and can be significant only when lateral acceleration of the vehicle and the steering angle are both large On the other hand, both steering and braking can significantly affect the yaw and lateral motions of the vehicle

LIMITS ON CONTROL INPUTS

In order to maintain realistic assumptions about the braking and steering actuators, limits on the magnitudes and rates of change of control inputs are considered Brake actuators are capable of locking the wheels on any surface, thus the actual limitation of the magnitudes

of brake forces on the tire road interface are the normal

tire forces times the surface friction coefficient If only

dry surface performance is considered, the limits on

brake forces are

R F j R L i N

F xij ij, , ; ,

Here N ij denote the tire normal forces, which according

to equation (25) depends on the directly measured

longitudinal and lateral accelerations Therefore, they

are computed on line The limits on the rates of change

of braking forces depend primarily on the capability of

the brake actuators in developing the actuating force

and on wheel inertia Since for the electric caliper the

speed of response is about the same in both directions,

the following limit is used:

max

  (32)

where F xrmax is the maximum rate of change of braking

force at the tire

The steering angle of the front wheels, Gf, has a hard

limitation, usually about 30 degrees (0.52 radians) in

each direction However, for control purposes we

introduce a speed-dependent limitation on the

magnitude of the steering angle, in order to avoid

excessive steering input at high speeds, which may

destabilize the vehicle The limitation used in this study

is as follows:

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 d

2 0

max

x u f

v

L K

G

Here Gmax is the maximum steering angle (about 0.5

radians, applied only at low speeds), L is the vehicle

wheelbase and K u0 is a selected understeer gradient,

which is larger than the understeer coefficient of the

vehicle in the linear range of handling This limitation

ensures that the steering angle is sufficient to achieve

the maximum lateral acceleration of the vehicle at any

speed, but is not much larger than that The limit on the

steer rate is a direct function of steering actuator

capability In this study, a fixed limit on the magnitude of

steer rate is used:

Grmax dGf dGrmax (34)

whereGrmax is the maximum steer rate

DESCRIPTION OF CONTROL ALGORITHM

The purpose of the control algorithm is to determine the

values of all control inputs, that is brake forces and front

steering angle, during brake actuator failure, so that the

optimal response of the vehicle in the yaw plane is

achieved Specifically, the vehicle should follow the

desired values of longitudinal and lateral velocities and

yaw rate, as well as derivatives of these variables as

closely as possible In some situations, the exact following of the desired trajectory is not possible due to limitations on the control signals and their rates of change Therefore, trade offs between longitudinal, lateral, and yaw responses may occur In order to perform such trade offs, the designer must specify weights on control of each state variable These weighting coefficients can be used to adapt the control system to specific conditions For example, during light braking the relative weight on longitudinal acceleration can be reduced, but increased during heavy braking

In addition, weights, or penalties, on actuator use must

be specified These can be used to determine the balance between using the brakes and steering actuator

to achieve desired motion of the vehicle One of the advantages of this approach is that the same algorithm can be used for vehicles with a BBW system only and for vehicles with both BBW and SBW systems In the former case, simply using a very high weight (penalty)

on the steering actuator use will prevent the control allocation algorithm from using the steering input to control the vehicle

In order to synthesize the control input, the control allocation algorithm uses the state equation in the form

of equation (6), in which an intercept term is present Since the state equation (27) describing vehicle motion

in the yaw plane has a more general form, the term

g(x,z) must be separated into a term linear in control input, B(z)u, and a non-linear intercept term H(z,u) In

order to apply this approach to the case of brake actuator failure, the following strategy is used:

1) Longitudinal acceleration is controlled with the

brake forces, F xij, by linearizing the equation of longitudinal motion about the current value of the front steer angle, Gfk, and ignoring Gfk in the

control influence matrix B(z) The steering

contribution to the longitudinal acceleration appears in the intercept term, H(x k ,z k ) This

prevents the control allocation algorithm from trying to use steering to control longitudinal acceleration, but the controller is still cognizant

of the steering effect on this variable

2) Lateral acceleration is controlled with the steering angle, Gfk, by linearizing this equation about the current values of the front brake

forces, F xLFk, F xRFk.

3) Yaw acceleration is controlled with the steering angle,Gfk , and the brake forces, F xLFk, F xRFk , F xLRk,

F xRRk, but by using only terms linear with respect

to brake forces The bilinear terms are handled

by linearizing the yaw equation about the current

values of the front axle brake forces F xLFk, F xRFk

Using this approach, the state equation for control synthesis assumes the form of equation (6), in which the vector of measured variables, the state and input vectors are given by equation (28) and the functions

)

u İ(z, z), f(x, and B(z) describing the vehicle dynamics, the intercept correction term, and the control influence

F xLR , and F xRR, and the steering angle, Gf, appear in equations (17b) and. .. measured and

estimated variables: longitudinal and lateral

accelerations and braking forces of a given axle, as

indicated by equation (22)

Substituting equations (18) and