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Robust decoupling, ideal steering dynamics and yaw stabilization of 4WS cars
Automatica. Vol. 30, No. 11, pp. 1761-1768.1994
Pergamon
0005-1098(94)E0010-F
Copyright~) 1994ElsevierScienceLtd Printed in Great Britain.All rightsreserved 0005-1098/94$7.00+ 0.00
Brief Paper
Robust Decoupling, Ideal Steering Dynamics and Yaw Stabilization of 4WS Cars* JURGEN ACKERMANNI" Key Words--Automobiles;
robust control; decoupling; nonlinear systems.
AlmtrKt--Four-wheel car steering is modeled by a single-track model with nonlinear tire characteristics. A generic control law for robust decoupling of lateral and yaw motion by yaw-rate feedback to front-wheel steering is derived. Ideal steering dynamics are achieved by velocityscheduled lateral acceleration feedback to front-wheel steering. For robust yaw stabilization a velocity-scheduled yaw-rate feedback to rear-wheel steering is given, by which the linearized system gets velocity-independent yaw eigenvalues.
Uncertain parameters are the velocity v = Ivl, v > 0 , the vehicle mass m = mr + mf, and the tire side-force characteristics which are uncertain nonlinear functions of vehicle states and inputs. Uncertainty results, for example, from changing operating conditions like dry or icy roads. Suppose the vehicle is equipped with a gyro for measuring the yaw rate r, an accelerometer at the front axle for measuring af, a sensor for the velocity v, and active frontand rear-wheel steering. Then the control system structure of Fig. 2 can be implemented. The steering wheel command is connected to a reference input af,~r for the lateral acceleration of the front axle. The velocity v can be used for gain-scheduling the controller. Disturbances acting on the car are, for example, side wind or /z-split braking as it occurs with a flat tire or when the right wheels run on ice remainders on the roadside and the left wheels run on dry pavement. Braking then induces a yaw rotation of the car. Undesired yaw motions may also be excited by fluctuations of the vertical wheel load as they occur with defective shock absorbers or in circular travel on an uneven road. The controller will be designed such that the closed-loop dynamics are as shown in Fig. 3. The lateral acceleration ae must follow closely the reference input afref, for frequencies with maximum frequency 1 Hz. A sine motion with a period of 1 s would be a very fast driver command at the steering wheel. The yaw dynamics should be well-damped, velocityindependent and not observable from at. The robustness aspect of this controller design is that the above properties must hold for all velocities v e [v-; v+], all vehicle masses m E [ m - ; m ÷] and for unknown nonlinear tire side-force characteristics. Ideally the disturbances should have no influence. If the above ideal behavior can be achieved with sufficient accuracy, then the driver only has to keep the mass point mf on top of his planned path via the lateral acceleration at, as illustrated by Fig. 4. The triangular decoupled structure of Fig. 3 indicates that the yaw rate r is still controllable from the steering wheel command. This influence on r is obviously necessary, otherwise the driver could only command a parallel lane change but could not enter into a curve. In stationary cornering on a circle with radius R, af is the centripetal acceleration afstat = 112/Rand the yaw rate is rstat = o/R, i.e. a commanded lateral acceleration arree implies rstat = a f t e r / I ) . The control law for robust triangular decoupling by feedback of the yaw rate to front-wheel steering is derived in Section 2. Section 3 treats the ideal steering dynamics by feedback of af to front-wheel steering, and Section 4 gives a transfer function interpretation for linearized tire characteristics. The yaw dynamics are analyzed in Section 5, and in Section 6 robust yaw stabilization by yaw-rate feedback to rear-wheel steering is shown. The details of the nonlinear four-wheel steering dynamics model are given in the Appendix. For the case of linear tire characteristics the main results of this paper have been published in Ackermann et al. (1993). The present paper extends the results to the case of uncertain nonlinear tire characteristics.
I. Introduction
ABOUT 5% of the newly registered cars in Japan have four-wheel steering (4WS) and recently a German car manufacturer also introduced 4WS. The advertised advantages are comfort for parking by steering the rear wheels opposite to the front wheels and safety for handling at medium and high speeds by steering front and rear wheels into the same direction. Thereby, a lane change with almost parallel shift of the vehicle center line becomes possible, i.e. the reaction of the lateral acceleration to a steering wheel command is faster and less yaw motion is induced. A survey of 4WS studies in the automotive industry is given by Furukawa et al. (1989). These studies primarily deal with the feedforward problem of commanding the rear-wheel steering angle from the steering wheel via a velocity-scheduled prefilter. A motivation for the present paper was curiosity as to how active front- and rear-wheel steering and gyros and accelerometers can be used from a feedback control point of view in order to achieve robustness with respect to uncertain operating conditions of the vehicle. Figure 1 shows a 4WS car with chassis coordinates (x, y) rotated by the yaw angle qJ with respect to an inertially fixed coordinate system (x0, Yo). Inputs to the system are the rear and front steering angles 8r and 8f. The left and right wheels are steered by the same angle. Outputs of interest are: /3 sideslip angle between vehicle center line and velocity vector v at center of gravity (CG). r = ~b yaw rate, af lateral acceleration at front axle.
*Received 15 April 1992; revised 14 December 1992; received in final form 22 November 1993. The original version of this paper was presented at the 12th IFAC World Congress which was held in Sydney, Australia, during 18-22 July 1993. The Published Proceedings of this IFAC Meeting may be ordered from: Elsevier Science Limited, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, U.K. This paper was recommended for publication in revised form by Associate Editor Frangois Delebecque under the direction of Editor Tamer Ba~ar. Corresponding author Professor JUrgen Ackermann. Tel. +49 8153 282432; Fax +49 8153 281847; E-mail [email protected] (internet). i'German Aerospace Research Establishment (DLR), Institute for Robotics and System Dynamics, Oberpfaffenhofen, D-82230 Wessling, Germany.
2. Robust triangular decoupling
Dynamic car steering models of different degrees of complexity exist in the automotive literature (Zomotor, 1761
1762
Brief Papers
~
a/
g
I I
, X
C
~Co
FIo. 1. Four-wheel steering car.
1987; Mitschke, 1990). Here the simplified model described in the Appendix is used. It is based on the classical linear Riekert-Schunck model (Riekert and Schunck, 1940), however it includes a nonlinear tire side-force model. With the assumptions explained in the Appendix the steering dynamics may be modeled by
[~t,]=[l/mvlr
',,m,r & ]
[ff(af)] LL(a,)J
[0]1 r
'1'
at, = a f - Of
The output, that we are interested in, is the front axle lateral acceleration af = lff(o/f)
l = -~Jf(af).
(4)
The decoupled steering subsystem (3), (4) is illustrated by Fig. 6. The yaw dynamics that are unobservable from at, will be further studied in Section 4.
~=a,--#r=a,--Of+lrlv. The state variable Of is the chassis sideslip angle at the front axle [see Appendix (A.18), (A.19) and Fig. A.3]. The goal of decoupling is to remove the influence of r on 0f and at, = 8 f - Of. This goal is achieved by the first main result of this paper, formulated as Theorem 1.
The desired steering dynamics shown in Fig. 3 are characterized by
(5)
at- ~ 42fref.
The ideal case is obtained by high gain feedback of the measured acceleration at, via
Theorem 1. With the control law ¢~f = Wf --
3. Ideal steering dynamics
r,
(2) Wt, = k s ( a f t e r , - at,).
the yaw motion (states r, 6 0 becomes unobservable from the front sideslip angle af = 6t, - 0f and af becomes uncontrollable from 8,.
Proof. The proof follows immediately by substituting ~f and 0t, into df = 6f - fit, = wt, - r - --~'-aff(at,) + r
a, =
(3)
l
Practically, the gain ks is limited by the actuator constraint on 18t,I. The actuator should be able to execute a steering command from the driver with maximum frequency 1 Hz. Uncertainties of the system of Fig. 6 are m, ff and v. The velocity v is easily measured and the velocity dependence can be removed by gain-scheduled feedback of at,.
Theorem 2. The control law
- m - - ~ o / / ~ , ~ 0 + w,.
1
Wf =
The yaw rate r and the steering angles 8f and 6r do not enter into &t, Q.E.D. Note that we have assumed v > 0. An implementation of the control law (2) is shown in Fig. 5, where the integrating effect is directly obtained by the electric or hydraulic steering motor with transfer function 1Is. Alternatively the integrator may be implemented in the controller.
v
yields a velocity-invariant steering system.
Proof. Substitute af from (4) into (6)
~-~ Controller~_~
(7)
wf= ksafref + ( ~ - kS) ~rrff(°tf)
] sidewind ~/~-split braking
al,,! alrel
(6)
ks(afref - af) + -13a f
-[
~ I
[
a!
Steering Dynamics Yaw Dynamics / welldamped velocityindependent
FIG. 2. Control system structure.
r ~
not observable from a!
FIG. 3. Ideal closed-loop dynamics.
Brief Papers
1763
~
!
I[ ill
~f
R
FIG. 6. Decoupled steering subsystem. J
FIG. 4. The path tracking task of the driver. system (A.18) for linear tire characteristics fr = ctotf, fr =CrOtr,
i.e. and wt into (3)
0 ][ Cf(~f--~f) "] F1 ] -1/mlfJLcr(-flf+lr/v)J - L0Jr
[~f]=[l/mvl r
L~J
I
L 1/m6
-1
=[ -cft/mot, L-crlml~ + c,lmlr
1[~f1
-crl/mvlrlL r J
[ c fl / mvl~ ] + [ cf/mlr jOf.
(8) The steering subsystem does not depend on v.
Q.E.D.
The block diagram of Fig. 7 illustrates the resulting steering dynamics. It contains the uncertain parameter m and the uncertain front tire characteristic ff(a). Their influence can be reduced by sufficiently high gain ks. In the automotive literature a lot of effort goes into detailed models for the tire force/} as a function of tire sideslip angle, vertical tire force, longitudinal slip, tire tread and pressure, and other variables. The result is unreliable, however, in view of the large variation in road adhesion between the dry road and the icy road. We have taken a robust feedback control approach, where we do not model the force but measure its effect in the form of the lateral acceleration af. We do not command the input af = 8 f - C f to the uncertain tire model by 8f, rather the resulting acceleration is commanded by at,~v This is possible without stability problems after robust decoupling of the steering subsystem by the control law (2).
(11)
The lateral acceleration at the front axle by (4) is
lcf af = -mlr r
lcf Oil = - -
mlr
tcf
a,=L- ~
0
(,~f -/3r)
(12)
][[3rf] . lcf 8f"
*ml--~
The transfer function gr(s) in
af(s) = gr(s)6f(s) is obtained as + d2s2) 1 + al$ + a2$2 lCfCrV2 kf CfCr12+ mv2(Crlr-- Cf/f)
a~( s ) =
kf(1 + dis
4. Steering transfer function
dl = 1_, d 2 - lem
For a linear interpretation of steering dynamics, iinearize the tire characteristics by ff(af)= cfaf, cf ~ [cF;c~]. The slope cf is called 'cornering stiffness' in the automotive literature. Then the transfer function of the system of Fig. 7 becomes
al
af(s) after(S)
1 mlr 1 + kscf---~l$
(9)
The steering transfer function can be made arbitrarily close to unity by high gain ks. It has been shown in McLean and Hoffmann (1973) that the frequency spectrum of human steering inputs does not extend above about 3 rad s- 1 (0.5 Hz) during normal driving, Therefore a 'high gain' ks may be chosen as ks = 30 ml~
cfl "
(10)
The transfer function (9) must be compared with the corresponding transfer function of the car with conventional steering, i.e. 8f = 8s, 8~ -= 0. For its computation, linearize the
13
(13)
Cr
lmv(cflf + crlr) CfCr12 +
m l I 2 ( C r / r -- c f l f )
lflrm2V2 a2 CfCr12+ mv2(Crlr-- Cf/f)
'
The conventional steering transfer function (13) has two zeros and two poles depending on the uncertain parameters v, m, cf, Cr. It should be much easier for the driver to control the front lateral acceleration via the new transfer function (9) with only one pole on the negative real axis depending on the uncertain parameters m and cf, and frequency response within the bandwidth of the steering commands. If the driver prefers to have the same stationary steering behaviour as in his conventional car, then the steering wheel command 8s may be connected by a velocity-scheduled prefilter kf(v) to afref, i.e.
afrei(t) = kf(v)Ss(t)
(14)
where kf(v) of (13) is chosen for nominal values of mass m and cornering stiffnesses Cr, Cf.
5. Yaw dynamics Actuator
So far we have investigated the steering subsystem. We now turn our attention to the total system including the yaw dynamics, that are unobservable from t~f. The open-loop dynamics are composed of the differential equations (A.18) and (2), i.e.
/
+.I = 11/m,, FIG. 5. Decoupling control law implementation.
8f_l
L
0
0
-I/L
r
_]
+
wf.
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Brief Papers af
after
Fio. 7. Velocity-independent steering dynamics.
for the lateral acceleration at the rear axle, the output equation is
Substitute wf from (6) to give =
L~fJ
1/ml, (1/v - ks)l/mlr
ri ,)l
-1/mlf 0
=
-
t O//r(.r)/mt q. r J
(17)
0 ilL
o
The system (15), (17) is illustrated by the block diagram of Fig. 8. The yaw dynamics, that are unobservable from af, are described by the following subsystem of (15)
xLfr(:~)J +[0]a'ef'Lks_ J Multiply the left by
[ ~ff] = [ - f r ( S r - Sf + af + lr/O)/ml' +
and substitute 8f - ~f = a, to give =[ 1/mlr [_Sfl L ( 1 / v - k s ) l / m b
[/f(~f) ]
0 i]
-1/mlf 0
1 [ ( 1 / u - ks)l] ff(°tf)[mlr+ [k0s] a'rev
(18)
The effect of an arbitrary initial condition r(O), 8f(O) in (18) is not visible from a, and af, therefore (18) is also called 'zero dynamics'. For linearized tire characteristics f~(ar)=Cra, the yaw dynamics equation is
-
× [r(r r)J + U< _i
[i'f]=[--crl/lmlr --
c ' /0m 4 1J [LrS]f j - [ C r / : l f ] ~r+ds'
(19)
Rearrange and substitute by (A.9), a, = 8 , - 8f + af + lrlv where the input from the steering subsystem is =
1 (1/u - ks)l
LSfd +
0
-
fr(Sr - 8f + af + lrlv)/mlf r
[-crlmlf]
(15)
afref.
0 a
The characteristic polynomial of (19) is
LksJ
_
2
Cr]
Cr
P l ( s ) - s +~mlf s'+ mlf"
With (4) and the corresponding equation 1 ~(a~)= ~/f f~(a~) ~,=~f
(16)
With the assumption v > 0 the unobservable subsystem is robustly stable because both coefficients are positive for all
afre!
/2r
I
v
(20)
h
v,
-.-.411.
m
FIG. 8. Block diagram of steering dynamics (state af) and yaw dynamics (states 8f and r).
Brief Papers admissible values of the uncertain parameters. Written in terms of damping D~ and natural frequency too, i.e.
is 1 + To(v)s F~(s, v) : kr(v) [1 + Tl(v)s][1 + T2(o)s] '
p(s) =s 2 + 2DloJoS + oJg, with
we have 2 - c~ t°° -- m l f
Ol '
/ = 2v
~ f-~ ~ mlf "
At high velocities v (e.g. on a German autobahn) the damping may be too small, such that safety and comfort are reduced. 6. Velocity-independent yaw dynamics There are several possibilities to improve the yaw damping, for example by feedback of the rear axle lateral acceleration ar=f~(a~)lm~=lf~(ar)/mlf to the front-wheel steering input afr~f. Such feedback would, however, destroy the decouping of the steering dynamics from the yaw dynamics. An elegant solution, that preserves decoupling, is possible in cars equipped with rear-wheel steering 8~ (Ackermann and Sienel, 1993). Theorem 3. The control law 8r=(ko-1)r+wr
(21)
yields velocity-independent yaw dynamics; ko is a tuning parameter for the yaw damping of the linearized system. Note that we have assumed v > 0. Practically, the control law (21) becomes active at a small velocity v = v- and l/v has an upper bound l/v-. Proo~ Substitute (21) into (18) and obtain
The velocity enters only into the coupling term from the steering subsystem but not into the yaw dynamics with states r and 8f. The linearized system (19) with the control law (21) becomes ~, =
1765
-1
0
J/8~J-L
o
jw,+,~. (23)
The characteristic polynomial is now kDCr
p2(s) = s ~ + - - s mlf
Cr
+--. mlf
(24)
Compared with (20) the natural frequency is unchanged too = Vcr/m~, but the damping k o ~Cr 02 = T X/m/~
(25)
is now velocity independent and can be tuned by ko. Q.E.D. The effect of the control law (21) is also seen in Fig. 8. The velocity-dependent feedback path l[v is cancelled and replaced by a velocity-independent feedback ko. 7. Prefilter for zero sideslip angle Donges et al. (1990) have used the requirement /3 -= 0 to calculate a prefilter for rear-wheel steering. The same approach is applied to the modified car steering dynamics with the three control laws (2), (6) and (21). The resulting prefilter for
Wr(S) =
Fr(S, v)afref(s)
kr(v) =
m l f o 2 - k o l c r v + llfcr Cry21
To(v)
ml~ltv - l f m v ~ + kocrlv - cfllf
TI" " mlr [ w ) = ksc--~fl r2(v) : t_~ v
The prefilter Fr(s, v) should be active only at medium and high velocities. For low velocities other operational considerations are more important than zero sideslip, 8. Summary A car with additional rear-wheel steering, measurement of the yaw rate r and the lateral acceleration af at the front axle is considered. Three robust feedback control laws and two prefilters have been derived. The results are summarized in Fig. 9. The control system structure was derived under the usual assumptions for the single-track model of steering dynamics with nonlinear tire characteristics. The additional assumption of a longitudinal mass distribution equivalent to concentrated masses at the front and rear axles was made. This assumption simplifies the derivations; it is, however, not essential, see (Ackermann, 1994). The main properties of the controller structure of Fig. 9 are: (i) Feedback of the yaw rate r via the generic controller 1/s to the front-wheel steering angle 8r provides triangular decoupling such that the yaw motion with states r, 8f is not observable from the front lateral acceleration af and af is not controllable from the rear-wheel steering angle ~Sr. This decoupling property is robust with respect to the uncertain parameters velocity o, mass m, and front and rear tire characteristics ft(a), fr(a). (ii) Feedback of af via 1/v to wt makes the steering dynamics velocity independent. Feedback of af via 'high gain' ks to wf allows ideal steering dynamics af -- arref at least for frequencies that can be commanded by the driver at the steering wheel input $s. (iii) The prefactor kf(v) is chosen such that the stationary cornering behavior shows the same velocity dependence as the uncontrolled car with nominal parameter values. (iv) Feedback of r to the rear-wheel steering angle 8r yields velocity-independent yaw dynamics. In the linearized system the controller parameter ko adjusts only the yaw damping and changes nothing else. Therefore it is not safety critical to tune ko during driving. (v) The prefilter Fr(s, v) is chosen such that for nominal values of mass m and cornering stiffnesses c , cf, the sideslip angle remains zero (/3 ~ 0 ) for all steering maneuvers, i.e. the vehicle longitudinal centerline remains tangential to the path. The above results were derived under some simplifying assumptions. In particular a single-track model with constant unknown velocity and small steering and sideslip angles was assumed. Further research is required on the robustness problems that arise if these assumptions are not satisfied. Also problems of interaction with other control systems, (ABS, traction control, active suspensions, automatic distance keeping in cruise control) and evaluations with driver models or in road tests with drivers, require further investigations. Finally the results are of interest also in automatic steering systems in which the driver is replaced by a feedback system employing a sensor for the reference track. Vehicles equipped with the control system described in this paper can be treated as individual mass points as illustrated in Fig. 4. Therefore all problems of track keeping, distance keeping,
1766
Brief Papers
~s
FIG. 9. Overall active car steering system.
formation of platoons etc. can be modeled as interactions between mass points rather than two-mass models.
on road level such that there are no couplings from pitch and roll motions. Also the heave motion is not modeled. The angles 3f and 8, are the front and rear steering angles. The distance between the CG and the front axle (rear axle) is If (lr) and together l = lr + If is the wheelbase. The velocity vector v has the absolute value v = [vl. We assume v > 0 because the vehicle is not controllable for v = 0. The angle /3 between the vehicle center line and v is called 'vehicle sideslip angle'. In the horizontal plane of Fig. A.1 an inertially fixed coordinate system (x0, Yo) is shown together with a vehicle fixed coordinate system (x, y) that is rotated by a 'yaw angle' ~. In the dynamic equations the yaw rate r := ~bwill appear as a state variable. The forces transmitted from the road surface via the wheels to the car chassis are represented in Fig. A.1 by the ° side forces ff and f~. Under acceleration or braking there are also longitudinal forces transmitted by the wheels. Further forces and torques result from aerodynamics, gravity (on slopes), return torque of the wheels etc. In chassis coordinates (x, y, z) we have the resulting forces fx and fy and the torque m~ with respect to a vertical (z-)axis through CG. Figure A.2 shows a block diagram of the model. Via the dynamics model the forces cause state variables/3, v, r. The equations of motions for three degrees of freedom in the horizontal plane are; (a) longitudinal motion
References Ackermann, J. (1994). Robust decoupling of car steering dynamics with arbitrary mass distribution. In Proc. Amer. Control Conf., Baltimore, pp. 1964-1968. Ackermann, J., A. Bartlett, D. Kaesbauer, W. Sienel and R. Steinhauser (1993). Robust Control: Analysis and Design
of Linear Control Systems with Uncertain Physical Parameters. Springer, London. Ackermann, J. and W. Sienel (1993). Robust yaw damping of cars with front and rear wheel steering. 1EEE Trans. on Control Systems Technology, 1, 15-20. Donges, E., R. Aufhammer, P. Fehrer and T. SeidenfuB (1990). Funktion und Sieherheitskonzept der Aktiven Hinterachskinematik von BMW. Automobiltechnische Zeitschrift, 10, 580-587. Furukawa, Y., N. Yuhara, S. Sano, H. Takeda and Y. Matsushita (1989). A review of four-wheel steering studies from the viewpoint of vehicle dynamics and control. Vehicle System Dynamics, 18, 151-186. McLean, J. R. and E. R. Hoffmann (1973). The effects of restricted preview on driver steering control and performance. Human Factors, 15, 421-430. Mitschke, M. (1990). Dynamik der Kraf~ahrzeuge, VoL C. Springer-Verlag, Berlin, Riekert, P. and T. Schunck (1940). Zur Fahrmechanik des gummibereiften Kraftfahrzeugs. Ingenieur Archly, 11, 210-224. Zomotor, A. (1987). Fahrwerktechnik: Fahrverhalten. VogelVerlag, WUrzburg.
-mv(B + ~) sin/3 + mr) cos/3 =f~ (b) lateral motion
my(f3 + ~k)cos B + m0 sin B =fy J~) = m~.
In this Appendix some results from the automotive literature (e.g. Mitschke, 1990; Zomotor, 1987), are summarized from the robust control point of view. The essential features of car steering dynamics in a horizontal plane are described by the 'single-track model' (or 'two-wheel model') by Riekert and Schunck (1940). It is obtained by lumping the two front wheels into one wheel in the center line of the car, the same is done with the two rear wheels, see Fig. A.1. The center of gravity (CG) is assumed
(A.3)
Here m is the vehicle mass and J is the moment of inertia w.r.t, the z-axis. These two parameters are both uncertain, but they do not vary independently. Thus we tie them together by assuming a mass distribution equivalent to concentrated masses mf and mr at the front and rear axles as illustrated by Fig. 1. Then m = mf + mr and J = mlflr. (A.4)
v0 _-
(A.2)
(c) yaw motion
Appendix. Four-wheel car steering model
:
(A.1)
~
f
~
ZO
FIG. A.1. Single-track model for car steering.
Brief Papers
.,
1767
,1[,.
I
P
Dynamics
8r
~"l
12
FIG. A.2. Block diagram of car steering dynamics.
[The case of an arbitrary mass distribution is treated in Ackermann (1994).] With r = ~bwe obtain from (A.1)-(A.4)
|
mlmlis
COS/3
sin/3 0//S/"
0
0
(A.7)
Figure A.3 illustrates the vehicle motion around a 'momentary pole' MP. The local velocity vectors in front (vf) and rear (vr) and at the CG (v) are oriented perpendicular to the connecting lines to the momentary pole. The velocity components in the direction of the longitudinal center line of the vehicle must be equal, i.e. vr cos/3, = v cos/3 = of cos/3f.
(A.8)
The velocity components perpendicular to the center line depend on the yaw rate • as vfsin/3f = v sin/3 + lfr Vr sin/3~ = v s i n / 3 - Irr.
(A.9)
The velocity terms of and v~ are eliminated by division by the corresponding terms from (A,8), thus the kinematic model is
tan/3r
-
v cos/3
lrr
tan/3
(A.12)
fr = fr(Otr)
mv(~+r)c°s/3]=[mL]. mlflri" J
o cos/3
f,= ff(af)
(A.6)
[
v sin/3
Note that all angles are counted positive in counterclockwise direction, thus/3r in Fig. A.2 has a negative value. The tire side forces
'y
from the second row of (A.5) is substituted into the first row yielding for cos 13 # 0
v sin/3 + lrr
(A.11)
ar = 8r -/3~.
lJLmzJ
sin/3 ¢
tan/3f =
ar = ~f - / 3 f
(A.,)
A constant velocity, f~ = 0, is assumed for the control system design. Thus v becomes another uncertain constant parameter. Now f~' = - COS/3
The lateral tire forces j~ and fr depend on the tire sideslip angles (see Fig. A.2)
ltr +
v
-
cos/3
lrr tan/3 - v cos------~"
(A.10)
satisfy f ( 0 ) = 0 and are assumed to be monotonic, i.e. f'(a)=df(a)/da>O for all a. Tire side forces change drastically between dry and icy roads. In stationary cornering we have f(astat + Aa) ~f(astat ) + Aotf'(Otstat), i.e. linear interpretations that we use occasionally refer to small deviations from stationary cornering with a linear gain f'(astat ) > 0 and f ( a ) depends on astat, i.e. on the curve radius. Also other unknown parameters enter into j~ and f~ which must be assumed as uncertain. In the design of a feedback steering system we do not try to model or estimate the uncertain tire forces. Instead the effects of these forces are measured by gyro and accelerometer and controlled by feedback of these signals. Finally the feedback structured system of Fig. A.2 is completed by projecting the tire forces into chassis coordinates by
JL,i-m t l=l"l 1 ]rff(otf)cossfl -IrJ ~r(Otr) COS 8rJ"
(a.13)
All other forces entering into fy and m z, e.g. acceleration, braking, aerodynamic forces are considered as external inputs and are therefore neglected in this analysis of dynamic coupling and stability. The entire model is obtained by subsituting (A.13) into (A.7) to give
[mv(13+r)cos/3]:[ll f mlflr~
J
1
-lrJLfr(ar) cos 8r]"
(A.14)
MP /
The model is now partially linearized by the assumption of small sideslip angles/3,/3f,/3r and small steering angles ~Sf,8r.
\
" \
M
[mv(~+r)l=[tl f mlflr i" J
where by (A.11) Otf=
l[h(a,)l
1 --lr J Lfr(Otr)]'
(A.15)
~f -- fir, a r = ~r - - / 3 r a n d t h e l i n e a r i z e d
form of (A.10) is /3f =/3 + lfr/v
(A.16)
/3, = [3 - lTr /v. FIG. A.3. Kinematic variables.
A simplification of (A.15) results by introducing the state
1768
Brief Papers where
variables
''lrn 1 JLrJ'
(A.17)
(A.19) ot~ = ~ - 13~ = 6~ - [3r + lr /v.
Then with the wheelbase / =/r + l~
[~,]=[,/mo,r L 1/mlr
0 lf,~of, q_ [10]r,
-1/mlrJ[f~(a,.)J
(A.18)
The system (A.18), (A.19) has the inputs 6f and 6r, the states 0f and r, and uncertainties in m, v, j~, and f~. Note that the yaw rate r enters only linearly and independently of the uncertainties into/3f. This coupling is removed by the linear decoupling control law (2).
Pergamon
0005-1098(94)E0010-F
Copyright~) 1994ElsevierScienceLtd Printed in Great Britain.All rightsreserved 0005-1098/94$7.00+ 0.00
Brief Paper
Robust Decoupling, Ideal Steering Dynamics and Yaw Stabilization of 4WS Cars* JURGEN ACKERMANNI" Key Words--Automobiles;
robust control; decoupling; nonlinear systems.
AlmtrKt--Four-wheel car steering is modeled by a single-track model with nonlinear tire characteristics. A generic control law for robust decoupling of lateral and yaw motion by yaw-rate feedback to front-wheel steering is derived. Ideal steering dynamics are achieved by velocityscheduled lateral acceleration feedback to front-wheel steering. For robust yaw stabilization a velocity-scheduled yaw-rate feedback to rear-wheel steering is given, by which the linearized system gets velocity-independent yaw eigenvalues.
Uncertain parameters are the velocity v = Ivl, v > 0 , the vehicle mass m = mr + mf, and the tire side-force characteristics which are uncertain nonlinear functions of vehicle states and inputs. Uncertainty results, for example, from changing operating conditions like dry or icy roads. Suppose the vehicle is equipped with a gyro for measuring the yaw rate r, an accelerometer at the front axle for measuring af, a sensor for the velocity v, and active frontand rear-wheel steering. Then the control system structure of Fig. 2 can be implemented. The steering wheel command is connected to a reference input af,~r for the lateral acceleration of the front axle. The velocity v can be used for gain-scheduling the controller. Disturbances acting on the car are, for example, side wind or /z-split braking as it occurs with a flat tire or when the right wheels run on ice remainders on the roadside and the left wheels run on dry pavement. Braking then induces a yaw rotation of the car. Undesired yaw motions may also be excited by fluctuations of the vertical wheel load as they occur with defective shock absorbers or in circular travel on an uneven road. The controller will be designed such that the closed-loop dynamics are as shown in Fig. 3. The lateral acceleration ae must follow closely the reference input afref, for frequencies with maximum frequency 1 Hz. A sine motion with a period of 1 s would be a very fast driver command at the steering wheel. The yaw dynamics should be well-damped, velocityindependent and not observable from at. The robustness aspect of this controller design is that the above properties must hold for all velocities v e [v-; v+], all vehicle masses m E [ m - ; m ÷] and for unknown nonlinear tire side-force characteristics. Ideally the disturbances should have no influence. If the above ideal behavior can be achieved with sufficient accuracy, then the driver only has to keep the mass point mf on top of his planned path via the lateral acceleration at, as illustrated by Fig. 4. The triangular decoupled structure of Fig. 3 indicates that the yaw rate r is still controllable from the steering wheel command. This influence on r is obviously necessary, otherwise the driver could only command a parallel lane change but could not enter into a curve. In stationary cornering on a circle with radius R, af is the centripetal acceleration afstat = 112/Rand the yaw rate is rstat = o/R, i.e. a commanded lateral acceleration arree implies rstat = a f t e r / I ) . The control law for robust triangular decoupling by feedback of the yaw rate to front-wheel steering is derived in Section 2. Section 3 treats the ideal steering dynamics by feedback of af to front-wheel steering, and Section 4 gives a transfer function interpretation for linearized tire characteristics. The yaw dynamics are analyzed in Section 5, and in Section 6 robust yaw stabilization by yaw-rate feedback to rear-wheel steering is shown. The details of the nonlinear four-wheel steering dynamics model are given in the Appendix. For the case of linear tire characteristics the main results of this paper have been published in Ackermann et al. (1993). The present paper extends the results to the case of uncertain nonlinear tire characteristics.
I. Introduction
ABOUT 5% of the newly registered cars in Japan have four-wheel steering (4WS) and recently a German car manufacturer also introduced 4WS. The advertised advantages are comfort for parking by steering the rear wheels opposite to the front wheels and safety for handling at medium and high speeds by steering front and rear wheels into the same direction. Thereby, a lane change with almost parallel shift of the vehicle center line becomes possible, i.e. the reaction of the lateral acceleration to a steering wheel command is faster and less yaw motion is induced. A survey of 4WS studies in the automotive industry is given by Furukawa et al. (1989). These studies primarily deal with the feedforward problem of commanding the rear-wheel steering angle from the steering wheel via a velocity-scheduled prefilter. A motivation for the present paper was curiosity as to how active front- and rear-wheel steering and gyros and accelerometers can be used from a feedback control point of view in order to achieve robustness with respect to uncertain operating conditions of the vehicle. Figure 1 shows a 4WS car with chassis coordinates (x, y) rotated by the yaw angle qJ with respect to an inertially fixed coordinate system (x0, Yo). Inputs to the system are the rear and front steering angles 8r and 8f. The left and right wheels are steered by the same angle. Outputs of interest are: /3 sideslip angle between vehicle center line and velocity vector v at center of gravity (CG). r = ~b yaw rate, af lateral acceleration at front axle.
*Received 15 April 1992; revised 14 December 1992; received in final form 22 November 1993. The original version of this paper was presented at the 12th IFAC World Congress which was held in Sydney, Australia, during 18-22 July 1993. The Published Proceedings of this IFAC Meeting may be ordered from: Elsevier Science Limited, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, U.K. This paper was recommended for publication in revised form by Associate Editor Frangois Delebecque under the direction of Editor Tamer Ba~ar. Corresponding author Professor JUrgen Ackermann. Tel. +49 8153 282432; Fax +49 8153 281847; E-mail [email protected] (internet). i'German Aerospace Research Establishment (DLR), Institute for Robotics and System Dynamics, Oberpfaffenhofen, D-82230 Wessling, Germany.
2. Robust triangular decoupling
Dynamic car steering models of different degrees of complexity exist in the automotive literature (Zomotor, 1761
1762
Brief Papers
~
a/
g
I I
, X
C
~Co
FIo. 1. Four-wheel steering car.
1987; Mitschke, 1990). Here the simplified model described in the Appendix is used. It is based on the classical linear Riekert-Schunck model (Riekert and Schunck, 1940), however it includes a nonlinear tire side-force model. With the assumptions explained in the Appendix the steering dynamics may be modeled by
[~t,]=[l/mvlr
',,m,r & ]
[ff(af)] LL(a,)J
[0]1 r
'1'
at, = a f - Of
The output, that we are interested in, is the front axle lateral acceleration af = lff(o/f)
l = -~Jf(af).
(4)
The decoupled steering subsystem (3), (4) is illustrated by Fig. 6. The yaw dynamics that are unobservable from at, will be further studied in Section 4.
~=a,--#r=a,--Of+lrlv. The state variable Of is the chassis sideslip angle at the front axle [see Appendix (A.18), (A.19) and Fig. A.3]. The goal of decoupling is to remove the influence of r on 0f and at, = 8 f - Of. This goal is achieved by the first main result of this paper, formulated as Theorem 1.
The desired steering dynamics shown in Fig. 3 are characterized by
(5)
at- ~ 42fref.
The ideal case is obtained by high gain feedback of the measured acceleration at, via
Theorem 1. With the control law ¢~f = Wf --
3. Ideal steering dynamics
r,
(2) Wt, = k s ( a f t e r , - at,).
the yaw motion (states r, 6 0 becomes unobservable from the front sideslip angle af = 6t, - 0f and af becomes uncontrollable from 8,.
Proof. The proof follows immediately by substituting ~f and 0t, into df = 6f - fit, = wt, - r - --~'-aff(at,) + r
a, =
(3)
l
Practically, the gain ks is limited by the actuator constraint on 18t,I. The actuator should be able to execute a steering command from the driver with maximum frequency 1 Hz. Uncertainties of the system of Fig. 6 are m, ff and v. The velocity v is easily measured and the velocity dependence can be removed by gain-scheduled feedback of at,.
Theorem 2. The control law
- m - - ~ o / / ~ , ~ 0 + w,.
1
Wf =
The yaw rate r and the steering angles 8f and 6r do not enter into &t, Q.E.D. Note that we have assumed v > 0. An implementation of the control law (2) is shown in Fig. 5, where the integrating effect is directly obtained by the electric or hydraulic steering motor with transfer function 1Is. Alternatively the integrator may be implemented in the controller.
v
yields a velocity-invariant steering system.
Proof. Substitute af from (4) into (6)
~-~ Controller~_~
(7)
wf= ksafref + ( ~ - kS) ~rrff(°tf)
] sidewind ~/~-split braking
al,,! alrel
(6)
ks(afref - af) + -13a f
-[
~ I
[
a!
Steering Dynamics Yaw Dynamics / welldamped velocityindependent
FIG. 2. Control system structure.
r ~
not observable from a!
FIG. 3. Ideal closed-loop dynamics.
Brief Papers
1763
~
!
I[ ill
~f
R
FIG. 6. Decoupled steering subsystem. J
FIG. 4. The path tracking task of the driver. system (A.18) for linear tire characteristics fr = ctotf, fr =CrOtr,
i.e. and wt into (3)
0 ][ Cf(~f--~f) "] F1 ] -1/mlfJLcr(-flf+lr/v)J - L0Jr
[~f]=[l/mvl r
L~J
I
L 1/m6
-1
=[ -cft/mot, L-crlml~ + c,lmlr
1[~f1
-crl/mvlrlL r J
[ c fl / mvl~ ] + [ cf/mlr jOf.
(8) The steering subsystem does not depend on v.
Q.E.D.
The block diagram of Fig. 7 illustrates the resulting steering dynamics. It contains the uncertain parameter m and the uncertain front tire characteristic ff(a). Their influence can be reduced by sufficiently high gain ks. In the automotive literature a lot of effort goes into detailed models for the tire force/} as a function of tire sideslip angle, vertical tire force, longitudinal slip, tire tread and pressure, and other variables. The result is unreliable, however, in view of the large variation in road adhesion between the dry road and the icy road. We have taken a robust feedback control approach, where we do not model the force but measure its effect in the form of the lateral acceleration af. We do not command the input af = 8 f - C f to the uncertain tire model by 8f, rather the resulting acceleration is commanded by at,~v This is possible without stability problems after robust decoupling of the steering subsystem by the control law (2).
(11)
The lateral acceleration at the front axle by (4) is
lcf af = -mlr r
lcf Oil = - -
mlr
tcf
a,=L- ~
0
(,~f -/3r)
(12)
][[3rf] . lcf 8f"
*ml--~
The transfer function gr(s) in
af(s) = gr(s)6f(s) is obtained as + d2s2) 1 + al$ + a2$2 lCfCrV2 kf CfCr12+ mv2(Crlr-- Cf/f)
a~( s ) =
kf(1 + dis
4. Steering transfer function
dl = 1_, d 2 - lem
For a linear interpretation of steering dynamics, iinearize the tire characteristics by ff(af)= cfaf, cf ~ [cF;c~]. The slope cf is called 'cornering stiffness' in the automotive literature. Then the transfer function of the system of Fig. 7 becomes
al
af(s) after(S)
1 mlr 1 + kscf---~l$
(9)
The steering transfer function can be made arbitrarily close to unity by high gain ks. It has been shown in McLean and Hoffmann (1973) that the frequency spectrum of human steering inputs does not extend above about 3 rad s- 1 (0.5 Hz) during normal driving, Therefore a 'high gain' ks may be chosen as ks = 30 ml~
cfl "
(10)
The transfer function (9) must be compared with the corresponding transfer function of the car with conventional steering, i.e. 8f = 8s, 8~ -= 0. For its computation, linearize the
13
(13)
Cr
lmv(cflf + crlr) CfCr12 +
m l I 2 ( C r / r -- c f l f )
lflrm2V2 a2 CfCr12+ mv2(Crlr-- Cf/f)
'
The conventional steering transfer function (13) has two zeros and two poles depending on the uncertain parameters v, m, cf, Cr. It should be much easier for the driver to control the front lateral acceleration via the new transfer function (9) with only one pole on the negative real axis depending on the uncertain parameters m and cf, and frequency response within the bandwidth of the steering commands. If the driver prefers to have the same stationary steering behaviour as in his conventional car, then the steering wheel command 8s may be connected by a velocity-scheduled prefilter kf(v) to afref, i.e.
afrei(t) = kf(v)Ss(t)
(14)
where kf(v) of (13) is chosen for nominal values of mass m and cornering stiffnesses Cr, Cf.
5. Yaw dynamics Actuator
So far we have investigated the steering subsystem. We now turn our attention to the total system including the yaw dynamics, that are unobservable from t~f. The open-loop dynamics are composed of the differential equations (A.18) and (2), i.e.
/
+.I = 11/m,, FIG. 5. Decoupling control law implementation.
8f_l
L
0
0
-I/L
r
_]
+
wf.
1764
Brief Papers af
after
Fio. 7. Velocity-independent steering dynamics.
for the lateral acceleration at the rear axle, the output equation is
Substitute wf from (6) to give =
L~fJ
1/ml, (1/v - ks)l/mlr
ri ,)l
-1/mlf 0
=
-
t O//r(.r)/mt q. r J
(17)
0 ilL
o
The system (15), (17) is illustrated by the block diagram of Fig. 8. The yaw dynamics, that are unobservable from af, are described by the following subsystem of (15)
xLfr(:~)J +[0]a'ef'Lks_ J Multiply the left by
[ ~ff] = [ - f r ( S r - Sf + af + lr/O)/ml' +
and substitute 8f - ~f = a, to give =[ 1/mlr [_Sfl L ( 1 / v - k s ) l / m b
[/f(~f) ]
0 i]
-1/mlf 0
1 [ ( 1 / u - ks)l] ff(°tf)[mlr+ [k0s] a'rev
(18)
The effect of an arbitrary initial condition r(O), 8f(O) in (18) is not visible from a, and af, therefore (18) is also called 'zero dynamics'. For linearized tire characteristics f~(ar)=Cra, the yaw dynamics equation is
-
× [r(r r)J + U< _i
[i'f]=[--crl/lmlr --
c ' /0m 4 1J [LrS]f j - [ C r / : l f ] ~r+ds'
(19)
Rearrange and substitute by (A.9), a, = 8 , - 8f + af + lrlv where the input from the steering subsystem is =
1 (1/u - ks)l
LSfd +
0
-
fr(Sr - 8f + af + lrlv)/mlf r
[-crlmlf]
(15)
afref.
0 a
The characteristic polynomial of (19) is
LksJ
_
2
Cr]
Cr
P l ( s ) - s +~mlf s'+ mlf"
With (4) and the corresponding equation 1 ~(a~)= ~/f f~(a~) ~,=~f
(16)
With the assumption v > 0 the unobservable subsystem is robustly stable because both coefficients are positive for all
afre!
/2r
I
v
(20)
h
v,
-.-.411.
m
FIG. 8. Block diagram of steering dynamics (state af) and yaw dynamics (states 8f and r).
Brief Papers admissible values of the uncertain parameters. Written in terms of damping D~ and natural frequency too, i.e.
is 1 + To(v)s F~(s, v) : kr(v) [1 + Tl(v)s][1 + T2(o)s] '
p(s) =s 2 + 2DloJoS + oJg, with
we have 2 - c~ t°° -- m l f
Ol '
/ = 2v
~ f-~ ~ mlf "
At high velocities v (e.g. on a German autobahn) the damping may be too small, such that safety and comfort are reduced. 6. Velocity-independent yaw dynamics There are several possibilities to improve the yaw damping, for example by feedback of the rear axle lateral acceleration ar=f~(a~)lm~=lf~(ar)/mlf to the front-wheel steering input afr~f. Such feedback would, however, destroy the decouping of the steering dynamics from the yaw dynamics. An elegant solution, that preserves decoupling, is possible in cars equipped with rear-wheel steering 8~ (Ackermann and Sienel, 1993). Theorem 3. The control law 8r=(ko-1)r+wr
(21)
yields velocity-independent yaw dynamics; ko is a tuning parameter for the yaw damping of the linearized system. Note that we have assumed v > 0. Practically, the control law (21) becomes active at a small velocity v = v- and l/v has an upper bound l/v-. Proo~ Substitute (21) into (18) and obtain
The velocity enters only into the coupling term from the steering subsystem but not into the yaw dynamics with states r and 8f. The linearized system (19) with the control law (21) becomes ~, =
1765
-1
0
J/8~J-L
o
jw,+,~. (23)
The characteristic polynomial is now kDCr
p2(s) = s ~ + - - s mlf
Cr
+--. mlf
(24)
Compared with (20) the natural frequency is unchanged too = Vcr/m~, but the damping k o ~Cr 02 = T X/m/~
(25)
is now velocity independent and can be tuned by ko. Q.E.D. The effect of the control law (21) is also seen in Fig. 8. The velocity-dependent feedback path l[v is cancelled and replaced by a velocity-independent feedback ko. 7. Prefilter for zero sideslip angle Donges et al. (1990) have used the requirement /3 -= 0 to calculate a prefilter for rear-wheel steering. The same approach is applied to the modified car steering dynamics with the three control laws (2), (6) and (21). The resulting prefilter for
Wr(S) =
Fr(S, v)afref(s)
kr(v) =
m l f o 2 - k o l c r v + llfcr Cry21
To(v)
ml~ltv - l f m v ~ + kocrlv - cfllf
TI" " mlr [ w ) = ksc--~fl r2(v) : t_~ v
The prefilter Fr(s, v) should be active only at medium and high velocities. For low velocities other operational considerations are more important than zero sideslip, 8. Summary A car with additional rear-wheel steering, measurement of the yaw rate r and the lateral acceleration af at the front axle is considered. Three robust feedback control laws and two prefilters have been derived. The results are summarized in Fig. 9. The control system structure was derived under the usual assumptions for the single-track model of steering dynamics with nonlinear tire characteristics. The additional assumption of a longitudinal mass distribution equivalent to concentrated masses at the front and rear axles was made. This assumption simplifies the derivations; it is, however, not essential, see (Ackermann, 1994). The main properties of the controller structure of Fig. 9 are: (i) Feedback of the yaw rate r via the generic controller 1/s to the front-wheel steering angle 8r provides triangular decoupling such that the yaw motion with states r, 8f is not observable from the front lateral acceleration af and af is not controllable from the rear-wheel steering angle ~Sr. This decoupling property is robust with respect to the uncertain parameters velocity o, mass m, and front and rear tire characteristics ft(a), fr(a). (ii) Feedback of af via 1/v to wt makes the steering dynamics velocity independent. Feedback of af via 'high gain' ks to wf allows ideal steering dynamics af -- arref at least for frequencies that can be commanded by the driver at the steering wheel input $s. (iii) The prefactor kf(v) is chosen such that the stationary cornering behavior shows the same velocity dependence as the uncontrolled car with nominal parameter values. (iv) Feedback of r to the rear-wheel steering angle 8r yields velocity-independent yaw dynamics. In the linearized system the controller parameter ko adjusts only the yaw damping and changes nothing else. Therefore it is not safety critical to tune ko during driving. (v) The prefilter Fr(s, v) is chosen such that for nominal values of mass m and cornering stiffnesses c , cf, the sideslip angle remains zero (/3 ~ 0 ) for all steering maneuvers, i.e. the vehicle longitudinal centerline remains tangential to the path. The above results were derived under some simplifying assumptions. In particular a single-track model with constant unknown velocity and small steering and sideslip angles was assumed. Further research is required on the robustness problems that arise if these assumptions are not satisfied. Also problems of interaction with other control systems, (ABS, traction control, active suspensions, automatic distance keeping in cruise control) and evaluations with driver models or in road tests with drivers, require further investigations. Finally the results are of interest also in automatic steering systems in which the driver is replaced by a feedback system employing a sensor for the reference track. Vehicles equipped with the control system described in this paper can be treated as individual mass points as illustrated in Fig. 4. Therefore all problems of track keeping, distance keeping,
1766
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~s
FIG. 9. Overall active car steering system.
formation of platoons etc. can be modeled as interactions between mass points rather than two-mass models.
on road level such that there are no couplings from pitch and roll motions. Also the heave motion is not modeled. The angles 3f and 8, are the front and rear steering angles. The distance between the CG and the front axle (rear axle) is If (lr) and together l = lr + If is the wheelbase. The velocity vector v has the absolute value v = [vl. We assume v > 0 because the vehicle is not controllable for v = 0. The angle /3 between the vehicle center line and v is called 'vehicle sideslip angle'. In the horizontal plane of Fig. A.1 an inertially fixed coordinate system (x0, Yo) is shown together with a vehicle fixed coordinate system (x, y) that is rotated by a 'yaw angle' ~. In the dynamic equations the yaw rate r := ~bwill appear as a state variable. The forces transmitted from the road surface via the wheels to the car chassis are represented in Fig. A.1 by the ° side forces ff and f~. Under acceleration or braking there are also longitudinal forces transmitted by the wheels. Further forces and torques result from aerodynamics, gravity (on slopes), return torque of the wheels etc. In chassis coordinates (x, y, z) we have the resulting forces fx and fy and the torque m~ with respect to a vertical (z-)axis through CG. Figure A.2 shows a block diagram of the model. Via the dynamics model the forces cause state variables/3, v, r. The equations of motions for three degrees of freedom in the horizontal plane are; (a) longitudinal motion
References Ackermann, J. (1994). Robust decoupling of car steering dynamics with arbitrary mass distribution. In Proc. Amer. Control Conf., Baltimore, pp. 1964-1968. Ackermann, J., A. Bartlett, D. Kaesbauer, W. Sienel and R. Steinhauser (1993). Robust Control: Analysis and Design
of Linear Control Systems with Uncertain Physical Parameters. Springer, London. Ackermann, J. and W. Sienel (1993). Robust yaw damping of cars with front and rear wheel steering. 1EEE Trans. on Control Systems Technology, 1, 15-20. Donges, E., R. Aufhammer, P. Fehrer and T. SeidenfuB (1990). Funktion und Sieherheitskonzept der Aktiven Hinterachskinematik von BMW. Automobiltechnische Zeitschrift, 10, 580-587. Furukawa, Y., N. Yuhara, S. Sano, H. Takeda and Y. Matsushita (1989). A review of four-wheel steering studies from the viewpoint of vehicle dynamics and control. Vehicle System Dynamics, 18, 151-186. McLean, J. R. and E. R. Hoffmann (1973). The effects of restricted preview on driver steering control and performance. Human Factors, 15, 421-430. Mitschke, M. (1990). Dynamik der Kraf~ahrzeuge, VoL C. Springer-Verlag, Berlin, Riekert, P. and T. Schunck (1940). Zur Fahrmechanik des gummibereiften Kraftfahrzeugs. Ingenieur Archly, 11, 210-224. Zomotor, A. (1987). Fahrwerktechnik: Fahrverhalten. VogelVerlag, WUrzburg.
-mv(B + ~) sin/3 + mr) cos/3 =f~ (b) lateral motion
my(f3 + ~k)cos B + m0 sin B =fy J~) = m~.
In this Appendix some results from the automotive literature (e.g. Mitschke, 1990; Zomotor, 1987), are summarized from the robust control point of view. The essential features of car steering dynamics in a horizontal plane are described by the 'single-track model' (or 'two-wheel model') by Riekert and Schunck (1940). It is obtained by lumping the two front wheels into one wheel in the center line of the car, the same is done with the two rear wheels, see Fig. A.1. The center of gravity (CG) is assumed
(A.3)
Here m is the vehicle mass and J is the moment of inertia w.r.t, the z-axis. These two parameters are both uncertain, but they do not vary independently. Thus we tie them together by assuming a mass distribution equivalent to concentrated masses mf and mr at the front and rear axles as illustrated by Fig. 1. Then m = mf + mr and J = mlflr. (A.4)
v0 _-
(A.2)
(c) yaw motion
Appendix. Four-wheel car steering model
:
(A.1)
~
f
~
ZO
FIG. A.1. Single-track model for car steering.
Brief Papers
.,
1767
,1[,.
I
P
Dynamics
8r
~"l
12
FIG. A.2. Block diagram of car steering dynamics.
[The case of an arbitrary mass distribution is treated in Ackermann (1994).] With r = ~bwe obtain from (A.1)-(A.4)
|
mlmlis
COS/3
sin/3 0//S/"
0
0
(A.7)
Figure A.3 illustrates the vehicle motion around a 'momentary pole' MP. The local velocity vectors in front (vf) and rear (vr) and at the CG (v) are oriented perpendicular to the connecting lines to the momentary pole. The velocity components in the direction of the longitudinal center line of the vehicle must be equal, i.e. vr cos/3, = v cos/3 = of cos/3f.
(A.8)
The velocity components perpendicular to the center line depend on the yaw rate • as vfsin/3f = v sin/3 + lfr Vr sin/3~ = v s i n / 3 - Irr.
(A.9)
The velocity terms of and v~ are eliminated by division by the corresponding terms from (A,8), thus the kinematic model is
tan/3r
-
v cos/3
lrr
tan/3
(A.12)
fr = fr(Otr)
mv(~+r)c°s/3]=[mL]. mlflri" J
o cos/3
f,= ff(af)
(A.6)
[
v sin/3
Note that all angles are counted positive in counterclockwise direction, thus/3r in Fig. A.2 has a negative value. The tire side forces
'y
from the second row of (A.5) is substituted into the first row yielding for cos 13 # 0
v sin/3 + lrr
(A.11)
ar = 8r -/3~.
lJLmzJ
sin/3 ¢
tan/3f =
ar = ~f - / 3 f
(A.,)
A constant velocity, f~ = 0, is assumed for the control system design. Thus v becomes another uncertain constant parameter. Now f~' = - COS/3
The lateral tire forces j~ and fr depend on the tire sideslip angles (see Fig. A.2)
ltr +
v
-
cos/3
lrr tan/3 - v cos------~"
(A.10)
satisfy f ( 0 ) = 0 and are assumed to be monotonic, i.e. f'(a)=df(a)/da>O for all a. Tire side forces change drastically between dry and icy roads. In stationary cornering we have f(astat + Aa) ~f(astat ) + Aotf'(Otstat), i.e. linear interpretations that we use occasionally refer to small deviations from stationary cornering with a linear gain f'(astat ) > 0 and f ( a ) depends on astat, i.e. on the curve radius. Also other unknown parameters enter into j~ and f~ which must be assumed as uncertain. In the design of a feedback steering system we do not try to model or estimate the uncertain tire forces. Instead the effects of these forces are measured by gyro and accelerometer and controlled by feedback of these signals. Finally the feedback structured system of Fig. A.2 is completed by projecting the tire forces into chassis coordinates by
JL,i-m t l=l"l 1 ]rff(otf)cossfl -IrJ ~r(Otr) COS 8rJ"
(a.13)
All other forces entering into fy and m z, e.g. acceleration, braking, aerodynamic forces are considered as external inputs and are therefore neglected in this analysis of dynamic coupling and stability. The entire model is obtained by subsituting (A.13) into (A.7) to give
[mv(13+r)cos/3]:[ll f mlflr~
J
1
-lrJLfr(ar) cos 8r]"
(A.14)
MP /
The model is now partially linearized by the assumption of small sideslip angles/3,/3f,/3r and small steering angles ~Sf,8r.
\
" \
M
[mv(~+r)l=[tl f mlflr i" J
where by (A.11) Otf=
l[h(a,)l
1 --lr J Lfr(Otr)]'
(A.15)
~f -- fir, a r = ~r - - / 3 r a n d t h e l i n e a r i z e d
form of (A.10) is /3f =/3 + lfr/v
(A.16)
/3, = [3 - lTr /v. FIG. A.3. Kinematic variables.
A simplification of (A.15) results by introducing the state
1768
Brief Papers where
variables
''lrn 1 JLrJ'
(A.17)
(A.19) ot~ = ~ - 13~ = 6~ - [3r + lr /v.
Then with the wheelbase / =/r + l~
[~,]=[,/mo,r L 1/mlr
0 lf,~of, q_ [10]r,
-1/mlrJ[f~(a,.)J
(A.18)
The system (A.18), (A.19) has the inputs 6f and 6r, the states 0f and r, and uncertainties in m, v, j~, and f~. Note that the yaw rate r enters only linearly and independently of the uncertainties into/3f. This coupling is removed by the linear decoupling control law (2).