art. bianchi

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The contribution of the present work is twofold. First, we pro-

pose a fullyintegrated controller, achieving the tracking of yaw

rate, lateral velocity and roll dynamics (angle and velocity), by

means of three actuators, in nominal conditions. Second, we

propose a methodology, based on a hybrid controller, for the

management of the actuators in saturation conditions, based on

fixed priorities on the states to be tracked, according to theirimportance for the vehicle attitude.

The paper is organized as follows. In Section 2, the mathemat-

ical model of the vehicle is presented, and the control problem

is stated. In Section 3, the nonlinear technique is developed

and the saturation management is described. In Section 4, the

proposed controller is tested with simulations and comparisons.

Some comments conclude the paper.

2. MATHEMATICAL MODEL AND PROBLEM

FORMULATION

For simplicity, we consider the model of a rearwheel driven

vehicle. The actuators considered in this work are

1. Active Front Steer (AFS), which imposes an incremental

steer angle on top of the drivers input. The control is then

actuated through the front axle tire characteristic.

2. Rear Torque Vectoring (RTV), which distributes the

torque in the rear axle, usually to improve vehicle trac-

tion, handling and stability. The control is then actuated

through the rear axle tire characteristics.

3. Semiactive suspensions (SAS), which are able to change

the damping coefficient of the shock absorber in a contin-

uous interval, differently from passive systems.

The mathematical model is derived under the following as-

sumptions, which are verified in a large number of situations

and which mitigate the complexity of the vehicle dynamics

The vehicle moves on a horizontal plane;

The longitudinal velocity is constant, so that vehicle shak-

ing/pitch motions can be neglected;

The steering system is rigid, so that the angular position

of the front wheels is uniquely determined by the steering

wheel position;

The wheels masses are much lower than the vehicle one,

so the steering action does not affect the position of thecentre of mass of the entire vehicle;

The vehicle takes large radius bends and the road wheel

angles are small (less than 10), i.e. the bend curveradius is much higher than the vehicle width;

The aerodynamic resistance and the wind lateral thrust are

not considered;

The tire vertical loads are constant;

The actuators are ideally modelled.

As a consequence of the previous assumptions, the vehicle

model has four degrees of freedom: lateral velocity, yaw rate,

roll angle and velocity. Following Guiggiani [2007], one can

consider the lateral acceleration of the vehicle

m(vy+ vxz) =(Fy,f+Fy,r) +mshdx (1)

where m is the vehicle mass, vy, z, xare the lateral, yaw, androll velocities, hd = h d,his the center of gravity height, disthe roll center height,ms is the sprung mass, is the roadtirefriction coefficient, andFy,f, Fy,r the lateral (front/rear) tireforces. The lateral forcesFy,f = Fy,f(f),Fy,r = Fy,r(r)depend on the slip angles

f =c+f,0= c+d+fx vy+lfz

vx

r =rxvy lrz

vxwhere c, d are the control, driver components of the wheelangle,f, r are front/rear sensitivities with respect to the rollanglex, andlf,lr are the distances from the center of gravityto the front/rear axle. A simple but accurate representation of

the lateral force functions is given by (Pacejka [2005])

Fy,f(f) =Cy,fsin(Ay,farctan(By,ff))

Fy,r(r) =Cy,rsin(Ay,rarctan(By,rr))(2)

where Ay,f, Ay,r , By,f, By,r , Cy,f, Cy,r are experimentalconstants.

The vehicle yaw dynamics can be expressed considering the

presence of RTV actuators

Jzz =(Fy,flf Fy,rlr) +Mz+Jzxx (3)

whereJz is the vehicle inertia momentum about thez axis,Jzxis the product of inertia about the axes z, x, andMzis RTV yawmoment.

The vehicle roll angular acceleration can be expressed as

x = x

Jrx = bxx (kx msghd)x+Jzxz

+mshd(vy+ vxz)

(4)

where Jx is the vehicle inertia momentum about the x axis,Jr = Jx + msh2d, bx is the suspension roll damping, kx isthe suspension roll stiffness, and g is the gravity accelerationconstant.

From (1), (3), (4) we get the mathematical model of a vehicle

with yaw, lateral and roll dynamics

z = 1

Jz,e

(Fy,flf Fy,rlr) +Mz

+kmkzhe(Fy,f+Fy,r)

kmkz

bxx+ (kx msghd)x

vy =zvx+ 1

me(Fy,f+Fy,r)

+kmkzhe

(Fy,flf Fy,rlr) +Mz

kmhe

bxx+ (kx msghd)x

x= km

bxx+ (kx msghd)x

+kmkz

(Fy,flf Fy,r lr) +Mz

+kmhe(Fy,f+Fy,r)

x= x

(5)


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wherekm= 1/Jx,eand

Jx,e=JrJ2zx

Jz

m2sh2d

m

me= m

1 +mskmhdhe, Jz,e =

Jz1 +kmkzJzx

he=ms

mhd, kz =

JzxJz .

As anticipated, we considervx constant. Moreover, the controlinputs that we consider areMz , and the differences

Fy,f =Fy,f Fy,f,0, Fy,f,0:= Fy,f(f,0)

bx= bx bx,0

with bx,0 the damping when the SAS system is not active.Clearly, the real active front input is the control angle c whichcan be determined by inverting (2), obtaining

c =

d+ vy+lfz

vx+F1y,f(

F) if|F| Fy,f(f,sat)

d+ vy+lfz

vx f,sat otherwise

with Fa fixed value to be imposed by the AFS.

The control aim is to track asymptotically some bounded refer-

ences, with bounded derivatives, for z,vy,x andx. Moreprecisely, the reference generator is

z,ref= 1

Jz,ref

ref(Fy,f,reflf Fy,r,reflr)

+kmkzheref(Fy,f,ref+ Fy,r,ref)

kmkz

bxx,ref+ (kx msghd)x,ref

vy,ref=z,refvx+ 1

meref(Fy,f,ref+ Fy,r,ref)

+kmkzhe


kmhe

bxx,ref+ (kx msghd)x,ref

x,ref=km

bx,refx,ref+ kx,refx,ref

+kmkz


+kmheref(Fy,f,ref+ Fy,r,ref)

x,ref=x,ref

(6)

where Jz,ref, ref are appropriate parameters and Fy,f,ref,Fy,r,refare ideal curves, depending on

f,ref=d+fx,refvy,ref+ lfz,ref

vx

r,ref=rx,refvy,ref lrz,ref

vx.

In particular, we set

Fy,j,ref(j,ref) =

=

Fy,j(j,ref) +F

y,j(j,ref)(j,ref+ j,ref)j,ref j,ref

withj = f, r, where j,ref is a limit value for the slip angle,above which the control action is needed.

3. DESIGN OF A STATEFEEDBACK LINEARIZING

CONTROL LAW WITH SATURATION MANAGEMENT

In this section a statefeedback linearizing control, imposing

desired behaviors for yaw, lateral and roll dynamics, is de-

signed. Then a saturation management algorithm is proposed.

3.1 StateFeedback Linearizing Control Law

Let us impose the following control requirements

z =d1:= z,ref k1(z z,ref)

vy =d2:= vy,ref k2(vy vy,ref)x= d3:= x,ref k3(x x,ref) k4(x x,ref)

(7)

ki > 0,i = 1, , 4. To fulfill (7), the following control lawhas to be imposed

u1

u2

u3

=

Fy,f

Mz

bx

=C1

r1

r2

r3

(8)

where

C=

kmkzhe+ lfJz,e

1Jz,e

kmkzx

1me

+kmkzhelf

kmkzhe kmhex

km(he+kzlf) kmkz kmx

(9)

and

r1= d1 kmkzheFy 1

Jz,eTy+kmkzDy

r2= d2+zvx 1

meFy kmkzheTy+ kmheDy

r3= d3 kmheFy kmkzTy+kmDy

Fy =(Fy,f,0+Fy,r)

Ty =(Fy,f,0lf Fy,rlr)

Dy =bx,0x+ (kx msghd)x.

In the following we assume that

kmk2zJz,e = 1, meh

2ekm= 1

which is reasonable since this is verified for an appropriate

choice of the vehicle parameters. Therefore, since the firstcolumn ofCdepends on and the third onx, Cis invertible ifand only if= 0andx= 0. The inverse of (9) is


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C1 =

0 1

me1meh2ekm

Jz,e1kmk2zJz,e

melf

1meh2ekm

1x

kzJz,e1kmk2zJz,e

1x

mehe1meh2ekm

1

mehe1meh2ekm

mehelf1meh2ekm

kzJz,e1kmk2zJz,e

1x

(1meh2

ek2

mk2

zJz,e)km(1meh2ekm)(1kmk

2zJz,e)

.

Therefore, it is clear that when 0 and/or x is zero,the control law (8) is not computable. Physically, when these

situations occur, some actuators saturate.

3.2 Saturation Management

Saturations may occur when = 0,x = 0. In the followingwe propose a hybrid approach to the design of a procedure to

manage saturation in the actuators. Withja = 1we denote theAFS, with ja = 2 the RTV, and with ja = 3 the SAS. If thejtha actuator saturates,ja = 1, 2, 3, the corresponding control isuja =uja,max.

Considering the control requirements expressed by (7), we

assign higher priority to the fulfillment of those for z, andlower priority to those forx. The actuators that do not saturatewill ensure the fulfillment of the requirements with higher

priority. To show the procedure in general, let

xi = di, i= 1, , m (10)

be the set of control requirements, generalizing (7), orderedaccording to the priority (first = higher priority, last = lower

priority), and let

Cu= C

u1...

um

=

r1...

rm

=R (11)

be the general equation for determining the control law satis-

fying (10). We partition u into a vector of nonsaturating andsaturating inputs

u=

unsus

, uns R

mns, us Rms, mns+ms = m.

We operate the corresponding partition on C, so that (11)becomes

Cns Cs

unsus

=R.

Finally, to select the firstmnscontrol requirementsCns Cs

=

Imnsmns 0mnsms

Cns Cs

R=

Imnsmns 0mnsms

R

so that one calculates the nonsaturating controls

uns= C1ns

R Csus

.

To illustrate the proposed saturation management, let us con-sider some examples in the case under study. Let us assume that

the SAS actuator, calculated according to (8), saturates. Hence,

Fy,f

Mz

=

kmkzhe+

lfJz,e

1Jz,e

1me

+kmkzhelf

kmkzhe

1

r1r2

bx,max

kmkzxkmhex

.

Note that if the saturation is due to the fact that x = 0, thiscontrol law reduces

Fy,f

Mz

=

kmkzhe+

lfJz,e

1Jz,e

1me

+kmkzhelf

kmkzhe

1

r1r2

whilebx= bx,max.

Analogously, if both SAS and AFS saturate, one has

Mz =Jz,e

r1 Fy,f,max

kmkzhe+

lfJz,e

bx,maxkmkzx

and if the saturations are due to= 0,x= 0, this control lawreduces to

Fy,f =Fy,f,max, Mz =Jz,er1, bx= bx,max.

4. SIMULATION RESULTS

In this section, we provide simulation results of the proposed

control technique. We consider two simulation sets. First, we

compare the proposed integrated solution with a linearizing

feedback controller (Borri [2007], Bianchi [2008]) based on a

lower number of actuators. Second, we show the improvement

ensured by the hybrid saturation management scheme with

respect to a classical fixed saturation scheme.

The parameters of the vehicle are

m= 1550 kg ms= 150 kg = 1

lf = 1.17 m lr = 1.43 m hd= 0.5 m

Jz = 2300 kg m2 Jx = 350 kg m2 Jzx = 50 kg m2

Ayf = 1.81 Byf = 7.2 Cyf= 8854

Ayr = 1.68 Byr = 11 Cyr = 8394

kx= 150, 000 Nm/rad bx,0= 7, 000 Nm rad/s

f =0.05 r = 0.05

while for the reference generation we have considered the

following set of values

Jz,ref=Jz ref= f,ref= 0.08

r,ref= 0.04 kx,ref=kx bx,ref= 11800.

The control inputs are restricted to the following intervals (see

Spelta [2010])

Fy,f[0.95Cyf Fy,f,0, 0.95Cyf Fy,f,0]

Mz [10000, 10000], bx [2500, 35000].

4.1 Comparison with other controllers

The first test maneuver is a step steer of120

with longitudinalvelocity of28m/s (100.8 km/h). It is useful to compare the per-formance of the controller with the AFS, RTV, SAS actuators,


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defined in Section 3.1, with a state feedback linearizing control

law with two actuators (AFS and RTV), as defined in Borri

[2007], Bianchi [2008], designed to track lateral velocity and

yaw rate. Thanks to the presence of the SAS, whose primary

goal is the reduction of the oscillating behavior, the proposed

controller obviously performs better results for the roll track-

ing, namely x, x track more closely the references x,ref,x,ref(see Figures 3, 4), as expected. Here the reference modelhas an increased value of the suspension roll damping (i.e.

bx,ref> bx,0). Moreover, the additional actuator influences thebehavior of the other states and inputs, as shown in Figures 1,

2. In particular, one can note an improved tracking of the lateral

velocity (Figure 2), while the yaw rate behavior is similar in

both cases (Figure 1), and is nonzero because of the AFS

saturation (Figure 5). An analysis of the control inputs also

shows higher values of RTV action in the presence of three

actuators.

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

30

35

(a)

Fig. 1. (a) z,ref(dashdot), zwith 3 actuators (solid), zwith2 actuators (dashed) [deg/s] vs time [s]

0 0.5 1 1.5 2 2.5 31.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

0.2(a)

Fig. 2. (a)vy,ref (dashdot), vy with 3 actuators (solid), vy with2 actuators (dashed) [m/s] vs time [s]

0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06(a)

0 0.5 1 1.5 2 2.5 38

6

4

2

0

2

4

6

8

10x 10

3 (b)

Fig. 3. (a) x,ref (dashdot), x with 3 actuators (solid), xwith 2 actuators (dashed) [deg/s] vs time [s]; (b) ex =x,refx with 3 actuators (solid),ex = x,refxwith 2 actuators (dashed) [deg/s] vs time [s]

4.2 Performance of the Saturation Management

In this section we compare the proposed controller Csm, withsaturation management, and a controller Cw, in which the

saturation is treated, as done usually in applications, simply bysaturating the input value, without a change of priority in the

control specifications.

The maneuver considered, shown in Figure 10.d, is a double

step steer of 120 with longitudinal velocity of33 m/s (118km/h). Figures 86 show the evolution of the state variables

with the two controllers. The algorithm of saturation man-

agement gives tracking priority to z , as shown in Figure 6:while the error withCw grows immediately, as consequence ofthe AFS saturation shown in Figure 11.a, the controller Csmsucceeds to reduce the errorez . This improvement is achievedthanks to the RTV compensation. Clearly, this is paid with a

larger error on lateral velocity tracking (see Fig. 7), because ofthe priority scheme. However, with Csm and with the priorityon the yaw rate tracking, the handling of the vehicle has been


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0 0.5 1 1.5 2 2.5 30.05

0

0.05

0.1

0.15

0.2

0.25

0.3(a)

0 0.5 1 1.5 2 2.5 30.08

0.06

0.04

0.02

0

0.02

0.04

0.06

0.08(b)

Fig.4. (a) x,ref(dashdot), xwith 3 actuators (solid), xwith2 actuators (dashed) [m/s] vs time [s]; (b)ex =x,refx with 3 actuators (solid), ex = x,ref x with 2actuators (dashed) [m/s] vs time [s]

improved, as testified by the spatial trajectory (see Fig. 12),

although this is notone of the control goals. In fact, an enhanced

yaw tracking improves the agility and cornering capabilities of

the vehicle. The roll tracking is displayed in Figures 9, 8, with a

performance that can be considered comparable in both cases.

Figure 10.a shows the total force of the front axle, Figure 10.b

the yaw moment, and Figure 10.c the damping coefficient vari-

ation. The control anglecdue to AFS is shown in Figure 10.d,which results to be lower, in absolute value, with Csm thanwith Cw, because most of work in tracking z is due to theadditional RTV action. This does not mean that the front axle

is not exploited at its maximum extent, since the AFS soon

saturates in both cases (see Fig. 11). We also note that the

SAS system saturates very soon after the AFS saturation, in

both cases, meaning that this actuator is not able to compensate

the lack in control action, due to the saturation of the other

actuators.

In general, we can conclude that the proposed algorithm for the

saturation management is able to achieve a better and deeper

0 0.5 1 1.5 2 2.5 30

5000

10000(a)

0 0.5 1 1.5 2 2.5 32000

1000

0

1000

(b)

0 0.5 1 1.5 2 2.5 31

0

1

2x 10

4 (c)

Fig. 5. (a) Fy,fwith 3 actuators (solid), Fy,fwith 2 actuators(dashed) [N] vs time [s]; (b) Mz with 3 actuators (solid),Mzwith 2 actuators (dashed) [Nm] vs time [s]; (c)bxwith3 actuators (solid), bx with 2 actuators (dashed) [Nmrad/s]vs time [s]

utilization of all the actuators in saturation conditions, resulting

in an improved handling and cornering behavior.

CONCLUSIONS

In this paper, we have analyzed a hybrid management of the

control saturations, in the attitude control of a vehicle which

considers also the roll dynamics. A hybrid control of three actu-

ators allowed solving a tracking control problem and managing

critical conditions, when multiple actuator saturations occur.Simulations results show the improvement of the proposed so-

lution with respect to existing controllers and simpler saturation

schemes.

This work suggests a number of future directions of research.

From the theoretical point of view, the hybrid controller poses

problems regarding the stability in presence of discontinuity

of the control action. Additionally, it would be interesting

to extend the simulation setup to more complex simulation

environments (e.g. Carsim), to show the robustness of our

approach in presence of unmodelled dynamics.

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0 1 2 3 4 5 650

40

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20

10

0

10

20

30

40(a)

0 1 2 3 4 5 610

8

6

4

2

0

2

4

6

8(b)

Fig. 6. (a)z,ref (dashdot), z with the controller Csm(solid),z with the controller Cw (dashed) [deg/s vs s]; (b) ez =z,ref zwith the controller Csm(solid),ez =z,refz with the controller Cw (dashed) [deg/s vs s]

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0 1 2 3 4 5 65

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2

3

4(a)

0 1 2 3 4 5 61.5

1

0.5

0

0.5

1

1.5

2

2.5(b)

Fig. 7. (a)vy,ref (dashdot),vy with the controller Csm(solid),vy with the controller Cw (dashed) [m/s vs s]; (b) evy =vy,refvywith the controller Csm(solid), evy =vy,refvywith the controller Cw(dashed) [m/s vs s]

0 1 2 3 4 5 60.1

0.08

0.06

0.04

0.02

0

0.02

0.04

0.06

0.08(a)

Fig. 8.x,ref(dashdot), xwith the controller Csm(solid), xwith the controller Cw(dashed) [deg/s vs s]


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0 1 2 3 4 5 60.6

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3(a)

Fig. 9.x,ref (dashdot), xwith the controller Csm(solid),x

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0 1 2 3 4 5 61

0.5

0

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1x 10

4 (a)

0 1 2 3 4 5 61

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0

0.5

1x 10

4 (b)

0 1 2 3 4 5 62

0

2

4x 10

4 (c)

0 1 2 3 4 5 615

10

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0

5

10

15(d)

Fig. 10. (a)Fy,f with the controller Csm (solid),Fy,fwith thecontroller Cw(dashed) [N vs s]; (b)Mzwith the controllerCsm(solid), Mzwith the controller Cw(dashed) [Nm vs s];(c) bxwith the controller Csm(solid), bxwith the controllerCw(dashed) [N m rad/s vs s]; (d) cwith the controller Csm(solid),c with the controller Cw (dashed),d (dashdot)[deg vs s]

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0 1 2 3 4 5 6

0

0.5

1

(a)

0 1 2 3 4 5 6

0

0.5

1

(b)

0 1 2 3 4 5 6

0

0.5

1

(c)

Fig. 11. Actuator saturation (1) or out of saturation (0) vstime [s], in the case of the controller Csm(solid) or withwith the controller Cw (stars): (a)Fy,f; (b)Mz; (c)bx

0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

30

35

Trajectory x (m)

Trajectoryy(m)

Fig. 12. Trajectory in the plane: reference (dashdot), trajectorywith the controller Csm (solid), trajectory with the con-troller Cw(dashed)

art. bianchi

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