A Method of Automatic Motion Control with Optimization

IFAC

Copyright (' IF AC Intelligent Autonomous Vehicles. Sapporo. Japan. 2001

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A METHOD OF AUTOMATIC MOTION CONTROL WITH OPTIMIZATION Katsumi Moriwaki The Uni\·ersit~· of Shiga Prefecture. School of Ellgilleering. Department of :\Iechanical Engilleering. Hikone. JAr..\:\ Abstract: Steerillg a car bv hand mealls that the driver plans a path b~' previe,,' alld controls the lateral deviatioll of the vehide from the planned pat h b~' the stPering wheel. III all automatic car steering svstPIIl. this path follmYing is automated. The deviation is kept small b~' feedback control via the stE'f'ring motors. The reference trajectorY may be cakulated from the data of a CCD camera and the compensation scheme will be derived b~' the data of a g~TO or a GPS . The desired system behavior in the car steering motion is primarily to obtain good damping and an almost disturbance decoupling property. A certain stabilit~· margin should be satisfied and the actuator actiyit.', should not be too high. This desired system behayior is to be made precise b~' formulating performance criteria. In the design process of controller. a design parameter is chosen for the optimization process. which results in good damping. our primar~' objectiw is to approximatel~' keep this damping. At the same time the lateral acceleration should be better attenuated in the representative response due to a ~'a\V rate initial value disturbance. In order to stud~' automatic car steering, the steering model must be extended. For the extended steering model with output equation whose elements can be measured bv equipped sensors, the optimal regulator is introduced so that the whic:le can tracks the reference path. Copyright:hi 2001 IFAC Ke.vwords: Autonomous vehicles. :\avigation. Optimal control. Servo State-space models

1

s~·stems.

Introduction

:\Iotion control of autonomous yehides ha.<; many problems to be considered according to the freedom of motion . Figure 1 shows the d~'namics model of a four "'heel whic:le \vhich is assumed to be a rigid bod~·. It has six degrees of freedom (Fl) up and dmnl motion along to a .::-axis. (F2) lateral motion along to a y-axis. (F3) longitudinal motion along to an .r-axis. (F 4) rolling motion around an .r-axis. (F5) pitching motion around a y-axis . and (FG) ~'a"'ing motion around a .::-axis. Among them . in order to consider the automatic steering system. It \vill be derived the mathematicalmodel "'ith three degrees of freedom i.e. (F2) lateral motion along to a y-axis. (F3) longitudinal motion along to an .r-axis. and (F6) va\\'ing motion around a :-axis. because it is the most important s~'stel1l for motion control of autonomous whicles. Steering a car b~' hand means that the driwr

,

Figure 1:

393

Rigid-bod~'

model of four "'heel vehides

y

o

Figure 2: Four wheel model for car steering

Figure 3: T,vo wheel model for car steering

plans a path by preview and controls the lateral deviation of the vehicle from the planned path by the steering wheel. In an automatic steering system of autonomous vehicles, this path following is automated. The lateral deviation from the reference path is kept small by feedback control via the steering motors. The reference trajectory may be calculated from the data of a TV camera or a CCD camera. In order to study automation of car steering, the steering model must be extended. The extended model must include not only velocities, but also the vehicle heading and lateral position of the sensor with respect to the reference path. For simplicity this extended model will only be derived using the linearized model of a nonlinear model that is valid for small deviations from a stationary circular path. In the next section (Section 2) we derive the extended linear steering model from the nonlinear model of four wheel vehicles, and in section 3 the optimal control is considered for reference path tracking. A simulation example is shown in section 4.

(resp. rear axle) is If (resp. Ir) and together I = 1/ + Ir is the wheel base. In the horizontal plane of Figure 2 an inertially fixed coordinate system (X . Y) is shown together with a vehicle fixed coordinate system (x, y) that is rotated by a " yawangle" 'If'. In the dynamic equations the yaw rate r := 0 will appear as a state variable. Assuming that 8 f = Pfl = Pf2 = 13 + Ifr/V - 6, Pr = Pr] = f3r2 = P - Irr/V, and 1.8fl « 1, lf3rl « 1, 161 « 1, then the " two wheel model" (Ellis, 1969) (Figure 3) can be regarded as the equivalent model to the four wheel model (Figure 2). The side forces if := 2Yf' ir := 2Yr are projected through the steering angle into chassis coodinate (x, y), where they appear as forces ix , i y and the torque m z around a z-axis which is pointing upward from the center of gravity (P).

2

Steering Model of tonomous Vehicles

(1 )

Via the d~'namics model the forces cause state \'ariables 13. V, r. The equations of motions for three degrees of freedom in t he horizontal plane are

Au-

The features of car steering dynamics in a horizontal plane are described b~' Figure 2 (Abe. 1992). In Figure 2. the angle 6 is the front steering angle and the angles !3fl. 3/ 2 . Br] . i3r2 are the sideslip angles of front tires and rear tires, respecti\'el~·. The angle J between the \'ehicle cent er line and the \'elocit~· vector V is called ,. \'ehicle sideslip angle". The cornering forces }f]' Yf2. }';.]. }~2 are the forces transmitted from the road surface \'ia the wheels to the car chassis. The distance between the cent er of gra\'ity (P) and the front axle

1. longitudinal motion

- mV(3+r)sin3+mf" cos3=ix(=O) (2) 2. lateral motion mF(3 + r) cos3 + mV sin J = 3.

394

~'aw

motion

iy

(3)

(4)

~I

It is obtaind that from (2) to (4)

(5)

The side force fy is known to be the nonlinear function of the tire sideslip angles ;3f. 3r. i.e .. (6)

Two wheel model (5) and (6) is nonlinear and can be linearized by additional assumptions (Ackermann. et al .. 1993). (AI) The sideslip angle is assumed to be small. Then, (5) becomes

[

mV(J.; mV r)

1 [1 01[f ] =

Ir

(3 0

0 1

(A2) The velocit~, is constant. it second row of (7) is eliminated.

y

Figure 4: Scheme of automatic vehicle steering The uncert.ain parameters in this model are mass m, moment of inertia I. velocity V and road friction coefficient Ji. Solving (10) for /3 and i' and rearranging terms yields the linear state space model

( 7)

m=

= O.

Then, the ( 11)

where

(8)

The velocity V is treated as an uncertain constant parameter. (A3) The nonlinear characteristic of (6) is approximat.ed by the tangent at {}f = .Br = O. i.e ..

b1 --

(9)

in '= •

The constant coefficients cf. cr are called .. cornering stiffness". and Ji is the adhesion coefficient bet\veen road surface and tire. Typical values of J1 are (Ackermann. et al.. 1993) 1 0.5 0.15

Ji Ji Ji

3

dr~'

road wet road ice.

!!! /-L'

Optimized Servo-Controllers for Automatic Vehicle Steering

In order to consider the problem of automatic car steering. the extended model of vehicle is introduced. The extended model must include not onl~' velocities, but also the Yehicle heading and the lateral position of the sensor with respect to the reference path. This extended model is derived using a nonlinear model that is valid for deviations from a stationar~' path. It is assumed that the reference path is given as an arc with radius Rref and cent er AI (See Figure 4) (Ackermann. et al.. 1993). For a straight path segment the radius

The linearized model follows from (7) to (9) and using (1) as

][

.!:..L

-mF'

(10)

395

is Rref = :x::. It is more cOllYenient to introduce the curyature Pref := 1/ Rref as input that the generates the reference path. The curyature is defined positive for left cornering and negative for right cornering. The radial line from the center !If passing through the cent er of grayity (P) of the vehicle intersects a unique point Z!II on the desired path. It is assumed that there is a small deyiation from the reference point Z/Il to the center of grayity which is the deyiation Yp and that a yehicle fixed coordinate system (x. y) is rotated from the inertially fixed coordinate system (X. }') b~' the yaw anglet:'o The tangent to the reference path at Z/Il is rotated by a reference yaw angle Ct with respect to X. Thus. the rate of change of yp is given by F sin(iJ + 64') where (3 is the vehicle sideslip angle and 6(' :=(. -li't is the angle between the tangent to the reference path at Z!II and the cent er line of the yehicle. \Vith the linearizatioll sin(;3 + 61.0) :::::: (3 + 6l!.' the deviation yp changes according to

The s~'stem (15) , (16) with input J'Pref and output r. Ys is shmvll to be controllable and obseryable. \\'e can. therefore, construct the servocontroller so that the output [rT. ylJT can be driven to [(FPref)T.O]T as t -. OC. From (15). (16). the vehicle s~'stem is able to be rewritten as Aexe(t) Cexe

+ B el'(t)

(17)

where

13 r Xe

=

6rj, Ys

-15(12)

Pref

If the sensor S is mounted in a distance ls in front of the center of gravity \vith ls « Rref, the measured deviation Ys from the reference path changes both with. yp and under the influence of the yaw rate r = l!'. Taking this into account, the rate of change of the measured deviation is

Ys

= V(,B

+ 6t::·) + lS1'

Ae

(13)

Determination of Ys requires knowledge of three variables ;3. rand 6t::·. The variables ;3 and rare given by (11). The angle 6t· will be obtained by

Be

=

(14) Combining (11). (13) and (14). the extended state space model is obtained as (Ackermann, et al.. 1993)

0 0 0 0 1 0

a12

a21

a22

0 V 0 0

ls 0 0

0 0 0 0 0 1

1

0 0 0 V 0 0

b1 b2 0 0 0 0

0 0 0 0 0 0

0

1

0 0 -V 0 0 0

0

0] . Ce = [ 0 0 0 010 1 0 0

:\ow consider the problem of optimal regulator for (17) with the performance index to be minimized

Je =

[L1

=

all

fo X. {(Ye -

re)T (Ye - re)

+ v T R ev }dt

(18)

where

ifs

re

=[

V r:Jef ] . R e

= 12

The optimal regulator for (17), (18) is given b~ ' (Furuta. et al.. 1984) (15)

(19)

Using sensors for the .va\\' rate l' (ex. a g~TO) and the deyiation Ys (ex. a GPS) , it can be assumed that the output equation for (15) is given b~'

r - \ ' Pref Ys

396

2.54 0.97 1.57 1.83 25.000 25.000 15 1.170 0.7 1.341

where If Ir

Is

Z=

all

al2

021

an

0 F 0 0

1 Is 1 0

0

0 0 0 0 0

0

1

0 0 0

V

bl

cf er

&2

0 0

0

-\"

m

0 0 0

0 0 0

1-

\" fl ."

[m] [m] [m]

[m] [:\' / rad] [:\' / rad] [m /sec] [kg] [m 2 ]

Table 1: Specification data of a passenger car

and Pe is the positive definite solution of the following Riccati equation

~ 1l L1

Using the notation

o

Ys

Then, the input for (17) is given by

according to (15). Then. the optimal control input for (23), (16) is given by

6 ,J,

Ys

8(t) ] = _ [ 9.17 [ Pref -2.12

r - V Pref Ys Finall~',

(21 ) the control input for (15), (16) is given b.y

8(t) ] = [ Pref

-1\"1

l ~ ,.1 + 6L Ys

-0.029] -0.327

+

J{ 2 (

io

[

r ] dt

+ [9.17 3( 0) 1'(0) 6l'( 0)

dt

Ys

1

- 1.68 0.145

- 0.0337 -0.807

-0.029] - 0.327

l ~~~? 1 61i'(0)

Ys(O) (24) The step response for the s~"stem (23). (16) with the controller (24) is shown in Figure 5.

(22)

l Ys(O)

By using the control input [8 T P~ef ] T gi\'en by (22), where the state vector of (15) is replaced with the observer state vector for the system (15). (16) (~Ioriwaki and Akasi. 1993). the automatic steering along by the reference path can be achieved.

4

- 0.0337 - 0.807

xli:, 1[~09~:6 =~~~~ 1I,' [;, 1 - 2.12

" +/\1

-1.68 0.145

5

Conclusions

It has been considered the problem of automatic steering of autonomous vehicles. The extended model of the steering motion is derived by linearization of the nonlinear vehicle model. The optimal regulator for the extended steering s~'stem is introduced so that the autonomous vehicle can be dri\'ed along the reference path .

Numerical Simulation

For a typical passenger car with a data given in Table 1. the extended state space model is obtained as

397

FroM: Pref 1-"

. .. . ...

~

.

u

.

r

U'

M

U

U

Time (.ee.)

Figure 5: Step response for a vehicle with automatic steering

REFERENCES Abe, M. (1992). Vehicle Dynamics and Control, Saikai-do Press, Tokyo. Ackermann, J., A. Bartlett, D. Kaesbauer, W. Sienel and R. Steinhauser (1993). Robust Control Systems with Uncertain Physical Parameters, Springer-Verlag, New York. Ellis, J. R. (1969). Vehicle Dynamics, London Business Book ltd., London. Furuta, K., S. Kawaji, T. Mita and S. Hara (1984). Mechanical Systems Control, Ohm Press, Tokyo. Moriwaki, K. and H. Akashi (1993). The Dynamic Regulation of Linear Discrete-time Systems with Unknown Input (A Method using Adaptive State Observer), Proceedings of 12th [FAC World Congress, 883 -886.

398