Dynamics and Robust Control of Wheeled Mobile Robot

Copyright © IFAC Intelligent Autonomous Vehicles, Madrid, Spain, 1998

DYNAMICS AND ROBUST CONTROL OF WHEELED MOBILE ROBOT

Zenon Hendzel, Wieslaw Zylski

Rzeszow University o/Technology, fV. Pola 2, 35-959 Rzeszow, Poland

Abstract: In the paper a model of 3-wheel mobile robot is proposed. The model has been reduced to the model with two substitute wheels, one driven wheel and one steered wheel. The motion phenomenon is described using Lagrange equations with multipliers, which are equivalent to dry friction forces that act on wheels of the model. The control algorithm is based on the input-output linearization method and variable structure control approach. The major contribution of this paper lies in application of new approach to control wheeled mobile robot along desired trajectory in the present of parametric and nonparametric modelling inaccuracies. Copyright © 1998 IFAC Keywords: Mobile robots, dynamic models, robust control.

only partially discussed, using very simplified models. In this paper, a model of 3-wheel mobile robot is proposed, which has been reduced to the model with two substitute wheels, one driven wheel and one steered wheel. Taking account of equations of non-holonomic constraints, motion phenomenon is described using Lagrange equations with multipliers. The equations that have been obtained are presented in so called reduced form, and they provide description of the motion with nonholonomic constraints. These equations make possible to calculate values of driving and steering moments, as well as multipliers, which are equivalent to dry friction forces that act on wheels of the model. Dynamic analysis of the system, that is presented in this paper can be applied in description of motion in respect to any wheeled mobile robot model. In recent years, the problems concerning wheeled mobile robot control created enormous interest, although literature on this subject is very scattered. Among other works, the papers (Kawasaki et al., 1994; Oelen and Amerongen, 1994) should be included. In those works a possibility of variations in operation conditions of the analysed mobile robots have been to some extent taken into consideration. In this paper the problem concerning tracking

1. INTRODUCTION Mobile wheeled robot it is a vehicle which moves along a desired trajectory. This type of motion is known as tracking motion and problems concerning control systems for these vehicles belong to tracking control class. Such vehicles are used chiefly for transport in automated industrial processes. Dynamic analysis of wheeled mobile robots is carried out mainly in order to obtain proper solution of the problems concerning system motion control. For description of system dynamics a simple model is often assumed, in which masses of many movable parts have been neglected. In the work (Bloch et aI. , 1992) dynamic non-holonomic system control is analysed, assuming single-wheel model of a mobile robot. In the work (Jagannathan et aI. , 1994) 2wheel model of a mobile robot has been assumed, and in dynamic description of this system, nonholonomic constraints are considered. In the paper (Baharin and Green, 1991) problems of dynamics are discussed using Lagrange equations and searching for effective computational algorithms for these equations, that could be used, for example, in description of motion of a wheeled mobile robot. In all above mentioned works dynamic problems are

147

control problem of wheeled mobile robots is discussed for such system, whose phenomena of its motion is not described in full by its mathematical model. This subject was analysed by the authors in works (Hendzel, 1995, 1996, 1997; Hendzel and Zylski, 1995). For instance, vehicle frame mass moment of inertia changes depending on mass of the load or its distribution. The drag arising at wheel turning is also variable. It should be noted that large allowable load (up to 500[kg]) has significant effect on realisation of wheeled mobile robot follow-up motion control, what has been confirmed by experiments (Hendzel, 1996). To take into account variations in parameters of the mathematical models of controlled system, when developing relevant control algorithm, it is required to use complex control methods, such as robust control method or adaptive control method.

Equations (I) result from the assumption that point A velocity vector coincides with direction of the AF axis, and that motion of the system is free of any slips, circumferential or lateral, that is to say velocities of the points of contact between substitute wheels and the floor equal zero. Motion of the model can be describe using Lagrange (Bloch at al ., 1992)

~(8E)T _(OE)T =Q+JT(q)A dt 8q

where: q = [X A 'Y A, J3,ex,\jI,cpf vector of generalised co-ordinates; E = E(q,q)

The model which has been assumed for analysis of system dynamics (Zylski, 1996a, b) is shown in Fig. I ; the wheels 1 and 2 of the real system have been replaced by the one substitute wheel, which centre is localised at the point A. At that time angle of rotation of this wheel is defined by the angle \jI. Angle of rotation of the wheel 3 is designated as ex. In such model there are two wheels, lz and 2z. Owing to this substitute model it was possible that dynamic analysis of the system to be carried out. Any specific attitude of such model as describe above requires to determine the following coordinates x A, YA, J3, \jI , ex, cp. Since analysed system

(m I + m 2 + m 3 + m 5 )x A + + (ml -

m2)11(~cosJ3 - ~2 sin J3)+

+ (m31 + -C I XA

m5Iz)(~sinJ3 + ~2 cosJ3) =

+A 1 +A 3

(ml +m z +m 3 +m 5 )YA + +(ml -

mz)II(~sinJ3 + pZ cosJ3)+

(m 31 + m 51z )(~ cos J3 -C 1YA+A2+

~ z sin J3 ) =

A4

(m] - mZXx A cosJ3 + YA sinJ3)11 +

is non-holonomic, so these co-ordinates are connected with the following equations of constraints, that have been imposed on velocities

Z

2

2 ] ..

1 +m5 12J3+ + [( ml+mZI]+m3 ) +(m3 1 + m 5I z)(-xA sinJ3+YA cosJ3) +

XA - rl\jJcosJ3 = 0 YA - fl \jJsinJ3=O r3acos(J3 +cp) = 0

kinetic energy of the

system; Q vector of generalised forces; J(q) Jacobian; A vector of Lagrange multipliers, and in the analysed system they are equivalents of the forces of friction, that lay in the plane determined by the points of contact between wheels and the floor. After some operations have been carried out, motion of the analysed model can be described by the following system of differential equations (Zylski, 1996a)

2. DYNAMIC EQUATIONS OF MOTION OF THE SYSTEM

xA -1~sinJ3 -

(2)

8q

+ [lxI + Ixz + IX3 + I z5 + (IzI +

Izz)hn~ +

+ (IzI -I z2 )h l \ji + I x3 q> =

(1)

= Ms - Mo sgnq, - czq, - A31sinJ3 + A4 1cosJ3

YA + l~cosJ3 - r3asin(J3 +cp) = 0

I z30' = - N 3f3 - A3r cos(J3 + cp) - A4 r sin(J3 + cp ) (Izl + I z2 )\V +(Izl - IZ2)hl~ = MD - N]f] - " -N 2fz -AlrcosJ3-1' 2rsinJ3 I x3 (~ + q> ) = Ms - M 0 sgn q, - c 2 q, (3) where: m l ,m 2, m 3,m 5 are the masses of particular parts, m3 is the mass of the wheel 3 and steering

gear 6, lxI ' Ix2 ,Ix3 , I zl ,Izz , I z3 , I z5 are the moments of mass-inertia for particular parts in relation to the respective axes, and IX3 takes also account of the moment of mass-inertia for the steering gear 6 in Fig. I. The substitute model of mobile robot.

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relation to the

Z6

axis. N] , Nz , N 3 are the loads of

compensating feed-back and primary feedback. In synthesis of the robust control algorithm for tracking motion of wheeled mobile robot, linearization of the system of input-output type has been used as a part compensating non-linear elements of the nominal system. On the other hand, for stabilisation of follow-up error, that is caused-among the others by incomplete compensation for dynamic characteristics of the controlled system, theory of sliding motion in systems of variable structure will be applied. Therefor, descriptions of 3-wheel mobile robot motion, that has been presented in section 2 above, will be expressed in state space in the following form m x=f(x,a)+ ~gi(x, a)Uj = f(x, a)+ G(x,a)u (7)

particular wheels, f] = fz = f3 = f are rolling friction coefficients

for

respective

wheels,

coefficients of viscous friction,

Mn

c] , c z

are

is driving

moment of the lz wheel, Ms is steering moment of the 2z wheel, Mo is the moment of resistance to turn of the 2z wheel, I, I] , I z are respective geometric dimensions of the system and r] = rz = r3 = rare radii of the respective wheels. Equation (3) is not suitable for design the control algorithm, it is necessary to adopt a special transformation that deconjugates multipliers from the moments and provides so called reduced form of description for motion of the system with nonholonomic constraints. Following (You and Chen, 1993) decompose vector of co-ordinates q as follows: q = [q] , q z

r.

l~]

analysed system it has been assumed that:

.]T h were x = [ q2 ,Q2

f(x,a)=[

G(x,a)=[g],gz]=

and

4

a5 =Nl

a3 =I z

a6 =I'vIo

RID

remammg

j (9)

j

0,0 ] o,~

rM2Z (x)Bzz (x)

variables

will

be

equal

a

It can be assumed that estimated values of parameter vector are known. Then estimated values for the functions that are present in equation (5) will be

We will assume that parameters, which appear in dynamic equations of motion, can have a value that is contained within the given range la; ~ y; ,

Me(x) = Mzz(x)la =

a

a Fe (x) = F22(x)la = a Ce(x) = C 22 (x)la =

ad

a;

U E

x=(\jI ,
~ =(21im+I~~ +t2ll1g) +2Ix +Iz5

where

Rn IS . state vector,

: - M 2i(x)[C22 (X) + F22 (x)] + z(1)

(5)

The matrices M 2Z , Cn , F22 ,1312 appearing in equation (5), results from equation (3).Parameters a; are defined as follows a] =r2(~ +Iz~) a =r z(2m+llIg +~)

az =Ix3

E

is vector of control signals, and y is vector of the system output signal. For the analysed systems n=4, m=2. The vector f(x,a) and matrix G(x,a) will result from equation (5)

Dynamic equation of motion (3) for the analysed system, in so called reduced form that is suitable for synthesis of follow-up motion control algorithm, is as follows M Z2 (qz,a)qz +C n (qz,4z , a)qz + + Fzz(qz,a) + z(t) = B22 (qzX M n , Msr

(8)

Yj = hj(x), i = l , ... , m

In the

is estimated value of i-th parameter, and

y; it is limitation of changes in i-th parameter

(10)

Let auxiliary vector of the control signal has the following form

value. A vector of unknown but limited disturbances z( t ), has been introduced into equation (1) . This vector takes allowance for non-parametric inaccuracies in mO<1elling, assuming that

Iz;(t)I~Zi ' Z; >0, i=I , 2.

Description of the real controlled system differs from nominal one, obtained after relationships (10) and (11) have been taken into account. Therefore inaccuracies of modelling have been included in description (7), defining it as description with regard to parametric and non-parametric inaccuracies.

3. CONTROL ALGORITHM

From mathematical point of view, algorithms of follow-up control have structure consisted of

149

m x=f(x,a)+M(x,a)+ :L[gj(x,a)+dgj(x,a)]Uj = j=1

(16)

= f( x,a) + M( x,a) +[G(x,a) + dG( x,a)]u

(12) Yl =hl(x)=q~ Y2 = h2(x) =q~

where

(13)

where Inaccuracies, that have been defined in (15), (16) will be recorded in the fom} given in relationship

G(x,a) = [gl(x,a),g2(x,a)], dG(x,a) = [dg, (x,a), dg 2(x, a)]

M(x)

= G(x,a)D(x)

dG(X) = G(x,a)E(x) Inaccuracies of modelling will be expressed as follows M(x,a) = f(x,a) - f(x,a) + z(t) dgj(x,a) = gj(x,a) - gj(x,a)

where D(x) and E(x) have the following form D(x) = M;;'(x)[Ce(x) + Fe(x) - z(t)]-

(14)

- Mi~ (X)[C22 (x) + F22 (x)] E(x) = Mii(x)Me(x) - I

A principle of compensation for non-linear part of

the nominal system (12) has been obtained for inaccuracies ~f = 0, ~gj = 0, the compensation is

(18)

(19)

The successive step consist in determination of the control signals v(x), which should ensure that kinematics parameters of motion, \jI and
based on input-output type linearization. Following procedure described in works (Isidory, 1985; Nijmeijer and van der Schaft, 1991 ; Hedrick, 1993) and others, vector of the output signals (13) should be differentiated with respect to time up to the moment when coefficient at the control signal is different from zero. Now, the problem of follow-up motion control consist in determination such control input signal v(x) that output signal Yj should followup with given signal

(17)

[\it,q;r =D(x)-E(x)be(x)+[I+E{x)]v(x)

Yjd . In order to ensure

(20)

and vector be ( x) is defined as follows

insensibility of the control algorithm to modelling inaccuracies control signal v(x) can be determined with regard to sliding motion properties in systems of variable structure. The problems connected with sliding motion theory in respect to variable-structure systems have been comprehensively discussed in many works (Utkin, 1977; Sira-Ramirez, 1988; Slotine and Li, 1991 ;Elmali and Olgac, 1992;) and others. Following procedure described, parametric inaccuracies M(x) and ~G(x) , which occur in

For this purpose, sliding functions should be defined as

where "'11 ' '" 21 are constant coefficients. If following expression

relationship (14), can be defined as follows

o M(x) =

0 M~I(X)[ Ce(x) + Fe (x)] - Mi~(x) x

x

(24)

(15)

will be added to and substrated from relationship (18), and if following denotations

[C22 (x) + F22 (x)] - M;;'(x)z(t)

150

Ll = M22(X) - Me(x)

(26)

M2i(x)=M~I(X)(I+Llt

(27)

a)

e - 0e

1- wtr-a d 1- - - - - - - - - - - - - - - - . - - - ljJ[rad / s]

::j

:

" . ee -~ - Iji I r-a d -! ~s -2 ] - - - - .- - - - - -. - - : ; /

will be introduced, the relationship (I8) will be transfonned into the fonn (Hendzel, 1996, 1997)

;

'

:1 ____________ __

' =1~'=:;z/

~= il1y e . ee 1'~ ----

1

D(x) = M _I(X)[C e(X) + Fe(x) - z(t) (28) e -(I+Llrl(8+C e (x)+Fe(x))

1 -

6 .ee

Yr - be(x) -

o

-

-

-

F

-

-

-

-

-

_tJ~ L~

-

15.00

1Q . 00

2 0 . e~

~\ \

/

"0~'{H' " ' . (

0ee

- 0.20

~

.

.

}\~

. . . .. .

nr · :

- - - --- -- --~

.

-

f

0.13121

i"

I

.- .

f

s _cae

- I

7

:

7

'21.(2)121

,

-

I

i"

-

15.012

l'

t[ s 1 f'

r

2~.0e

Fig.2. Desired trajectory. For simulation, following parameters have been taken

values

of

design

k(x)sgn(S)] (30)

/"11

where Yr = [A.lICl'A.21C2f · On the other hand, driving and steering motor torque values have been detennined from the following relationship

[:::]{~ :.1::]

,

c . e0

~ q> [ra d) ~ q> [ra d ' s )

(29)

gain k j (x) > 0, i = 1,2 , under which sliding motion takes place, have been determined (Hendzel, 1996, 1997). Finally, control signals have been detennined

B2~Me(x)[y d -

....•..........~....

'" " 'I "'l - - - - - - - - - - - - - - - - - - - - - - - --

Vector 8 and matrix Ll can be determined basing on the system of equations (27), (28), (29) and using equations (10) and (5), in the function of parameter limits Yi and then coefficients of sliding controller

= [ MU] Ms

___ _

b)

whereas relationship (19) can be written as

E(x)=M~I(x)(I+LltMe(x)-I

. . / O.lIJl '

L

=6

v A = 0.5 <1>1 = 0.04

1.21 = 6 is = 120 <1>2 = 0.04

in = 49 r=0.15

Because continuous control has been introduced into the control algorithm (31), so follow-up errors, should meet following conditions (Hendzel, 1996) lell~O.0066[rad] , le21~0.0066[rad]. For

(31)

simulation, non-parametric inaccuracies have been taken as signals generated by random - number generator, of zero average values and the following standard deviation 1'\ = 4.2 [Nm]

where in is reduction ratio in the driving unit, and is is reduction ratio in wheel 2z turning system. Knowledge of these torques allows to detennine currents that are applied for controlling the electric motors used in real systems.

8 z2 = 14.65 [Nm]. The aim of this test was to verify if proposed control algorithm is susceptible to nonparametric disturbances, such as, for instance, measuring noise or error resulted from numerical differentiation of non-measurable signals (for instance
4. EXAMPLE

Verification of the control algorithm (31) will be carried out for three periods of mobile robot motion, such as: starting period, travel at constant speed of point A, when point H is moving along straight-line trajectory, and turning-round at constant speed of point A, when point H is moving along circular trajectory. In fig.2a,b changes in angular parameters have been shown, taking they as the magnitudes defining desired trajectory for follow-up motion control. It has been assumed that wheeled mobile robot carries unknown mass Llm 5 .

disturbance

signal

limitations

ZI = 5 [Nm] ,

Z2 = 15 [Nm] . In fig.3a,b,c selected results of computer simulation are shown, in respect to travel of the mobile robot along desired trajectory of motion, with initial load ~5 = 0 [kg], and then, during 7[s], takes place sudden change in loading, up to ~5 = 500 [kg].

151

b)

e . 121 1 e

T - - - :j~ [rad]

-I -

-

-

-

~ -

-

,

Hedrick, 1.K. (1993). Analysis and control of nonlinear systems. Journal of Dynamic Systems, Measurement and Control, Vol. 115, pp. 351361. Hendzel, Z. (1995). Robustness problem in control of wheeled mobile robot. Journal of Theoretical and Applied Mechanics, Vol. 33 , No. I, pp. 157169. Hendzel, Z. and W . Zylski (1995). Robust motion control of wheeled mobile robots with nonholonomical constraints. CAE-Techniques, Proceedings of 2nd International Scientific Colloquium, Bielefeld, pp. 287-298. Hendzel, Z. (1996). Tracking control of wheeled mobile robots. Edit. Office of Rzeszow University of Technology, Rzeszow, (in Polish). Hendzel, Z. (1997). Robust tracking control of wheeled mobile robot. The Archive of Mechanical Ingineering. Vol. XLIV, Z. I , pp. 43-62 . Isidory, A (1985). Non-linear Control Systems: An Introduction. Spriner-Verlag, Berlin. Jagannathan, S., S.Q Zhu and F.L. Lewis (1994). Path planning and control of a mobile base with nonholonomoc constraints. Robotica, Vol.l2, pp. 529-539. Kawasaki, Y , Z. Iwai and S. Haramaki (1994). Decoupling and robust adaptive control of omnidirectional, automated guide vehicle based on nonguided line navigation system. ISME International Journal. Vol. 37, No. I , pp. 107114. Nijmeijer, H. and AJ. van der Schaft (1991). Nonlinear Dynamical Control Systems. SpringerVerlag, New York. Oelen, W. and 1. Amerongen (1994). Robust tracking control of two-degrees-of-freedom mobile robots. Control Engineering Practice, Vol. 2, No. 2, pp. 333-340. Sira-Ramirez, H. (1988). Differential geometric methods in variable-structure control. International Journal Control, Vol. 48, No. 4, pp. 1359-1390. Slotine, J.J. and W. Li (1991). Applied non-linear control, Prentice Hall, New Jersey. Utkin, I. U. (1977). Variable system with sliding modes. IEEE Transactions on Automatic Control, Vol. 22, No. 2, pp. 212-222 . You, L. and B. Chen (1993). Tracking control designs for both holonomic and nonholonomic constrained mechanical systems: a unified viewpoint. International Journal Control, Vol. 58, No. 3, pp. 587-612. Zylski, W. (1996a). Kinematics and dynamics of wheeled mobile robots. Edit. Office of Rzeszow University of Technology, Rzeszow, (in Polish). Zylski, W. (1996b). Dynamics of wheeled mobile robots. The Archive of Mechanical Ingineering. Vol. XLIV, Z. 1, pp.351-364.

___ __ _ __ " - !

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t

'

i

I

e. ees:r - - - - -, - - -

- - I

- - - - -, - - - - - ",

:

I i ~

" "' ' ' 'ij\Jr~:~L ~I- ~ -----:- ----~ , 1

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- 121.1211216

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- 1-

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- 1!

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-'!-,.~~~h-.,.-~~-++~,, ~,~~,,+1~~..;L!IS?r-]I

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Fig.3. Graphs of control errors and control signal. 5. CONCLUSION The proposed algorithm for 3-wheel mobile robot motion control has been designed using method for input/output system linearisation, and properties of sliding motion in a system of variable structure. Basin on numerical analysis, it has been demonstrated that the tested control algorithm is insensitive to assumed inaccuracies in modelling. The results obtained have been confirmed by experimental tests (Hendzel, 1996, 1997).

REFERENCES Baharin,I.B. and RJ. Green (1991). Computationally-effective recursive lagrangian formulation of manipulation dynamics, International Journal Control, Vol. 54, No, I, pp, 195-214. Bloch, AM., M. Reyhanoglu, and N.H. McClamroch (1992). Control and stabilization of nonholonomic dynamic systems, IEEE Transactions on Automatic Control, Vol. 37, No. 11 , 1992, pp. 1746-1756. Elmali, H. and N. Olgac (1992). Robust output tracking control of nonlinear MIMO systems via sliding mode technique, Automatica, Vo1.28, No.l , pp. 145-151.

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