Sliding Mode Tracking Control for a Single-Link Flexible Robot Arm

Copyright ® IFAC Robot Control, Vienna, Austria, 2000

SLIDING MODE TRACKING CONTROL FOR A SINGLE-LINK FLEXIBLE ROBOT ARM

Dadi Hisseine and Boris Lohmann

University of Bremen, Institute ofAutomation Technology Kufsteiner Strasse, FBI / NWI, 28359 Bremen, Germany e-mail: [email protected]@iat.uni-bremen.de

Abstract: In this paper two different sliding mode tracking approaches for controlling single link-flexible manipulators are presented. On the basis of an approximated finitedimensional nonlinear flexible model, the first robust controller is derived using classical sliding mode techniques and the second control strategy using continuous sliding mode approach . The performances of the proposed control law are demonstrated by simulations and experiments carried out with the single-link flexible robot arm of our Institute. Another new aspect of this article consists of the extension of the continuous sliding mode approach to tracking problems and the practical implementation of the developed control concepts. Copyright @ 2000 IFAC Keywords: Sliding mode control, Robust control, Nonlinear control, Robot control, Flexible arms.

1. INTRODUCTION Trajectory control of flexible-link robot manipulators is a challenging problem. Due to the inherent unstable zero dynamics behavior of the end-effector position of flexible-link robots, the simultaneous achievement of high level performance and robustness is not straightforward. In order to be able to counteract the undesirable effects of flexibility and to avoid the non-minimum phase obstacle, advanced robot control techniques should be investigated on the basis of a more complete dynamic. For control design purposes, different approaches exist for the of dynamics of flexible-link description manipulators. Among others, exact solution approaches (Fliess, et al.,1997) which deal with the partial differential equations of system dynamics, are available as well as approximation approaches such for example Lagrangian-assumed modes method (Book, 1984). A variety of control techniques have been used by different publications to control lightweight flexible manipulators. Among such methods, one finds singular perturbation (Siciliano and Book, 1988), feedback linearization (lsidori, 1995), nonlinear regulation ( De Luca, 1998) as

many other eXlstmg tools for solving trajectory tracking problems in nonlinear systems. Based on the state space formulation, the sliding mode control (Utkin, 1992), also called variable structure control, is a nonlinear robust controller design method. Outstanding properties of sliding mode control such as robustness to modeling uncertainties and disturbances are desirable by robot manipulators, since the complexity of the manipulator dynamics makes the exact knowledge of the dynamics infeasible, as well as in rejecting effects due to Coulomb and viscous friction. Unlike distinguishing properties, the use of switched control encounter the drawback of chattering phenomenon. In this paper, two non linear robust approaches using different sliding mode design methods are presented. First, a nonlinear robust controller is presented, which is derived using classical sliding mode techniques. In order to overcome the undesirable control chattering due to the high frequency switching, here the control action is smoothed according to the concept of "boundary layer"

333

Using Lagrange's Formulation or Hamilton's Principle, we obtain a fourth-order partial differential equation:

(Slotine, 1984) which approximates the ideal relais characteristics to satured amplifier characteristics. Second, a continuous sliding mode tracking controller is proposed. This second proposed control strategy satisfies the sliding condition using a continuous sliding mode approach (Zhou and Fisher, 1992) which avoids the discontinuity in the classical sliding mode methods for counteracting the chattering problem and for simultaneously improving the robustness to modeling errors and disturbances. In order to achieve perfect tracking while maintaining robust closed-loop performance, the flexible robot system under consideration should be minimum phase. Two alternative system outputs (tip position or joint variable) can be considered for trajectory tracking. The minimum phase behavior may be achieved for example by output redefinition (Madhavan and Singh, 1991). By a suitable choice of the sliding surface (naturally appropriate for the respective design approach), it is shown here that the tracking problem can be sol ved by considering the joint variable as the minimum phase system output. Simulations studies and experiments (with a considerable amount of parametric uncertainties) on the link-flexible robot arm of our Institute are demonstrated the tracking features and the robustness of the designed controllers. The new aspect of this article consists of the extension of the sliding mode concept under consideration to tracking problems and the practical implementation of the designed controllers to a lightweight flexible robot.

(2) with 4 boundary conditions as given 1999) .

III

(Hisseine,

This partial differential equation of the system dynamics can be either directly solved (exact solution approach such in (F1iess, 1997)) or using approximation approaches such as assumed modes methods (Book, 1984). In the following, the flexiblelink manipulator is modeled by an approximation approach using assumed mode methods:

w(x'!) = fq1j (x) .SJt ):= q/ ·0

(3)

i=l

By consideration of a finite number m of modal terms, the system dynamics are derived using Lagrange Formulation (Book, 1984):

2. MODELING The single link flexible robot arm under consideration, presented in figure 1, is modelled as an Euler-Bernoulli beam. The motion occurs only in the horizontal plane and the arm does not undergo torsional deformations.

where

M(6)

matriX,

e

the positive definite symmetric inertia

the joint variable, 6

= [SI ...

SJT

the

vector of modal amplitudes, Co and Ca coriolis and centrifugal terms respectively, F the structural damping matrix, K the stiffness matrix and u the input torque at the joint. The angular tip position is . gIven by where

~~ ·0 e =e + -,

(I),

,ip

L

(5)

=~(x=L) .

3. ROBUST NONLINEAR CONTROL USING A CLASSICAL SLIDING MODE APPROACH

3.1 Preliminary remarks Let a dynamical system be given in the form:

x:= f(x)+ g(x).u +d(t), x,

with the assumption: bounded modeling uncertainties in f(x), g(x) and unknown (bounded) disturbances

t

Fig . 1: Flexible-link robot arm

d(t). The sliding surface is given and denoted by

The motion of a point along the beam, is given by

y(x,t)= x·e(t)+ w(x,t).

(6)

s:=

(1)

334

s(x).

(7)

So, the sliding mode controller has the following general structure:

where

(8) where u eq (x) is the so-called equivalent control, which stabilizes the nominal system and u, (x) is the discontinuous

flIljB,8,~,6)= -hll (~)min . Co (8,~,6)~

robust control component, which

fulfils the sliding condition S5<0.

+H;2(~)max ·L(8,~)max

(9)

Remark: Finite reaching time is guaranteed by modifying the sliding condition (9) to

, s5<-1]·lsl,1]>0 .

+Kmax ·o+Fmax

.6]'

j(B,8,~,6)= -~. (~) . c8 (8,~,6)+ - 8;z(0)' ~o (8,~)+ K' 0 + F·

(10)

6]'

and represents the width of the boundary layer. Note that the symbol (A.) signifies the nominal value. So, the control law can be now given as follows:

3.2 Nonlinear robust controller design The sliding mode control forces the system trajectory to reach and stay on the sliding surface.

(16)

Let e be the tracking error: e = B- Bd . We select here the sliding surface according to the approach in (Yeung and Chen, 1990):

s = e+ f vdt,

-

By consideration of a two-mode flexible robot , the sliding surface (12) can be modified according to

e

s = + c.e + PllO. + P120Z +

(11)

+ f[cze+ pz/5. + PzzOz +c, 'e/ where v =c.e +c 2e +P.~ +P2~+ c, ·e/, and

e/ = f e,ipdt = f(B ,iP -Bd}tit.

e/ = f e,ipdt = f(B-B d +

where

ldt'

(17)

±if'i(L)8i rtt. L

i=.

By assuming small deflections (so neglecting the expressions c 8 and Co in (4», the constant

In order to compensate error deviations (for more details, see (Yeung and Chen, 1990», the additional integral term e I is introduced.

coefficients c.' c 2 ' PII' P.2' P2l' P22 and c, in (18) can be determined by means of following approach:

From 5 = 0 , we obtain the equivalent control:

[ (13)

M~"

M 22 ,11

M 22.12

M 22.Z.

M 22.22

0

0

0

Cz

0 + 0

(14) In order to obtain smooth control action, a boundary layer (Slotine, 1984) is introduced around the sliding surface and from that, the robust control part (14) is modified to:

-1

(15)

335

~I J.l

0

M. 2•Z

+[~

Starting from the reaching condition (10), we derive the discontinuous robust control component as follows:

0

PII

PI2

FI,)

0

0

F 2. 2

0

0

..

o O2 o e/

o ~. o O2

Tl 1

P2.

P2Z

K •.1 0

0

c,

K 2.2

-~ _ if'ze

0

L

+

(18)

e/

0 0

L

+

:}o

O2 e/

For the following, we select (for the nominal case) suitable eingevalues (pole placement) as:

The first row of above equation is resulted through the following sliding condition .

.

s = ii + c1e + P"OI + P1202 + c 2e + + P2A + P22 02 +CI 'e,

=

°

(A, ,A2'~' A4 ,As ,A6'~ ) =

(19)

(22)

(- 3.2, - 5,- 3, - 4, - 40, - 45,- 3.5)

Note that the second and third equation within (18) do not precisely describe the system dynamics (see system equation (4)) due to the term Bd it 0; from there, the occuring error deviations have to be compensated by the use of the additional integral component in the last row of (18).

From this, the following design parameters are estimated:

The characteristic equation of (18) is given by the following expression:

For more interpretations, see the next section for experimental results.

(cl 'C2' P", P12. P21, P22, Cl) = (8.00,14.44,0,99,16.66,313.47, - 3836.88,10.23) (23)

(20) 3.3 Experimental results

where

l

A? +CIA+C2]

v

= 1

M A2 12.1 M A2

V

'2

=

The physical parameters of the considered flexible robot system are: L = 1.155 m , pA = 2660 x 3.210-4 Kg Im ,

p"A + P21 M 22,,,A2 + FI,IA + KI,I

El = 710 10 x1.71 10-9 Nm 2 , 1m = 3.210- 3 Kgm 2 ,

M 2Z,21A2 _ fP1• L

12.2

-1

•

•

ML =5Kg and 1L =7.9610

-3

2

Kgm .

We choose the desired reference trajectory as an

V3

=

A PI2 + P22 M 22,12A2 2 M 22,22 A + F2,2 A + K 2,2

[~ and v 4 = ~

_ fP2e L

°

1

angular motion from 8 d (0) =

.

the velocity profile

Cl

P"

aT(A PI2 =b(A) , where a(A) = P21 P22 Cl

8AT) = 1[ , 2

with

8d (t) = ; ( 1- co{ ~ t)) . The

/l.

sliding mode control law (16) has been tested by means of experiments carried out with the one-link flexible robot arm of our Institute (see Figure 1).

After the evaluation of the above determinant, the characteristic equation can be represented in the following form:

Cz

to

a l (A) a z(A) a3(A) a 4(A) ,

Remark: Pole placement in (22) is performed for the nominal case (i.e. tip-mass it L = 5Kg ). By payload variations, one can ascertain migration of closed-loop poles: For M = 2kg :

(A,I2,A3/4,AsI6'~)M,=Hg

(21)

=

(-1.19 ± 0.77 ),-1.7 ± 8.23),-68,91 ± 107.27 ),-5.17)

as (A) a6 (A) a7(A)

For M = 7.5kg:

(A,/2'~/ 4,AsI6'~)M,=7.5kg

=

(- 0.79 ± 1.87 ),-4.53 ± 4.87 ),-40.12 ± 74.97 ),-1.18)

and a; (A) and b(A) are polynomial functions of A. Note that the degree in A of (20) is seven (generally 2m + 3, here m = 2). In order to determine the sliding surface design parameters Cl' c Z' p", PI2'

It can be experimentally demonstrated, that the sliding mode controller is robust against payload variations. Figures 2-4 show effectively the trac1cing performance of the designed controller.

PZI' P22 and Cl while maintaining stabilizing requirement, seven distinct stable eigenvalues A; (i.e. Re(A; )< 0), i = 1,2"",7 , must be suitable selected.

336

there exists an unobservable subsystem. the so-called zero dynamics). It can be easily shown. that the zeros dynamics associated with this output is stable. i.e. the

-...---:,==========1

~ 1.6 ......... t'!

: ; 1.4

Measured values / '

go

above system is minimum phase. With x

/ .:"---Smuated vatues

'" 1.2

c:

:2.

i

1

i

0.8

0.6

11 = (0 ~

I

t

et.

=(8

and the joint angle as the alternative

system output y = Xl' the system equation (4) can be represented into the so-called Bymes-Isidori normal form (Isidori. 1995):

i

!

0 .4

i = f{x.,.} + g{x.,.}. u

0.2

(24 )

re = 'I'{,.} -O.2~O-7-~--:----:--;----:~~-=--::-9-!10

An input control torque. which is capable of exactly reproducing a given trajectory Xd (t) • can be derived by means of continuous sliding mode techniques (Zhou and Fisher. 1992). The sliding mode control forces the system trajectory to reach and stay on the sliding surface. We select the sliding surface as

tine (s)

Fig. 2: Joint angle (rad) 1.8r--_-...--.......-......---....-......----.--.--_-,

~ _1 .6

15

1. 4

follows: s = eT e. where e

Q.l .2

'"3

tracking error and e

1

::>

~O. 8

Let

V{X}

={Cl

=x - Xd

represents the

lY with Cl > 0 .

be a Lyapunov function :

V{x} =.!.S2 . (25)

2 For the fulfillment of the sliding condition. it must be the following valid:

0.6 0.4

0.2

V{x}= s·s = s.(eTf +e Tg·u _eT Xd )

o -O.20!--7-~~-~---:-~6;--~---:~-9=-~10

= s ·e Tg . ((eT gt ·eTf +u _(eT gte Tid )~o

time(s)

Fig. 3: Angular tip position (rad) 10'.--_-......-__-

_

_ _ . - - . - -_

(26) The sliding condition (26) is satisfied by the choice of the continuous tracking control law

_ . _ -__- ,

E

~8

u = _(eT gtleTf + (eT g)-leT id -a· s ·e Tg

= -qJ{X) - a· a{x}

. a>

0

(27) such that the time derivative of the Lyapunov function is negative definite.

represents the steady state. .1°i0 -7---:;--~--7-"7---:;--~---:=--9=-~10

Extension of control law (27) for robustness to modeling uncertainties: In order to counteract the inherent chattering phenomen in classical sliding mode control. Slotine (1984) have introduced the concept of "boundary layer". According to this approach and in order to deal explicitly with parameter variations and disturbances. the concept of boundary layer equivalence (Zhou and Fisher, 1992) for the continuous sliding mode control can be introduced. Here, the boundary layer is imposed on

time (s)

Fig. 4: Input control torque (Nm)

4. TRACKING CONTROL USING CONTINUOUS SLIDING MODE APPROACH 4.1 Continuous sliding mode Controller

For the purpose of design. we will transform the previous system (4) into the normal form using differential geometric methods. By choosing the motor position angle as output variable. the linkflexible robot system has a relative degree of 2. (i.e.

the switching curve s ·e T g

=0

instead of directly on

the sliding mode s{x} = o. In the presence of

337

r{x)

bounded modeling errors in equation (27) yields:

and

g{x)

of (24), Despite of considerable parameter uncertamtles, experimental results, which are shown in figures (56), demonstrate the effectiveness of the proposed continuous sliding mode approach. The joint angle's time evolution by use of simulation and experiments is closely related to the plots in figure (2).

(28)

5. CONCLUSION

where A represents the width of the boundary layer and can be interpreted as the allowable deviation from the ideal sliding due to model errors. The sliding mode controller (28) guarantees that the sliding condition is satisfied even in the presence of model uncertainties.

Two different sliding mode control approaches for the trajectory control of a link-flexible robot have been investigated in theory, simulation and experiments. The presented methodologies form in each case an efficient and robust tool for solving the trajectory tracking problem for single link-flexible manipulators.

4.2 Experimental results The sliding mode control law (27) has been tested, both by means· of simulations using a two-mode non linear model (4) of the link-flexible robot arm and by experiments. We choose the reference trajectory denoted by xit) = (Bd (t) angular motion from

Bd{O)=O

to

Od (1)

r

BAT) = 1[

REFERENCES Book, WJ. (1984). Recursive Lagrangian Dynamics of flexible manipulator arms. Int. J. Robotics Research, 3, 87-101. De Luca, A. (1998). Trajectory control of flexible manipulators. In: Control Problems in Robotics and automation . (B. Siciliano and KP.Valvanis. (Ed», 83-104. Springer-Verlag, London. Aiess, M., H. Mounier, P. Rouchon and J. Rudolph (1997). Systemes lineaires sur les operateurs de Mikusinski et Commande d'une poutre flexible. ESAlM: Proceedings, 2, 183-193. Hisseine, D. (1999). Trajektorienfolgeregelung fur einen flexiblen Roboterarm. Workshop GMAFachaussschujJ 1.4, "Theoretische Verfahren tier Regelung stechnik", Thun/Switzerland. Isidori, A. (1995). Nonlinear control systems. 3nd ed., Springer-Verlag, London. Kuczynski, A. (1999). Erprobung robuster Regelungskonzepte an einem elastischen Roboterarm. Diplomarbeit 08R01 , lA T, Universitiit Bremen. Madhavan, S. Kand S. N. Singh (1991). Inverse trajectory control and zero-dynamics sensitivity of an elastic manipulator. Int. J. Robotics Automation, 6, 179-191. Siciliano, B. and WJ. Book (1988). A singular perturbation approach to control of lightweight flexible manipulators. Int. J. robotics research, 7, 79-90. Slotine, U. (1984). Sliding controller design for non linear systems. Int. 1. Control, 40, 421-434. Utkin, V.1. [1992]. Sliding mode in control optimization. Springer-V erlag, Berlin. Yeung, KS . and Y.P. Chen (1990). Sliding mode controller design of a single-link flexible manipulator under gravity. Int. 1. Control, 52, 101-117,1990. Zhou, F. and D.G. Fisher (1992). Continuous sliding mode control. Int. J. Control, 55,313-327.

as an ,

2 the velocity profile as given in preceding section.

with

'0 1.6i'--~--'-----:r=========~ I (!!

81.4 :~

0.1.2

.g. l
'5

g>

'"

1

0.8

0.6 0.4 0.2

°O~'-~~~~------5--~6--------~9~10 time (s)

Fig. 5: Angular tip position (rad)

5

6

8

9

10

lime (s)

Fig. 6: Input torque (Nm)

338