Tracking Digital Controller for Robot Manipulator

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TRACKING DIGITAL CONTROLLER FOR ROBOT MANIPULATOR Valery D. Yurkevich, Boris

v.

Shatalov

Automation Department, Novosibirsk State Technical University £0, K.Marx ave., Novosibirsk, 630092, Russia, E-mail: [email protected], [email protected]

Abstract: The problem of the robust digital controller design to solve a tracking problem for robotic manipulators is studied. The design method based on the formation of two-time scale motions in the closed loop system is used. It has been shown that if a sufficient time-scale separation between the fast and slow modes in the discussed closed loop system and stability of fast modes are provided then slow modes have the desired form and thus the output transient performance indices are insensitive to parameter variations and external disturbances of the robotic manipulator. Numerical simulations of the two-link robotic manipulator are presented. Copyright ©2001 IFAC Keywords: sampled-data systems, output regulation, disturbance rejection, singular perturbation method, robot manipulator

LM is application of higher order derivatives jointly with a high gain in the control law. Singular perturbation method is used in LM to analyse the closed loop system properties.

1. INTRODUCTION

Various methods are used today to solve the problem of a controller design for robotic manipulators. For example, Variable Structure Systems (VSS) theory (Utkin, 1978; Young, 1978; Slotine and Li, 1991), adaptive approachs (Abdallah et al, 1994), learning controllers (Park P.H. et al., 1996), controllaws based on the Non-linear Inverse Dynamics (NID) method (Timofeev, 1980) are used. The control law based on NID method may be used if the dynamics of the system are exactly known. Therefore, an algorithmic approach to solution of the inverse dynamics problem should be used under condition of incomplete information about varying parameters and external disturbances of the robotic manipulator. For example, a solution of the inverse dynamics problem based on the Gradient Descent Method (GDM) may be used. Another way of the algorithmic solution of the NID problem is the application of the Localization Method (LM) (Vostrikov, 1977; Vostrikov, 1990; Vostrikov and Yurkevich, 1991). The peculiarity of

In the present paper the so called Dynamic Contraction Method (DCM) is applied which is a generalization and further development of LM (Yurkevich, 1993) . At the same time, the proposed approach can also be seen as a generalization of GDM. In particular, the structure of the control law discussed here follows from a higher order local optimization procedure (Tsypkin, 1968). As opposed to LM, DCM allows an integral action to be incorporated in the control loop without increasing the controller's order. In this paper, an approach (Yurkevich, 1993; Yurkevich, 1999) is used which is based on limiting properties of the discrete-time model of the control loop. The paper is organized as follows. First, a model of the robotic manipulator is defined, next a background of the discussed method is summarized, and finally the universal digital control law is

487

3. SYSTEM WITH ZERO-ORDER HOLD

presented. Numerical simulations of the two-link robotic manipulator are presented.

In accordance with (Yurkevich, 1999) let us find the approxima.te difference equation which allows us to describe the behavior of output sample points when the sampling period T is sufficiently small.

2. PROBLEM STATEMENT

= tiT

Let us introduce a new time scale r and denote

2.1 Model of robot manipulator

Let us consider a nonlinear time-varying dynamical system in the following form

= Kov

U

where

Ko

in (1)

= B- 1 .

(4)

Here v is the output of a zero-order hold. Then

(5)

dXi/dr = TXH1 'V i = 1, m-I , dxm/dr=T{f(x,w) Xm = f(x, w)

+ B(x)u,

Accordingly

(1)

rFy/drm = Tm{f(x,w)

y=Xl

where x = {xi , x2," " x~}' , Xj E RP 'V j = 1, m and w is the vector of unknown parameters and external disturbances. We assume also that dw/dt is bounded for all its components.

If T

+ B(y)u

-+

-+

dw/dr

0,

-+

(7)

0

x(r) ~ const, w(r) ~ const.

(2)

Accordingly, f(x(r),w(r»

~

(8)

const.

So, if the sampling period T is suffieciently small then it may be assumed that at least during the sampling period T the condition f(x,w) = const for kT :::; t < (k + l)T is satisfied. Accordingly, as a result of the Z-transformation of (6) it follows that

Assumption 2.1 Let us assume that the condition det B (x) f. 0 'V x E Ox is satisfied where x E Ox C Rmp and Ox is a bounded set.

£m(Z)

Yi ,k = m! (z -l)m T

Assumption 2.2 Let us assume that a series connection of a zero-order hold (ZOH) with a sampling period T and the system in the form (1) takes place. Accordingly, if kT :::; t < (k + l)T then u(t) = Uk where {Uk}f::o is a discrete input sequence.

m{

f.i.,k + Vi ,k

}

(9)

where Yi,k = Yi(t)1 t = kT ,Vi,k = vi(t)1 t = kT' f.i.,k = f.i.(x(t), w(t»1 t = kT and £i(Z) are Euler polynomials (Sobolev, 1977; Astrom et al., 1984; Blachuta, 1999) where £I(Z)

2.2 Control problem

El ,;

=

fl ,lZ1-1

= t(-l);-P/

+ El ,2Z1- 2 + ... + EI ,I,

(10)

G~~) , j = D,l = 1,2,~1l)

p=1

The robotic manipulator control system is being designed to provide the condition

lim ek = 0

(6)

and in a new time scale r we have that

where y = {Yl , Y2, · .. 'YP}' is the output available for measurement, u = {U1. 11,2, ... , Up}' is the control vector (vector of joint torques in (2».

k-+oo

+ v} .

0 then dx/dr

If m = 2, then from (1) the p-link manipulation robot model follows (Spong and Vidyasagar, 1989)

jj = f(y, iJ)

+ v}.

£1(l) = l! , l=1,2 , . ..

(12)

Then for T small enough the behavior of Yi ,k can be a.pproximately discribed by the difference equation

(3)

where ek = e(t = kT) is the error of the reference input realization, ek = rk -Yk, Yk = y(t = kT) is the sample point of output y(t) . rk = r(t = kT) is the sample point of reference input. Moreover, the controlled transients ek -+ 0 should have desired performance indices. These transients should not depend on the external disturbances and varying parameters of the system (1) .

Y. ,k

=

f)

_l)j+1

(m~ j)

Yi,k-j

+

3=1

m

+.L'Pffi 'L" Em, m!j

{ h,k-;

+ Vi ,k-j }

j=l

where !i.k-j are unknown values for all i, j .

488

(13)

Remark 3.1 1fT = 0 then from (19) the differ-

Proof. Obviously limT-+O{Y"k - Yi,k-I}/T = y~l} (t)1 t = kT Similarly, the exp.(20) follows from (19) and remark 3.1.

ence equation Yi,k =

f)-l)j+1 ( m ~

j)

Yi,k-j

(14)

J=1

4.2 Insensitivity condition

follows where its characteristic polynomial equals to (z - l)m.

Let us denote

(21)

4. CONTROL LAW STRUCTURE

where Fi,k = Fi(Yi,k,R;,k) ' Then the desired behavior of Yi,k is assured if and only if the following holds

4.1 Desired difference equation

Let the stable differential equation (m)

Yi

= F.•(Yi(m-I) "",Yi,ri(m-I) , ... ,r, )

(

15

)

for all k = 0, 1, ... IT (22) is met then the output transient performance indices of Yi,k are insensitive to external disturbances, inherent dynamical properties and parameter variations in the system (1).

follows from the continuous-tUne transfer function Cd (

) -

, s -

M, ISm-1 + ... + M'o 1,m" (16) sm + ad1.,m-l sm-I + ... + a'l,O 4

where parameters of ct( s) are selected on the base of the required output transient performance indices of y.(t) and ,0 = b,4,o'

4.3 Control law

at

To fulfil the requirement of (22) let us form the control law in the form of the difference equation (Yurkevich, 1993)

From the Z-transformation of a series connection of a zero-order hold and a continuous-time system with the transfer function ct (s) the pulse transfer function follows

m

V',k

= 2:di,jVi,k-j + .Ai(T)

where m

j=1

.A.(T)

= 5../rm, 5., =I 0,

where Yk = {Yk-a, ... , Yk-l}', Rk = {rk-a,"" rk-rJ'·

As a result the closed-loop system equations are given by

Yi,k

(19)

= t(-l)j+1

(m~ j)

Yi,k-j +

J=1 m

where r',k = Y',k at the equilibrium for all i = 1,p.

+ rm 'L" -fm,j - I { !i,k-j + Vi,k-j } , m.

Theorem 4.1 From (19) and remark 3.1 it follows that

(26)

j=1

m

Vi,k

lim p.(Y.\,' k D. k T) = T-+O ' -'Lt, '

= 2: d"jV"k-j + .A,(T) e[k

(27)

j=1

r. ,... , (m-I)

(25)

j ) Y',k-j +

+Tm F.(Yi,k, R;,k, T)

_ F. ( (m-I) i Yi , ... , Vi ,

= l,p.

5.1 Closed-loop system

3=1

-

i

5. MAIN RESULTS

Similar to (13) the exp o (18) can be rewritten in the form

(m~

(24)

2:di,j = I,

(18)

_l)j+1

(23)

(17)

d _ -d m-I d where Bi (z) - bi,IZ + ... + -bdi ,m_l z + -bi,m' ) - -d m + ... + ai,m_I -d -d From (17) A d( z + a"m' i z - ai,Oz we obtain the desired stable difference equation

E(

e[k

j=1

Ht!-( ) = Bt(z) 1 z At(z)

Yi,k =

(22)

ef'k = 0 1,

ri

)

It =

where i = !,p. From (19) and (21) it follows that the closed-loop system eqns.(26), (27) may be reVI'Titten in the form

()

kT 20

489

Yi,k =

f(

_1);+1 (

m'":...

j) Yi,k-j +

is fulfilled if and only if

)=1

-

.Ai = 1 -

m

+ T"' L €m,~ {fi,k-j + Vi,k-j}, m.

(28)

di,j

j=1

m

V'k= t,

Vj

· k_·+ L{d··-.A·-}V m. -fm,j 1,.1" I

'I,

d ~ d /3.,1 - P',2 - ... - /3i,m, ~

-

= Pi,j + .Ai€m,j{m!} -

= I,m

and

1

,

(34) (35)

= l,p.

Vi

J

j=l

m

+'xi{Fi(Yo,k,Rt,k,T) -

Proof. Substituting (34) ,(35) into (31) yields (33). Hence the proof is complete.

L €;;:f fi ,k-j}(29) j=1

where i = alli , j .

r,p and

Corollary 5.1 The requirement h ,k-j

are unknown values for (36)

In accordance with (5) and (6) if T -+ 0 then the rate of output transients of (28) is decreased, i.e. Yi,k - Yi,k-j -+ 0 V j = I , m. Accordingly, the fast and slow modes appear in the closed loop system (28), (29) where a time-scale separation between the fast and slow modes is represented by parameter T .

is satisfied if and only if do,j = €m,j{m!}-\ V j = 1, m

and

'xi = 1

(37)

V i = 1, p

Proof. Substituting (37) into (31) yields (36) . 5.2 Fast-motion subsystem Theorem 5.1 If T follows that

-+

5.4 Slow-motion subsystem

0 then from (28), (29) it

Theorem 5.3 If a steady state (more precisely, a quasi steady state) for the FMS (30) takes place, i.e.

Vi,k - V',k-j = 0 V j = 1, m , is the fast-motion subsystem (FMS) equation of the i-th channel where fi,k - li,k-j ~ 0, Yi ,k VJ' = to rn,)·{m!}-l Y'I,· k- .1' ~ 0 and/3'1.,j. = d1,j . -)..€ I,m.

(38)

then Vi ,k = vf.k where

Proof. If T -+ 0 then from (28),(29) and (6) the FMS equation (30) follows. Proof. The expr.(39) follows from (12) and (24).

Remark 5.1 From (30) follows that the characteristic polynomial of FMS of the i-th channel has the form

Theorem 5.4 If (33) holds and T -+ 0 then the SMS equation of Yi,k in the closed loop system (28), (29) is the same as (19).

A FMS,1·(z) = zm - /3.t., 1Zm-1 - ... - /3.",ffl (31)

Proof. From (33) it follows that FMS is stable. If T -+ 0 then after fast damping of FMS transients in (28), (29) we have that (38) and (39) are fulfilled . Substituting (38), (39) into (28) yields the SMS equation which is the same as (19) .

5.3 Control law parameters Let us form the desired characteristic polynomial A~MS , i(Z) in the form

Theorem 5.5 1fT -+ 0 then vf.k -vf(t)\ t 0, where

= kT -+

(40) where the roots of (32) are selected in a some small neighbourhood of zero in accordance with the requirements to the admissible transients in FMS .

is the NID problem solution. Proof. If T -+ 0 then from (8) it follows that li,k -Ii,k-j -+ 0 V j = 1, m . In accordance with (12) and (20) we have that from (39) the expr.(40) follows.

Theorem 5.2 The condition

490

Assuming a small sampling period, robust decoupled output tracking with desired continuous-time dynamics is accomplished. It has been shown, if a sufficient time-scale separation between fast and slow modes in the discussed closed loop system and stability of FMS are provided then SMS equation has the desired form and thus after fast damping of fast transients we have that the output transient performance indices are insensitive to parameter variations of the system and external disturbances.

6. SIMULATION RESULTS Let us consider a 2-link manipulation robot model (Utkin, 1978; Young, 1978) displayed in Fig.1

= X2, :£2 = (a22/a)(,812x~ +

Xl

2,812 X2X4 + "(19 + Ul) -

-(a12/ a )( -,812X~ + "(29

+ U2),

(41)

X3 =X4, :£4 = -(al2/a)GBI2x~ + 2,812 X2X4 + "(lg + Ul) + +(all/a)( -,812X~ + "(2g + U2)

REFERENCES

where Yl = Xl, Y2 = X3, X = {Xl,X2,X3,X4}' = {e,B ,rp,rp}" u = {Ul,U2}' is the vector of joint torques and

all = (ml + m2)li + m21~ + 2m2ltl2 COSX3, a12 = m21~

+ m21l12 COSX3,

a22m21~,

= alJ a22 - ai2' ,812 = m2h 12 sin X3, "(1 = -(ml + m2)Lt cos X3 + m2h COS(Xl + "(2 = -m2h COS(Xl + X3). a

X3),

The parameters ml, m2, 9, Lt, h are assigned by

ml = 2.5[kg], m2 = l[kg], 2 9 = 9.8[m/s ], Lt = 12 = 0.5[m] . We assume that the requirement (4) for Ko is satisfied. Require that the controlled outputs Yi(t) behave as a response of type 1 systems which are given by transfer functions

Gf(s) = (4s + 4)/(s2 + 4s + 4) Vi = l,2 (42) The simulation results of (41) during time interval t E [0, IOO]ser_ are displayed in Figs.2, 4 a.nd during time interva.l t E [0, l ]sec. are rli'lplayed in Figs.5, 7 for sampling period T = 0.02 sec. with the following control law Vi,k

= 0. 5Vi,k_1 + 0.5Vi,k-2 + +Ai{ -Yi.,k - at,lYi,k-l - at,2Yi.,k-2 + (43)

= K O{Vl,k,V2,k}', Ai = 2500, af,1 = -1.9215788, a,d,2 = 0.9231163,

where {Ul,k,U2,k}' -d

-d

bi,I = 0.0776421, bi ,2 = -0.0761046,

. t

-

= 1,2.

7. CONCLUSIONS A resulting output feedback digital controller for robotic manipulator has a simple form of a combination of low-order linear dynamical discretetime systems and a matrix whose entries depend nonlinearly on certain known process variables.

491

Abdallah, C., et al (1994) . Survey of robust control for rigid robots. IEEE Control System Ma.qazine, 11,24-30. Astrom, KJ. , P.Hagander, and J.Sternby (1984) . Zeros of sampled systems. Automatica, 20, 31-38. Btachuta, M.J. (1999). On zeros of sampled systems. IEEE Tmns. on Auto. Control, AC-44, 1229-1234. Fehrmann, R., W. Mutschkin and R. Neumann (1989). Entwurf und praktische erprobung cincs digitalen PL-reglers. Messen Steuern Regeln, VEB Verlag Technik, Berlin, 32(2), 7072. Park, P.R., T.-Y. Kuc and J.S.Lee (1996) Adaptive learning control of uncertain robotic systems. Int. J. Control, 65, no .5, 725--744. Slotine, J.-J. and W. Li (1991). Applied non-linear control. Prentice Hall. Sobolev, S.L. (1977) . On the roots of Euler polynomials. Dokl. Akad. Nauk SSSR. 235; English trans. in Soviet Math. Dokl., 1977, 18, 935938. Spong, M. and M.Vidyasagar (1989) Robot Dynamics and Control. John Wiley and Sons. Timofeev, A.V. (1980) Adaptive control system design with prescribed motions. Leningrad, Energia. Tsypkin, Ya.Z. (1968). Adaptation and learning in automatic control systems. ~auka, Moscow. Utkin, V.I. (1978). Sliding modes and their application in variable structure systems (In Russian) . Mir Publishers, Moskow. Vostrikov, A.S. (1977). On the synthesis of control units of dynamic systems. Systems Science, Techn. Univ., Wrodaw, 3(2), 195-205. Vostrikov, A.S. (1990). Synthesis of nonlinear systems by means of localization method. Novosibirsk State University, Novosibirsk. Vostrikov, A.S. and V.D. Yurkevich (1991). Decoupling of multi-channel non-linear timevarying systems by derivative feedback. J. on Systems Science, Wro claw , 17(4),21-33. Young, K-KD. (1978). Controller design for a manipulator using theory of variable struc-

ture systems. IEEE Trans., vol.SMC-8, 101109. Yurkevich, V .D. (1993) . Sampled control system synthesis by means of dynamic contraction method. In:Prepr. of the 2nd Int. IFAC Workshop on Nonsmooth and Discontinuous Problems of Control and Optimization, (24-30 May 1993): Chelyabinsk. Russia, 157-158. Yurkevich, V.D. (1995). A new approach to design of control systems under uncertainty: dynamic contraction method. In:Prepr. of the :lord IFAC Symp . on NOLCOS. Tahoe City, California, 2, 443-448. Yurkevich, V.D . (1999). Robust two-time-scale discrete-time system design. In: Prepr. of the 14-th World Congress IFAC'99, Beijing, China, G: 343-348.

0.1 0.05

-0.05 -0.1

t-_~_ _~_..)..L~_~_----=!

o

20

40

60

80

sec

0.8

sec

Fig. 4. Ramp-wise rf'lipOnSe of errors

o

0.2

0.4

0.6

Fig. 5. Ramp-wise response of outputs Fig. 1. The manipulation robot

10 8

12.7

6

12.6

4

12.5

2

12.4 12.3

20

40

60

80

sec

o

0.2

0.4

0.6

0.8

sec

Fig. 2. Ramp-wise response of outputs Fig. 6. Ramp-wise response of control

'Ill

5.1 ~.,,--~--~-~--~----. 5.05 5 4.95

o

20

40

60

80

sec

4 .9

Fig. 3. Ramp-wise respOU::ie of controls

o

0.2

0.4

0.6

0.8

Fig. 7. Ramp-wise response of control

492

'1.£2

sec