Globally adaptive decentralized control of time-varying robot manipulators

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Proceedings o f the 2003 IEEE loterostional Conference on Robotics &Automation

Taipei, Taiwan, September 14-19, 2003

Globally Adaptive Decentralized Control of Time-Varying

Robot Manipulators

Su-Hau

Hsu' and Li-Chen

Fu','

'Dept. of

Electrical

Engineering

'Dept. of

Computer

Science 8

Information Engineering National Taiwan University, Taipei, Taiwan.

TEL: 886-2-23622209;

FAX:

886-2-23657887

E-mail: lichen@ccms.ntu.edu.tw

Abstract

-

In this paper, we develop a globally adaptive decentralized control scheme of time-varying robot manipulators for trajectory tracking control. Since the proposed adaptive control law is in a decentralized manner, only low-cost hardware is required for implementation. Furthermore, even if the time-varying parameters of the robot manipulator change arbitrarily fast, both the position and velocity tracking errors of the manipulators will converge to zero after em- ployment of the proposed adaptive control law. For practical implementation, the adaptive law can be combined with a leakage term so that there is no chattering in the motion of the manipulator but at the price a residual set of the tracking errors as mentioned above, of which the size can be however made smaller by use of some larger design parameters. Finally, in order to illustrate the performance of the proposed scheme, simulation r e s u l t s are also provided, which turn out to he quite satisfactory.

1. INTRODUCTION

Research in the area of trajectory tracking control of robot manipulators has been widespread. Many industrial tasks of the robot manipulator such as material handling and pan assembly involve such a problem. On the other hand, in many situations, some of the unknown parameters such as the mass of the payload or the mass of the links, may be time varying. For example, robotic pouring and tilling operations belong to this case. A large number of control design exist in the literature that works well for both cases with the both known and unknown constant pa- rameters [Slotine and Li, 7; Spong, 91. However, it is worth noting that, because the time-varying parameters of robot manipulators enhance complexity of the overall systems, the control schemes developed for the time-invariant robot manipulators are not suitable for the time-varying ones [Reed and Ioannou, 61.

During recent years, quite a few research works have

been devoted to the trajectory tracking control of time-varying robot manipulators. In Tao [12], a robust adaptive controller is proposed to guarantee asymptotic tracking in the presence of time-invariant parameters. However, the transient performance is related to the speed of parameter variations and cannot he arbitrarily improved. In Tomei [14], a robust adaptive control law is proposed which ensures the arbitraxy transient performance. In Song and Middleton

[PI,

a robust switching-type control law is proposed for the robot manipulators with time- varying parameters. In this research, the property of Hadarmad product is applied such that the dynamic model can be expressed in a parameterized form and the zero asymptotical tracking error can he obtained. In Su [ll], an adaptive sliding mode control is proposed which guarantees asymptotic convergence of the tracking error. In the work by Pagilla et al. [ 5 ] , with hounded time-varying parameters an adaptive control law is designed, of which the effectiveness is dem- onstrated by providing experimental results. In the literature, different control laws have been proposed for the tracking control of time-varying robot manipulators; however, most of them are in centralized manner. Since the decentralized controller structure is adopted by the majority of modern robots in favor of its computation simplicity and low-cost hardware setup [Fu, 2; Liu, 3; Tang et. al., 131,

how to best improve the tracking performance of time-varying robot manipulators through decentralized control becomes an interesting research topic.

In this study, a globally adaptive decentralized trajectory control law is proposed for time-varying robotic

manipulators.

11. PROBLEM STATEMENT

In this section. the dynamic model of a time-varying robot manipulator is introduced and the difference between

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the time-invariant ease and the time-varying one is also described. The present dynamic model is derived under on the assumption that the generalized constraints are independent of time but the payload mass can he time varying. It is also under on the assumption that the time-varying payload mass has no contribution to producing the generalized force. The dynamical model of robot manipulators is derived by Enler-Lagrange equations [Spong and Vidyasa- gar, IO]. Let q = (ql,

... ,

q,JT be the set of position vari- ables of n joints of the robot manipulator. The scalar func-

tions

K

: R" x R" x [O, -) R and

P

: R" x [O, -)

+

R denote the kinetic energy and the potential energy, respec- tively, where apparently the kinetic energy K : R" x R" x [0,

-)

+

R can he explicitly expressed as defined as follows

K

= i q T M ( q , f ) q (1)

where M : R" x [0, -)

+

R"" is the inertial matrix of the robot manipulator. It is worth noting that the inertial matrix M and the potential energy P are explicitly dependent on f such that the robot manipulator is called the time-varying one. The Lagrangian L is obtained by taking the difference

between the kinetic energy K and potential energy P , i.e., L

=

K

- P . After substituting the Lagrangian L into the

Euler-Lagrange equations, the overall dynamics of the robot manipulator can he obtained as follows [Tao, 121:

M ( q , t ) i ; + ~ 4 + C ( q , q , f ) q + g ( q , f ) = z (2)

where the (k, j ) element, 1 S k, j S n , of C : R" X R" X [0, -)

+

RnXn 1s:

c -

-Ai(!%+&-?

aq; ab ( 3 )

i-l

with mu being the ( i . j ) element of

M,

and g(q, I) = W ( q ,

r)/aq. This dynamic model has the following properties that

will be used in the controller design [Su, 1 I]:

Property I:The inertial matrix M i s a symmetric and positive

definite matrix,

which satisfies

p , l <

M <

bMI,

Vq

E R", V I E [0, -), for some constants ~r,, p M > 0.

Property 2: The matrix C satisfies IlCllS pcllqli, Vq

E Rn, V

4

E R", V f E [0, -), for some constant pc > 0

Property3: z T ( n ; ( - % - 2 C ) z = 0 , V z ~ R " , V ~ E CO,

-)

It is worth noting that, if the inertial matrix M is not explicitly independent of 1, then Property 3 can be rewritten as:

zT(fi

-2c)z = 0 , VZ E R", which is well-known in the literature ofrobot manipulators.

Property 4: The vector g satisfies llgll< pG, Vqe R",

Vf E [O, -), for some constant pG > 0

Compared with the time-invariant robot manipulators, be- sides that the parameters in the dynamic model of the time-variant robot manipulators are Considered as time functions, additional terms related to the rate of change of time-varying parameters are also derived such that Property

3 of time-varying robot manipulators is different from that

of time-invariant robot manipulators.

Now let q d f ) denote the desired position trajectory for tracking, which is generally chosen twice differentiable to guarantee smoothness of the motion. Let TT = [T,,

.. .

, TJ.

Iftheithelement,iE{l,

._.

,n},ofthecontrolinputisonly

a function ofjoint configuration and velocity of the ith joint, the control input 7 is called a decentralized control. In this study, a decentralized control scheme is to be developed such that q(t)

+

q&) andq(t)

-+

q r ( f ) as f

+

-.

111. CONTROLLER DESIGN

In this section, one adaptive decentralized control schemes will he developed for the control of a general d i n k rigid time-varying manipulator. Let qdf) denote the desired position trajectory. The following error signals are defined as e = q - qd and s

=

d + Ae where the feedback gain matrix A is a positive definite matrix. Now the dy- namics defined hy the signals e and s can be derived as:

d = -Ae+ s (4a) (4b) M(q,t)S = -C(q,

4,

f)s+.r-Nf, 4.4)

W,

q,

4)

= M ( q , f)(& + A 4 + C(q,

4.

f ) ( Q d + where ( 5 ) aM(1,'l ' + g ( 9 ) - 7 4

behaves as the perturbation term In the following, without

loss of

generality, several technical

assumptions

are

made to pose the problem in a tractable manner.

Assumption I : The feedback gain matrix A is a di- agonal positive definite matrix; that is, A = diag

(XI, ...,

1,)

> O f o r h , > O , i e { l ,

...

,n}.

Assumpfion 2: The desired position trajectory 461) and its time derivatives q d ( f ) , q d ( t ) are all bounded time-varying signals.

Assumpfion 3: The time-varying parameters of robot

manipulators are hounded and smooth functions in time. Furthermore, there is a positive constant q

>

0 such that

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In this study, we start with investigating the following lemma. In this sequel, we adopts the truncated L., norm

defined as

Ibllr,.,

E maxl,,sup,e~o, &(t)l for all real vec- tor-valued functions x E c"[0, for some T > 0. To assess the tendency of growth, the following lemma will provide

us useful facts to develop the control scheme in this study.

Lemma 1: Under Assumption 3, ifthere exists a constant T

> 0 such that 11s exists, then there arepositive constants

a,,

az,

a3,

and% such that f o r t E [0,

TJ

:

Ile(t)ll ~al//eoll+a2ll~ll,,~ (6)

Il@)lls

a3lleoll+ a 4 1 1 ~ l l T , ~ (7)

where eo = e(0) is the initial condition of e, and positive constants

PI,

8,

and p 3 are such thatfor t E [0, r ] :

Ilv(t,

4.4)11

s

PI

+ B ~ / / ~ / l ~ , ~

+ D 3 l l ~ l l : , ~

v ( t , q, q ) = v ( t , q , q ) - + + s

(8) where V ( t , q , q ) E [O, -) x R" x R"

+

R" is defined asfol-

lows:

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-

Proof: Consider the linear system defined by

e = - h e + s , e(0) = e , . Since the matrix -A is Hunvitz, we

obtain (6) for t E [O,

rl

where

ai

and

az

are some positive parameters. Furthermore, according to the definition of the linear systems, we then obtain (7) for t E [O,

TJ

where a, and a4 are appropriate positive parameters. Next, according tu the definition of the perturbation term v in ( 5 ) and As-

sumption 3, we have (8) for I E 10, U , where

PI,

p2,

and

p,

Now, an adaptive decentralized control scheme is presented to solve this tracking control problem. Consider the control law

rT

= [T,,

..

., r,]

given as follows:

-ti;-+,

- e , ~ , s i - 8 , ~ , s ~ if

~ ( - , I s ~ I ~ E ~

1

sgn(sj) -e,_,sj

-ei-&

if i,_,Js,l> E~

fur i = 1,

. . . ,

n, where the parameter €Ij

,

, i E (1,

._.

, n }

needs to he adjusted on line, and Of-, > 0, i E { 1 ,

. . . ,

n } and j ~ { 2 , 3}, and sgn(-) denotes the signum function [Khalil,

41. Furthermore, the auxiliary signal Eir i E { I ,

._.

, n } , satisfies:

are some chosen positive parameters.

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Ti =

E < = - P E . I ,, E , @ ) > 0 (11)

where p i > 0. It is worth noting that the time-varying signal

E,, i E { 1,

.

. .

, n } . are always positive, which are used as the time-varying boundary layers. In order to account for the parametric uncertainty from the manipulator and the desired position trajectories, we choose the adaptive law as

follows

k ,

=

Y,&,I

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where b j - , ( 0 ) L O , and the adaptive gain " f j - ] > 0, i ~ { l ,

.., , n}. With (lO)-(l2), the closed-loop system becomes differential equations with discontinuous right-hand sides [Filippov, I]. A considerable amount of works has been camed out to deal with situations like this. Here, we base our results for these differential equations on Filippov's concept. Note that the adaptive control law (10)-(12) is

apparently in a decentralized manner, and its performance can be summarized into the following theorem.

Theorem 2 (Adaptive Decentralized Control Scheme): Under Assumption 1-3, consider the error dynamics of the robotic manipulator with the adaptive control law (10)-(12) defined above. Then, a// signal are bounded, and, both, the

position tracking error e(t) and the velocity tracking error

$ 1 ) will converge to zero as t

+

-.

Proof: The proof proceeds in the following two steps. Step 1: Prove the signal s(t) is ultimately bounded. Con- sider the Lyapunov-like function as follows:

V ( t ) = + T M S (13)

Consider IIsIIT,,- < I, for some finite T > 0. Then, after taking the time derivative of (13) along s ( t ) for t E [O,

T J ,

we oh- tain the following result:

d=s'r-sV~sT.r+II~II(PI

+ P J ,

+P,l,') (14) where the function

V

is defined as that in (9). Then, the following two different cases will he obtained

Case1:

i 3 j + l < ~ ,

f o r t s

[ n , q ,

. ,

+11sIl,(P,

+ P A

+ P J 3

where

ej,

,,,m = min(81 j ,

...,

0._,}, j

E R

3). From both

Case 1 and 2, we can conclude that for I E [0, 7J:

v s

-~2.m,"l14: (17)

when Ilsllz t [""'*~~,":"""]"', which implies that there exists lZ > 0 such that for I E [O,

T J ,

lIs(t)

)Iz

5 /, < /, if lI is suf- ficiently large. Similar to the previous argument, 1, can he

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defined as l2

=E[

""'+~~,~,'""'

1 '"

where II is sufficiently large, and hence one can show that s(t) is actually ultimately bounded so that there exists l, > 0 such that 1, =

Step 2: Prove all signals are bounded and the signals

+

0 as t

+

m. Now consider the Lyapunov-like function as fol-

lows:

supz.ro,-,

/I

s(t) 112.

V(t) = jsTMs+

x[+y::,(e,-l

-

e:-$

+ p ; ' E i ] (18)

i=,

where 8: I , i E { 1,

. . . ,

n } is the desirable hut unknown parameter. After taking the time derivative of (18) along the solution trajectories of the closed-loop system, the following two different cases will he obtained

Case I: Bi-IJsi)

<

ci

for t E IO,

-1,

where

2, we conclude that for sufficiently large

= m i n i 8; I ,

...

,e.-,).

From both Case 1 and we obtain:

c's

-e*,mi.llsll:

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for t E [0, -), which implies all signals are bounded and

Ils(r)ll, is L2. Finally, to show the zero convergence of the

tracking error e(t) and e ( t ) , we need to show Ils(t)lIz is uniformly continuous and then apply Barbilat's Lemma [4].

Sufficiently, we investigate houndedness of the signal i ( t ) from (4),

(S),

and (IO), and easily we can verify such a condition. Hence, the zero convergence is thus insured.

W

R e m a r k In Theorem 2, we would like to emphasize that here in OUI proposed scheme, the control gain I&-,, i E { I,

. . . ,

n}

and j E { 1, 2}, only needs to be chosen positive, and the adaptive decentralized control law (10)-(12) will then successfully drive the tracking errors e(<) and e ( ( ) of the closed-loop system to converge to zero globally. Hence, this control law can he claimed to be a global one.

Due to the numerical noise caused by convergence of

&i, i E { I,

... ,

n}. to zero, practical implementation of the

scheme should he considered, e.g., using addition of the leakage term [ 2 ] . Now, consider another modified adaptive law as follows:

e i . , = Y j ~ , ( l ~ ! l - I S ~ s . J (22)

where 6,-,(0) t 0 and the adaptive gains y(-, > 0, i E { I ,

. . . ,

n } , and the leakage-term constant IS > 0. Note that the adaptive law (22) is apparently in a decentralized structure, too. The performance of adaptive control law (IO), (1 I ) and (22) can he summarized into the following corollary. Corollary 3 (Adaptive Decentralized Control Scheme with the leakage term): Under Assumption 1-3, consider

the error dynamics of the robotic manipulator with the adaptive control law (IO), (11) and (22). Then, a / / signal are bounded, and, theposition tracking error e(t) will con-

verge to a residue set whose sire can be reduced by use of

larger O Z . ~ ~ , and y1,en, where = minlOz.I,

... ,

OZ-,,},

andy,,,i.=min{yl.l,

... ,yn.ll.

Proof: This proof is similar to that in the proof of Theorem

2, and, hence, is neglected here.

W

IV. SIMULATION

RESULTS

In order to demonstrate the performance of the proposed adaptive decentralized controller, several numerical results are provided now. A two-link planar robot manipu- late with two revolute joints is considered here [Spong and Vidyasagar, IO; Su, I I]. Let, for i = 1, 2, d, denote the mass of link i, 1, denote the length of link i , lCi denote the distance from the previous joint to the center of mass of link i, and I, denotes the moment of inertia of Link i about an axis com- ing out of the page, passing center of mass of Link i. The inertial matrix Min ( 1 ) has four elements

m,, =dllzl + d 2 ( l : + / ~ ~ + 2 l , / , , c o s y , ) + I ,

+I,

m,, = m , , = d , ( / ~ 2 + l ~ c o s q 2 ) + 1 2 (23)

mZ2 = d21:2

+I,

and the potential energy P i s as follows:

where go = 9.8 is the gravitational magnitude. The mat& C in (2) has four elements as follows:

P(q, t ) =

(4L

+ m,l,)g, sin y1 + m,l,,g, s i n k , + q 2 ) (24)

(25) where h = -dz

h

lc2 sin y2. If we consier on an unknown time-varying load camed by the robot as part of the second link, then the parameters m2, l,,, and 1, will explicitly he time-dependent. In this case, the time rate of the change of the inertia matrix Mbecomes

CII = 42h, ci2 =

(4,

+4,)h,

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where 0, = d2 l:z, 02 = d2 I f , = I2, 04 = d2 l2

IC>. which

satisfies Assumption 3. We assume that the two physical links are thin cylinders with uniform density and identical length. The parameters used in this simulation study are listed as follows: d , = 10, /I = 1, /,I = 0.5, II = 516, 4 = 5 tAd2(t), 12= 1, lcz=0.5 + A h ( / ) , 12= 5d21:/12 where Ad2(()

= 0.5cos(0.21), Al,(t) = 0.2cos(0.2t). For practical consid- eration, the adaptive control scheme with the leakage term is employed. The desired trajectory q d ( t ) = [qdl(t), qm(t)lT is defined by qdl(t) = sin t, qm(t) = sin 1. The initial condi- tions used in the numerical study are ql(0) = --P, q2 (0) = 0,

k,(o)=

m)=

0,

m=

q2w=

0, 0,~,(0)=0,.,(0)= 0, ~ ~ ( 0 ) = E2(0) = 1. The control gains, adaptive gain constants, and auxiliary signals are, respectively, defined as hi = h2 =

6, 01.2 = 02-2 = 50, 01-3 = 02-3 = I , ? ] = ~2 = 50, o = 0.5, and

pI = p 2 = 0.5. After compute simulations, Figures 1 and 2 illustrate the numerical results of tracking for Links 1 and 2, respectively. From Figures (la), (2a), (Ib) and (2b), it can be concluded is found that the position and velocity tracking errors approach to a residual set as the time approaches infinity. Figures (IC) and (2c) demonstrate that the updated parameters due to the adaptive law are kept bounded with time. Finally, Figures (Id) and (2d) illustrate that the two control inputs pertaining to this adaptive control scheme also remain bounded.

V. CONCLUSION

In this paper, we develop a globally adaptive decentralized control scheme for the time-varying robot manipu- lator performing tasks of trajectoly tracking. Because the proposed adaptive control law is in a decentralized manner, only low-cost hardware is required for implement. Even if the time-varying parameters of robot manipulators change fast, both the position and velocity tracking error of robot manipulators will also converge to zero by use of the proposed adaptive control law. For practical implementation mentioned, the adaptive law can be combined with a leakage term such that there is no chatting in the control to manipulators and the tracking errors of joint positions and velocities of robot manipulator will converge to residual set whose size can be made smaller by use of larger design parameters. Finally, in order to illustrate the performance of the proposed scheme, satisfactory simulation results are

also provided. All the simulation results are satisfying.

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G

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mm.,*./,

(la) Position tracking error e,

nnr<.-i

(lb) Velocity tracking error e,

I r n ,

, , , , ,

, , , ,

,

m

Y

e4

(IC) Estimated parameter

6,

I r n .Irn .m I) I

.

6 8 10 12 I. 16 I8 m n m / a r / kc

(Id) Control input T~

Numerical results of Link 1 Fig. 1

d

T r n , * 4

(2a) Position tracking error e ,

nrner,

(2b) Velocity tracking error e, Irn rn M Y m .a 4 0 2 I 6 8 I Y 1) I. 16 >I( = mrnta=, ~Irn (2c) Estimated parameters

6,

,

(2d) Control input 5,

Numerical results of Link 2 Fig. 2