AD-A245 716

Hybrid Efficient Control Algorithmsfor Robot Manipulators

Ju-Jang Lee 1

CMU-RI-TR-91-30

, FEB 11199253

The Robotics InstituteCarnegie Mellon University

Pittsburgh, Pennsylvania 15213

.4- . .... November 1991

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01991 Carnegie Mellon University

'Visiting Scientist, Dept. of Electrical Engineering, Korea Advanced Institute of Science and Technology

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Hybrid Efficient Control Algorithms for Robot Manipulators

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Ju-Jang Lee

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The Robotics InstituteCarnegie Mellon University CMU-RI-TR-91-30Pittsburgh, PA 15213

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13. ABSTRACT MaxArnnICOfworos)

In this report we discuss accurate and robust sliding mode tracking control for highly nonlinear robot manipulatorsusing a disturbance observer. To eliminate the chattering problem existed in conventional Sliding Mode Control (SMC)approach, which is caused by modeling errors and uncertainties, the efficient compensation of the disturbance observerhas been introduced.

The proposed sliding mode control is presented in two theorems. The bounded stability of the proposed controlmethod is proven and the efficiency of the control algorithms has been demonstrated by simulations for a positiontracking control of a two-link robot subject to parameter and payload uncertainties.

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Contents

1 Introduction 1

2 Mathematical Work 2

3 Efficient Sliding Mode Control for Robot Manipulators 5

4 Modified Sliding Control for Robot Manipulators 8

5 Numerical Simulation 10

7 Conclusions 21

References 22

-lI

List of Tables

Table. 1. Numerical comparison between CTM and proposed SMC. 12

ii

List of Figures

Fig. 1. The block diagram of the proposed SMC .............. S

Fig. 2-1. Tracking errors for CTM algorithm ................ 13

Fig. 2-2. Corresponding control inputs by CTM algorithm ...... .14

Fig. 3-1. Tracking error for Theorem 1 of proposed SMC ....... .15

Fig. 3-2. Corresponding control inputs by Theorem 1 of proposed

SMC ............................................ 16

Fig. 3-3. Corresponding sliding surfaces for Theorem 1 of proposed

SMC ............................................ 17

Fig. 4-1. Tracking error for Theorem 2 of proposed SMC ....... .IS

Fig. 4-2. Corresponding control inputs by Theorem 2 of proposed

SMC ............................................ 19

Fig. 4-3. Corresponding sliding surfaces for Theorem 2 of proposed

SMC ............................................ 20

III

ABSTRACT

In is report, we discuss accurate and robust sliding mode tracking control for

highly nonlinear robot manipulators using a disturbance observer. To eliminate the

chattering problem existed in conventional Sliding Mode Control (SMC) approach.

which is caused by modeling errors and uncertainties. The efficient compensation

of the disturbance observer has been introduced.

The proposed sliding mode control is presented in two theorems. The bounded

stability of the proposed control method is proven and the efficiency of the control

algorithms has been demonstrated by simulations for a position tracking control

of a two-link robot subject to parameter and payload uncertainties.

iv

1 Introduction

The overall robot control problem can be divided into two phases, trajectory planing andcontrol phases. The trajectory planing is the planing of the desired time trajectory froman initial position to a desired position of robot manipulators with collision avoidance ifnecessary. The control phase is the tracking control of robot manipulators to track the de-sired trajectory planned in the trajectory planning phase. This tracking control phase isthe field of our concern in this paper. The strategies for designing a robust and accuratetracking controller for a highly nonlinear robot manipulator have been studied enthusiasti-cally in order to extend its application fields. There are several approaches to attempt toobtain the desired tracking performance such as, Computed Torque Method (CTM) [1-3],Adaptive Control [4, 5], Sliding Mode Control (SMC) [6-13], and so on [14]. Each methodhas its merits and shortcomings. The CTM normally provides a feasible controller if theexact knowledge of the manipulator dynamics is available. However, for a large amoultof applications, it is impossible to obtain the complete dynamic model of robots, due tomodeling uncertainties, parameter variation and unknown payloads. These uncertainties,especially the error of inertia matrix, may result in the instability of robot systems. All ofhighly nonlinear model dynamics are taken into account in order to calculate the controlinput which is a hard load in view of the computation time. Thus the optimal trade-off ismade between approximate modeling of the robot manipulator and its output performance.rhe CTM is sensitive to modeling errors which come from linearization and approximationo' nonlinearities, uncertainties in physical systems, and disturbances [3].

In order to give the robustness to modeling uncertainties and parameter variations, adap-tive algorithms for a highly nonlinear robot manipulator are considered such as Self-Tuningand Model-Reference Methods but essentially n cc, an adaptation mechanism for the pa-rameter identification resulting in a heavy computation burden, complexities, and a highcost for the digital implementation. As another robust controller different from the adaptiveapproach, SMC using Variable Structure Systems (VSS) for robot arms is studied by manyresearchers recently [6-13]. The desired performance with a simple control structure canbe obtained in the existence of the acceptable model error and unknown payload using the"sliding mode". In the sliding mode, it is well known that the controlled system is robustto the bounded parameter variations and disturbances [17, 18]. The other advantages of aSNIC are that the output performance 7an be predetermined by a sliding surface and thereis no overshoot in regulations. The first application of SMC to robot manipulators seems tobe in the work of Young dealing with a set point regulation problem [6]. A modification ofthe Young controller was presented by Morgan [8]. Other SMC of robot manipulator may befound [10-12]. The SMC unfortunately has a problem that the motion trajectory of system,:hanges frequently in the vicinity of the sliding surface due to a high frequency switchingof the input which is called as "chattering". Because in order to guarantee the existence ofthe sliding mode, the control has imperatively the discontinuities designed by the maximumbound of model errors or uncertainties. The conventional SMCs for robot manipulators usethe feedforward compensation by using an available model dynamics, there exist inevitablechattering due to model errors or uncertainties in system parameters. To alleviate thisproblem bounded layer method or saturation function [9] are used instead of discontinuous

I

parts. These methods are based on the try-and-error not a systematic approach and notenough to obtain the desired performance with the satisfactory tracking error. Thus as pos-sible as the exact compensation of the high nonlinear interactions in robot link systems isessential to obtain the good performance without chattering. Using the multirate samplingconcept, a SMC with an off-line feedforward compensation for the nonlinear interaction ofrobot manipulators using an available model is presented.

In this paper, design of a sliding mode control algorithm with an efficient on-line com-pensation for robot manipulators is studied for robust and accurate tracking of the desiredtrajectories. The proposed control consists of two parts of the efficient compensation andthe sliding mode only based on the nominal inertia matrix of robot manipulators. Thiscompensation technique is responsible for not only highly nonlinear interactions but alsomodeling errors of the inertia matrix using the acceleration information and the nominalinertia matrix. The sliding mode control is applied to the compensated dynamics of robotmanipulators to obtain robust and accurate tracking performance with the correction ofcompensation errors in view of the real implementation using the microprocessor. The sta-bility of the proposed SMC is deeply analyzed in sense of Lyapunov, summarized in twomain theorems with a preliminary lemma. Based on this analysis, the given desired track-ing specifications can be satisfied by a proposed SMC. The design of the proposed SMIC isindependent of the uncertainty parameter space. Thus there can be no chattering due tomodeling errors or uncertainties.

The rest of the paper is organized as follows. Section 2 gives the mathematical basis fora stable SMC design by Lemma 1. In Section 2, a new SMC with the efficient compensationis proposed, and its stability is analyzed in Theorem 1. A modified control is then presentedin Section 3 and another more efficient approach is proposed in Section 4. Its stability issummarized in Theorem 2. The simulation studies for the proposed algorithms are carriedout in Section 5 and conclusions are presented in Section 6.

2 Mathematical Work

The motion equations of an n degree-of-freedom manipulator can be derived using theLagrange-Euler formulation as

J(q(t), 0) .4(t) + D(q(t), 4(t), 0) = r(t) ()

where J(q(t), 0) E R" " is a symmetric positive definite inertia matrix. D(q(t), q(t), 6) E Rnis called a smooth generalized disturbances (SGD) vector including the centrifugal and Corio-[is terms H(q(t),q(t), ) E Rn, Coulomb and viscous or any other frictions F(q(t),q(t), o) ER", gravity terms G(q(t), 6) E R".

D(q(t), 4(t), 0) = H(q(t),4(t), ) + F(q(t), 4(t), 0) + G(q(t),8)

where r(t) E R" is an input vector, q(t), 4(t), and 4(t) E Rn are the generalized position.velocity, and acceleration vector, respectively, 6 is a vector composed of the parameters ofrobot manipulators (i.e. the masses, lengths, offset angles, and inertia of links). Exact mod-eling of robot dynamics is difficult because of parameter uncertainties, unknown frictions.

2

payloads variations. Therefore, by using estimated parameters in the model, we express themodel of the robot system (1) as follows

J[q(t), .q(t) + D(q(t), 4(t), 01 = -r(t). (2)

The objective of robot trajectory control is to follow the desired trajectories qd(t), d(t),

qd(t) computed in planning phase. Let's define the state vector X(t) E R 2n for SMC as

X(t) [Xl(t)T X 2 (t)T]T (3)

where X 1 (t) and X2(t) E Rn are the trajectory errors and its derivative

Xi(t) e(t) = (qd(t) - q(t))

X2 (t) -i (t) = (4d(t) - 4 (t)).

Then the state equation of the robot system becomes(t) 001] [ 0 -.()= 0 0 J(q(t), 0) - 1 7()+[J(q(t),)-. D(q(t), 4(t), 0) + 4 It)"

(5)An augmented sliding surface vector s(t) is defined as

s(t) X 2 (t) + K, . Xi(t) + K, - I X(t)dt (6)

where K and K. E R n " are gain matrices. The ideal sliding dynamics defined by Equation1

(5) is the trajectories q*, 4', 4" E R" which satisfy the following equation

X;(t) + K . X;(t) + Kp . X;(t) = 0. (7)

Ork-(t) = A . X*(t) (S)

where

(qd(t) - q*(t)) T In 0 1 1 ~X*(t)= [(d(t)_(t))T]T ER A= [ -K -K, 2n

We can choose K, and K. so that all the eigenvalues of A have negative real parts, whichgiiarantees the exponential stability of the system (7), i.e., there exist positive constants Nand K such that

1 e A 1< K (C)

where is the induced Euclidean norm [15].

Now, we will state the following lemma as a prerequisite to the two main theorems.Lemma. : If the sliding surface defined by Equation (6) satisfies 11 s(t) 11< ' for iny

f > to and 1I X(to) I!< "y/K is satisfied at the initial time, then

Il x I !_< 1 1 0a)

SX 2(t) 115 E2 (10b)

is satisfied for all t > to where el and 62 are positive constants defined as following

K K4-.i = 'Y 2=7-[1+z---]I, Z-II [KpK.]I (I 0c)

Proof [16]Define a new state, Xo(t), to be an integral of the state X, (t), i.e.,

Xo(t) - X(u)du, Xo(to) = 0. (11)

then Xo(t), the augmented sliding surface (6) can be rewritten as

s(t) = X 2(t) + K, . X(t) + Kp. Xo(t). (12)

If we define a new state vector X(t) = [Xo(t) T , XI(t)TIT, thus we have the following state-space equation

X(t)= A' X(t) +[O]s(t) ' (13)

Where s(t) may be considered as bounded disturbances, I s(t) 11 7. The solution ofEquation (13) is

(t) = eA(t-t°) "(t°) + ti:[e(t) [ 0 ] s(u)]du.(14)

From Equation (9) and boundedness of s(t), the norm of X(t) becomes

,, X(t) 11< K +() t + I eA(t-u) 11 [1 °ft'1 "0

-- 7 + [1K e((tto)

K-<y"- for all t> t, (15)

which drives X 1(t) to satisfy the following inequality

II x 1 (t) 1_< K

From Equation (12) we have

X 2(t) = s(t) - [KpK] .(t). (16)

then by using the result of Equation (15)

II X2(t)I1-,* [1 + Z KIK

4

for all t > t, which proves (10b) and therefore we complete our proof.Lemma I implies that the tracking error of the trajectory and its derivative are uniformly

bounded, provided that the sliding surface is bounded for all time t > to. Using this results

of Lemma i, we can give the specifications of tracking errors dependent on convergence ratewhich is determined by the sliding surface, (9). In the next section, we are going to designa controller with efficient compensation which guarantees the boundedness of s(t), i.e., if11 s(t) 11_ -t for a given y, then the trajectory error is bounded in virtue of Lemma 1.

3 Efficient Sliding Mode Control for Robot Manipu-lators

There are a variety of disturbances acting on robot manipulators, which make robust trackilgcontrol highly desirable. It is commonly noted that the generalized nonlinear disturbances.D(q(t), 4(t), 0), should be compensated to improve performance. For sliding mode control ofthe robot system (4), the equivalent control of the augmented sliding surface (5) is obtained[17, 181

req(t) = D(q(t), 4(t), 0) + J(q(t), 0). (4d(t) + KX 2 + KOX 1 ) (17)

The SGD D(q(t), 4(t), ) is included in an equivalent control, Tq(t). SGD is generally com-plex, and difficult to be calculated directly due to uncertainties in the model.

In the paper, using the efficient estimation method so called disturbance observer, weconsider the following continuous control input, r(t)

r(t) = r(t) + r(t) (IS)

WVhere r(t) is the compensation term for SGD as well as the error of nominal inertia matrix.is not the direct calculation from D(q(t), 4(t), €) in the model (2) but the efficient estimation

of the generalized disturbance, D(q(t), 4(t), 0) only using the nominal inertia matrix, ...of the model (2) and an available acceleration information which can be obtained from thespeed information by means of Euler method.

T (t) = 'r(t) - JN"-qt

=D(q(t), 4(t), 0)4- AJ(q(t),0).- 4(t) -J(q(t), ().-A4l(t) - A(t ) (9

where AJ(q(t), 0) is the deviation between the real inertia matrix and its nominal value... q is the acceleration information error to the real acceleration value, r(t) is the cont rol

input delay error resulting from the digital control are defined by

AJ(q((t), €) = J(q(t), €) - JN

Ar(t) = 7(t - h) - -r(t)

where h is the sampling time. The second term in Equation (16) is defined as

;,(t) = (fq(t) + 7-(t)) (20)

5

where fq(t) is the modified equivalent control for the compensated dynamics of Equation(1), and is designed so that the error dynamics of the controlled system has the slidingsurface dynamics defined by Equation (7). It is defined as

trq(t) = J.V. (4d(t) + K. X 2 + Kp. X). (21)

7,(t) is the feedback term of the sliding surface resulting from the compensation error

rx(t) = JN {k,, " s(t) + kx2 O(t)}, 0a(t) - (t)0 +22

If we apply the input control torque given by Equations (18)-(22) to the robot system (5).we obtain the following equation

,Q(t) = -J-'(q(t), 0). (AJ(q(t), ) - 4(t) - At(t)) + Ai4(t) + qd

-J-1 (q(t), )JN.-[4ld+ K,.X2 +Kp.-X1 + klst+x (t) [[+-Js~t + X,11SMt 1 (23)

and the dynamics of s(t) is expressed in the following simple form

(t) = ni(t) - [kxl ,s(t) + k, 2 s(t) (2-1)11 S(t) 1 +6

where nt(t) E R' is the resultant disturbance vector given by

n,(t) = n I(A4(t), A',

=JN • J(q(t),) A4 + JN' " A7(t). (25)

For some positive constants el and E2 defined in (lOc), let the constant N be defined asfollows

N= max{ 1nl, (A4(t), AT(t), ) 1I; q(t) E B(ei; qd(t)) A 4(t) E B(E2 ; 4d(t)) } (26)t,q ,q,€

where the matrix norm is defined as the induced Euclidean norm, and for a positive numberp > 0 and a vector v E R n the boundary set defined by as

B(p; v) = {w E R;II - II- P}. (27

In Equation (25), the resultant disturbances are mainly dependent on the acceleration infor-mation error and the control input computation delay error other than system uncertaintiesOr modeling errors of robot manipulators. The disturbance observer can compensate od, -dling errors of the inertia matrix besides SGD. Thus the SMC design is independent ofmaximum bound of modeling errors in the parameter space but dependent on only the ac-celeration information error and the control time delay due to the digital implementation.Since a control input is continuous and accurate acceleration can be obtained, the maximumOf resultant disturbances, N becomes very small, thus we can design a new SMC withoutchattering.

6

Now, a stability property of the system (5) with control laws (18) -(22) will be stated inthe next theorem.

Theorem 1 Consider the robot system given by Equations (18)-(22). Assume that fo,some positive -y, i s(t,) 11_ y and I x(t.) l1< 7/tc are satisfied at the initial time t = t,, and

if the gain k. 2 satisfieskx2 > N - kxl • 6 (28)

for given k., and 6, then the global control system is uniformly bounded (i.e. the solutionX is uniformly bounded at origin in state space) for all t > t, except 11 s(t) II> q1 where r1is defined by

[6 + ( ] 6 N~71 a21+ 012 ai kX2

ProofIf we take V(t) = 1/2sT(t)s(t) as a Lyapnuov candidate and differentiate with respect 1o

time, it followsdV(t) = ST (t). (t)

dts(t) (30)

= T(t). ni(t) - ST(t) {k,. s(t) + k I s(t) II(3

If we use the matrix inequality

II xTAy Il11x I " II y I" A II

it follows dt) II s(t) 112 (31)dV(t) <I 11 s(t) I ni(t)I1 -kH ," II s(t) 1 - kx2 II s(t)II +3

So that II ni(t) 1< N is satisfied from the definite of N, and we can rewrite Equation (31)as

dV t) 11 s(t)II *{N - k I,. 11 s(t) II -k-2 II s(t) II 1dt -I <z t 11SM1 +6

I s(t) II-kx . {11 s(t) 112 +2. a v II s(t) II - 01} (32)

II s(t) II +6

at t = ti. If the gain k. 1 , and k. 2 satisfy the condition (27),

dV(t)dt

at f > t, until 11 s(t) 11 a + fl - a, which completes the proof of Theorem 1.Theorem 1 guarantees the uniform bounded stability of the proposed continuous S\IC

(18)-(22) for robot manipulators. This controller is shown in Fig. 1, its structure is simple.The sampling time can be as small as possible so that the acceleration information is accurateenough, therefore, the maximum values N is very small. If a smaller 6 is used in controlalgorithm (22), then the lower bound of qh is decreased. The lower bound r71 can be decreasedby increase of k., for given 6 and N so that 77, is sufficiently small in comparison to -,. If

7

the initial position error is small which is reasonable in case of the known initial state ofrobot manipulators, it is evident that s(t) remains closer to the sliding surface s(t) = 0 andthe trajectory error is also small, hence we may assume small -Y for the desired specificationsof the small trajectory errors. The feedback control (18)-(22) designed by Theorem 1 andLemma 1 maintains the stability of the system with the prescribed performance. To designthe proposed sliding mode controller, firstly the desired sliding surface defining which definesthe desired error dynamics is chosen, i.e., the gains, K and K, are specified. Then the gains,k,, and kx 2, in Equation (22) are selected based in Theorem 1. The design does not needthe information of maximum bound of system parameter variations or uncertainties becauseof the efficient on-line compensation. The proposed SMC can be designed to be accurateand robust against to parameter uncertainties.

+I+ ) Robot q(t)qd(t)--'" + -N[- q(t)

+ Manipulator

Tkes(t)+k o T2( r(t) C t l

Is(t)=Xs+KIX I I dK v +KpfXldt '0-T0NQ(t1

X1Di sturbianc~e: Observer::":

Trajectory Track-ng Controller

Fig. 1. The block diagram of the proposed SMC.

1 Is()1l + 5 for Theorem I

(1() "ts(() for Theorem 2

is(t) + 6

4 Modified Sliding Control for Robot Manipulators

In this section, we present a modified control algorithm from the control law of the previoussection so that the efficient computation is possible in the case of the digital implementation.The modification replaces 7r,(t) in Equation (22) by

rx(t) = -JN" (kxl" s(t) + kx2 b(t)) (I)

8

where the i - th element of vector Ob(t) = [l(t), 0 2 (t),-.. ,,(t)]T is defined by

Ot) = s2(t)I s (t) 1 +(:

If we apply the control given by Equations (18)-(21) and (34) to the system (1), then

(t) = ni(t) - k2- s(t) - ki -cb(t) (36)

where the disturbance vector ni(t) is also given by Equation (25). Let's denote its i - helement by nji(t). For some positive numbers y, e and > 0, let the constant N, be defined

as follows

= sup {n,[n 1 (Ai4(t), Ar(t), 11; q(t) E E(Ei; qd(t) A q(t) E E(E2 : q(t))} (37)i,z,w~y, i

where the neighborhood set is defined as

E(p; v) = {w E R; 11 w - v I < p} (38)

for any scalar p and vector v E Rn, where 11 • 1o is the infinity norm of the vector which isdefined as the maximum absolute value of its components.

Theorem 2 : Consider the robot system (5) with controls given by (18)-(21) and (31).Assume that for some -y > 0, 1 S(to) II.o_ -y and 11 x(to) Il.:_ y/yc is satisfied at the initialtime t,. If the gain k, 2 satisfies

kX2 :_ N. - 6. k(3)

for given k. 1 and 6, then the global control system is uniformly bounded (i.e. the solu ionX is uniformly bounded at origin in state space) for all t > t, until 11 s(t) 11_ q2 where r/2 is

defined by72= a2+ 0 2 C 1 = 2 '

ProofIf we take V(t) = 1/2sr(t)s(t) as a Lyapnuov candidate and differentiate it with respect

to time, it followsdV(t) = sT(t) . S(t)dt

= {sT. ni(t) - ST(t) (k. s(t) + k, 2 s(t). ab(t))}. (.41)

We can rewrite the above equation as

dX'(t) _s____)_diq -{Z s,(t) n,(t) - Z k,. s,(t)2 - 8s (t) 2 (12)

dt <2 sj(t) 1 +6

So it followsdv (t) < {E Is(t) .nI -(t) t - k<, .si(t)2 -,x -/ Sj(t)

dV-t) I _S() +

From the definition of N , and norm inequality of 11 s(t) 11!i s(t) I11, we can rewrite theequation (43) as

dV(t) <11 s(t) 11.0 IN. - kxl1 11 s(t) 11.0 -kxz IS(t) 10dt -11I(t) Iloo+ 6dkt---'d S( IL- N - . I I (t) 1 + - 2' .11 S (t) 1 -0 1 ( 4

II s(t) II 4= II s(t) 110 +6 "{jIs(t) llO +2. ,2.11Is(t)112 -13}.(4

If the gain k, 1 , and and k. 2 satisfy the condition (39),

dV(t)

dt

at t > t0 , until I s(t)1> a + fl - a 2, the proof of Theorem 2 is completed.The desired performance of the modified controller for robot manipulators can be ob-

tained by using Theorem 2. The feedback sliding mode control (18)-(21) and (34) maintainthe uniformly bounded stability of the system as far as the condition (39) is satisfied, butTheorem 2 is only a sufficient condition for stability. Under the same conditions, N definedin Equation (37) is always larger than N defined in Equation (26). Thus as shown in Equa-tions (39) and (40), for the same trajectory error the gains of the modified controller aredesigned to be larger than those of the SMC defined in section 3. Under the same conditionsfor the both controls, the error of the modified version be smaller than that of the originalSMC because of II s(t) 11>1 si(t) I.

5 Numerical Simulation

The numerical simulations are performed for the purpose that is to show accurate androbust trajectory tracking property of the proposed algorithms compared with the CTM.The SCARA-type two degree-of-freedom manipulator dynamics [1] used for simulation areas follows

rIM 1+ IM t52 + mn2C2 IM2 + IM2C2 .q,±r2 it , 22 [M2 q2[ m]c[2 ] ~ q ]

+ -n2S2q2 - rn2S2q12 + Mm1 gC 1 + rM2gC 12 + M 2gC 1+12 [ 2 1 2 2 2 1- 2 (46)j 2D l I Ij 2gC12I

where C,, S, and C,, imply cos(qi), sin(qi) and cos(q + qj), respectively. The parametersare m = m 2 = 0.782[kg], l = 0.23[m] and g = 9.8[m/sec2 . The reasonable modeling erroref robot dynamics is produced for each mass and length. Moreover, unknown payload isreflected to m 2. To test the robustness to modeling uncertainties and unknown payload.simulation studies are carried for the CTM and two proposed SMCs under three case condi-tions, i.e.,(i):no modeling error and no unknown payload, (ii):10 [%] modeling error and 0.5[kg] unknown payload, and (iii): 10 [%] modeling error and 1 [kg] unknown payload. Thedesired trajectory for each link was

(qf - qi) (qf - qi) -si'n(irt)1 180qd(t) -qi+ ( - - [degree] (47)

10

where the initial position qi = [0.0 0 .0 ]T [degree], and the final position qf = [60. - 60.]T

[degree]. We take the execution time of the trajectory tracking as 2 seconds. The necessarytime for the computation of model (2) was selected to be 2 [msec], and this computation timeis considered as a time delay of input for the simulation of the CTM. The gain k. = 400 - I.k, = 40 • I were used in the sliding surface (6) so that the ideal sliding dynamics haveits eigenvalues to be assigned to -20. The corresponding constants defined in (9) , K andn become 14.7 and 10. Using the results of Lemma 1, the trajectory error, X 1, and itsderivative, X2 , are bounded as c = 1.47 .- and 62 = 592 .y for a given -y of the constraint tothe sliding surface. The smaller bound of the sliding surface, Y, the smaller trajectory errorand its derivative. For 0.01 [degree] maximum tracking error, -y was selected to be 0.006S.

Now, in order to obtain the above prescribed tracking performance, the proposed SNICsare designed to make the sliding surface be bounded by -. Assuming the initial positionof the robot manipulator is known, the conditions, IIX(to)Il < -y/t and IIs(to)H < - aresatisfied at initial time t = to. The controller gains k~l and k, 2 were selected to be 50 and10 which satisfy the conditions (39) and (45) for given 8 = 0.05 and N or NV so that r/iand q,2 are sufficiently small with respect to the selected -' by Theorem I and Theorem 2.For comparison of the three case conditions, the N and Noo are summarized in Table 1. A sthe payload increases, the value N and No, increased. As a results the lower bound of A',2becomes larger such that the stability of the system is maintained. The simulation blockdiagram of proposed algorithms are shown in Fig. 1. The proposed trajectory trackingcontroller is consisted of the modified equivalent control, -Tq(t) (21), feedback of the slidingsurface, 7,(t) (22) and disturbance observer, r(t) (19), terms.

The gains K = 40. 1, K, = 400- I were used in the CTM algorithm. The simulationresults of the CTM are shown in Fig. 2. The tracking errors for the CTM are shown in

Fig. 2-1 (a) for link one and (b) for link two, and its corresponding control inputs are shownin Fig. 2-2 (a) for link one and in Fig. 2-2 (b) for link two. As the payload increases.the root-mean-square (rms) error is rapidly increased from 0.0073 [degree] to 3.6801 [degreejusing the CTM algorithm. Thus the CTM need to be added the robust algorithm against

model uncertainties and unknown payloads.The results of the proposed SMC simulations are shown in Fig. 3 for Theorem 1 and

in Fig. 4 for Theorem 2. The tracking errors of the proposed SMC are presented in Fig.:3-1 and Fig. 4-1. The tracking errors by Theorem 2 are smaller than those by TheoremI because the norm of the sliding surface in (22) is greater than the infinity norm of thesliding surface in (35). The correspondent continuous control inputs and sliding surfaces aredepicted in Fig. 3-2 and Fig. 3-3 for Theorem 1 and in Fig. 4-2 and Fig. 4-3 for Theorem2. The simulation results in view of maximum and rms values are summarized in Table t.Using the proposed algorithm, the resultant error maintains nearly unchanged Especially.

note that using the proposed algorithm the tracking error is not increased so much as theincrease of the palyload error, for the case Theorem 1 and Theorem 2.

From the simulations,we have found that the proposed two algorithms provides a better

performance than the CTM algorithm with respect to the tracking errors. Moreover, thetracking errors of the proposed algorithms satisfy the prescribed specifications as shown inFig. 3-1 and Fig. 4-1 subject to parameter and payload uncertainties.

11

Table I. Numerical comparison between CTM and proposed SMC

Error Error Control Control Surface Surface N N-Method CASE (max) (rns) (max) (rms) (max) (rns)

-degree]. [degree] [Nmj [Nm] L

1 0.0066 0.0073 3.6438 3.7569

CTM 2 3.6375 3.6801 5.7958 5.83001 6.3872 6.4713 8.0360 8.0453

1 0.0004 0.0005 3.6438 3.7569 0.0015 0.0016 0.3281

THM 1 2 0.0008 0.0010 5.3968 5.6014 0.0031 0.0034 0.4862

SMC 3 0.0011 0.0016 7.1498 7.4476 0.0041 0.0044 0.6462 -

1 0.0002 0.0002 3.6439 3.7569 0.0008 0.0008 0.4681

TIHM 2 2 0.0004 0.0005 5.3968 5.6014 0.0015 0.0016 0.6911

3 0.0005 0.0006 7.1498 7.4476 0.0020 0.0021 0.9153

12

10

9

~ 7 (i) : CASE 1a) 6 (di): CASE 2

M (iii): CASE 3

cc 2

z 0

< -2

I- -3

- 0 0.2 0.4 0.6 0.8 1 '12 1.4 1.6 1.8 2

TIME [sec](a) link one

10

CM 7

0 1.3.TIE[sc

.33

10

9

'-C S

z

0

3

z0 2

1J

0--

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

TIME [sec](a) link to

41

0.001

00008

,oooooe - )0) 0.00070) 0.0006(D

-~00005-

0 00040.0003

a: 0.0002o 0.00010a:

Uj-0.00011

UD -0.0002 -1Z -0000

-00004 (i) :CASE1

< -00006 - (ii) "CASE 2-000- (iii) "CASE 3

-00009 -

-00010 02 04 06 08 1 12 14 16 18 2

TIME [sec](a) link one

a 0102

00025

0) 00015

0)

c)CCJ

Uj

z (i) :CASE 1

O 0001 (ii) CASE 2S(iii) CASE 3

-00015 -4

-00,020 02 04 06 08 1 12 14 16 16 2

TIME [sec]

(b) link two

Fig. 3-1 Tracking error for Theorem I of proposed SMC

(i) case 1 no model error and no unknown payload(ii) case 2: 10 [%] model error and 0.5 [kgl unknown payload(ii) case 3 : 10 [%] model error and 1.0 [kg] unknown payload

15

10

9-

(i) CASE 1

0 4

3-z0 2

0 02 04 06 08 1 1,2 1 4 1 6 1 8 2

TIME [sec](a) link one

4

(i) CASE 13 (ii): CASE 2

E (iii) CASE3z 2i

-J0

z0

0

-10 02 04 06 00 1 1.2 1A 16 a a 2

TIME [sec](b) link two

Fig. 3-2 Corresponding control inputs by Theorem I of proposed SMC

16

0.005

0.004

0.003 - (i): CASE 1LIJ (Ji) :CASE 2

0.002 (iii): CASE 3U-

CC 0.001

Cl) 0

0 -0.001-

- -0.002

Cn -0.003

-0.004

-00050 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

TIME [sec](a) link one

0.005

0.004 iiC 0,003 (i0 0.002

UL-a: 0001

Cn 0

0 -0.001

o -0002 - (i) CASE 1__j (ii) CASE 2

W -0.003 - (iii) CASE 3-0 004

-00050 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 118 2

TIME [sec](b) link two

Fig. 3-3 Corresponding sliding surfaces for Theorem 1 of proposed SMC.

17

0.00 i

00008

Q) 0.0007

00006S0 0005

-M 0.0004

0.0003

S0.0002

o 0.0001(x 0

CC -0,0001wU -00002

-0.0003z -00004 (i): CASE 1-0,000 (ii) "CASE 2C-)

.~-00006CC -0,00000 - (iii) • CASE 3

-0.0006-0 0000

-0.001 . . . . . .

0 02 04 06 08 1 1,2 14 16 1.8 2

TIME [sec]

(a) link one

0001

000090 00080 0007

00006

< 0 00050 0004

0 0003

0 0002C 00001

0_

-r00001UJ -0 0002

oD 0 0003Z -00004 (i):CASE 1. -0.o005 (ii) "CASE 2

O -000o- (iii) CASE 3-00006-00001

0.2 04 06 0.8 1 12 1 4 1.6 18 2

TIME [sec]

(b) link two

Fig. 4-1 Tracking errors for Theorem 2 of proposed SMC

(i) case 1 : no model error and no unknown payload(ii) case 2: 10 [%] model error and 0.5 [kg] unknown payload

(iii) case 3 : 10 [%] model error and 1.0 [kg] unknown payload

18

10

9

T 7

z 30

0C-1

TIME [sec]

(a) link one

4*

(i) CASE 13 (ii) CASE 2

E (iii): CASE 3z ii

2

__j (6i)0

z0

0

0 02 04 06 O8 1 12 14 16s 1a 2

TIME [sec]

(b) link two

Fig. 4-2 Corresponding control inputs by Theorem 2 of propoed SMC

19

0.005

0004

0

-0002-

-000

.00050 02 04 06 08 1 12 1 4 1 6 18 2

TIME [sec]

(a) link one

0 005

0004-

0002

U-i -

Ul) 0

- -0001

-~-0002 (i) CASE 1-003 (i i) -CASE 2

-000 (iii) CASE 3-0004 -

-0.0m50 02 04 06 08 1 1 2 1 4 1 6 1 8 2

TIME [sec]

(b) link twoFig. 4-3 Corresponding sliding surface for Thieorem 2 of proposed SMIC

20

6 Conclusions

In this paper, continuous sliding control algorithms have been proposed for accurate androbust tracking control of robot manipulator and their stability properties are analyzed illdetail. As a preliminary work, the relationship between the maximum bound of the trackingerror and that of the sliding surface is presented in Lemma 1. The continuous sliding modecontrol algorithm with disturbance observer is proposed and its uniform bounded stabilityproperty is proven in Theorem 1. To improve efficiency, a modification to the proposedalgorithm has been preceded and its stability property is presented in Theorem 2. Undorthese control algorithms, the tracking errors can be reduced to the prescribed range withoutchattering problems based on the above stability analysis.

The simulation results have shown that the proposed algorithms are very efficient for ttfetrajectory tracking and robust to the modeling inaccuracy and unknown payloads.

21

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