Synthesis of a Multivariable Control System with Variable Structure for a Six-Link Manipulator

Copyright © IFAC Large Scale Systems, Rio Patras , Greece, 1998

SYNTHESIS OF A MULTIVARIABLE CONTROL SYSTEM WIm VARIABLE STRUCTURE FOR A SIX-LINK MANIPULATOR

L Markina t, A. Zakrevskij

*

t Belarusian State University, 220080 Minsk. Belarus. Scorina Av.4.Fax: 375-1 72-208821 :j: Institute of Engineering Cybernetics NAN B. 220012. Minsk. BeJarus. Surganova Str.6. Fax: 375-172-318403. [email protected]

Abstract. A control system for a robot-manipulator is synthesised on the base of a feedback presenting the gripper location and the approach vector which can be softwareimplemented by means of solving the problems of direct kinematics and multivariable control in a class of variable structure systems. That enables not to regard the inverse kinematics problems. Taking into account deviations of the actual manipulator parameters from the prescribed ones when solving the direct kinematics problem or using a sensor for the manipulator gripper location considerably increase the dynamic accuracy of the robot manipulations. That is confinned by experiments with the industrial robot PUMA 560. Copyright © 1998 IFAC

Key Words: Multivariable control system; manipulator; variable structure; synthesis.

1. INTRODUCTION

helps to cope with this effect but causes the appearance of a non-zero steady-state error even when there

The conventional approach to a manipulator control system design consists in the movement control for each joint using electromechanical servomechanisms. Such an approach does not allow to take into account coupling of manipulator degrees of freedom leading to non-linear differential equations descnbing dynamic of the system as a whole. That decreases the accuracy and speed of positioning and reduces the area of the robot application (Po, et al. , 1987). Among different methods which take into account non-linear properties of the manipulator model, the methods of adaptive control play an important role. They can be divided into two basic classes: methods using large amplification coefficients for the compensation of non-linear effects (Lim and Eslami, 1987) and methods of "the calculated moment" (Sadegh and Horowitz, 1987). The first ones leaves unformalimf the choice of some parameters and coefficients in the adaptation contour, and there arises rather frequently the effect of control action trembling. Introducing the zone of insensitivity into the laws of control and adaptation

are no disturbances. The methods of "the calculated moment" require a rather complete information about the manipulator model, and sometimes (Craid, et al., 1986) - measuring of angle accelerations or using special filter algorithms for restoration of these accelerations (Middleton and Goodvin, 1986). Sometimes the second method of Lyapunov is used for the synthesis of manipulator control systems (Dubowsky and DesForses, 1979; Durrant-Shyte, 1985). It is known, that this method could guarantee their steadiness but transient responses can turn out to be 1msatisfa<;tory. There are widely used nowadays control algorithms with variable structures (Young, 1988) attracting the attention of designers by their simplicity and properties of robustness. The advantage of this approach consists in the possibility of obtaining prescribed transient responses and providing at the same time the regulator steadiness. Sliding modes in systems with variable structures are highly efficient and often are used for non-linear decoupling of joints (DeCario, et al.• 1988) ..

623

inference of analytical expressions for the matrices of non-linear multijoint controllers with variable structures on the base of the proposed S}nthesis conception.

In recent time some papers were published regarding manipulator control in Cartesian space (Utkin, 1974; Sadegh and Horowitz, 1990). Unlike the robot joint space control the Cartesian space control directly determines the end-effector location in Cartesian space which simplifies the problems of planning trajectories and going round obstacles (KhaW and Chevallereau, 1987; Chem-Yuan and Shay-Ping, 1991). It is not necessary by that to map the trajectory of the tool centre point onto the joint space, i.e. to solve the inverse kinematics problem which is non-linear, rather complicated and having nonunique solutions. In this case the target gripper locatio is compared with the actual one, and a control signal is calculated which enables to obtain the linear differential equations in regard to the error vector. The co-ordinate vector of actual position is determined by solving the direct kinematics problem. using information from angle sensors located in the manipulator joint links. Cartesian space control provides a steady accuracy of the gripper displacement over the whole space, whereas even small errors in joint space controllers can lead to essential control errors in Cartesian space (Lukyanets, et al., 1991).

Hereafter index 1 is used to denote matrices and vectors for the bottom three-link controller, and index 11 - the same for the top one. Neglecting the gravitational force vector, let us present the manipulator model as

iji = Ji-J(qXMsi(U i - Vi)- Bi(q,q)qd , where i=1, 11,

ij = [ijJ '

ijIl

q = [q/ ' qII

r-

(I)

r,q= [q! ' qnr' =[ r= r-

vectors of joint co-ordinates, joint

u! '

velocity and joint acceleration; U vector of relative control actions;

v

U II

[V! ' VII

vector of relative velocities;

(2)

In this paper, a control system is suggested which could be implemented as a digital controller intended for using in the robot "00D-2.5 AE" (Mogilev, Belarus). It is proposed to decouple the manipulator control system. into two parts related to the top and bottom manipulator triplelinks. That enables to reduce the time for computing control signals - by diminishing the dimensionality of the processed matrices.

- matrix of initial driving torque's;

The response error is calculated by measuring gen-

eralised co-ordinates, finding corresponding Cartesian co-ordinates and comparing the target vector with the actual one. The accuracy of measurements

- inertia ma~

JJl(q) J12(q) J13(q)] /J(q) = J21 (q) J22 (q) J23 (q) [J lq) J (q) J (q) 3 32 33

depends on the accuracy of generalised co-ordinate sensors and the manipulator mechanical errors.

2. SYNfHESIS OF CONfROL FOR A SIX-LINK MANIPULATOR

- inertia matrix for the top hree-link part;

There is described below the synthesis of an algorithm for the controller of a multijoint manipulator with the program-implemented gripper location feedback. The problem of synthesis is reduced to constructing such a control in a class of systems with variable structures which enables to transform an intricate non-linear system of n-order equations into a system of independent known non-linear differential equations of 2-order in regard to the components of the control error vector. Due to the complexity of the synthesis of control for a six-link manipulator there arises the problem. of the controller decomposition into the two ones - for "bottom" and "top" three-link parts accordingly, and that of designing the coupling between them. There is given below the

(3)

- matrix which takes into account the influence of centrifugal and Coriolis forces. Let us consider coupling of degrees of freedom to be unessential for the three last ones, hence the inertia matrix for the bottom part can be presented as follows:

Target control vector

X in ,

output vector

Xo

and

error vector E are presented below as block vectors

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containing prescribed position and orientation vectors, actual position and orientation vectors, and error position and orientation vectors:

= I;R2 '

RlI

[Cl

=~

I;

The gripper position vector X can be expressed as follows :

R)

-Sl Cl

0

(11)

~r; 1

0

-S23

1

s"o ],

0

C23

[-s,C,C, - C,S,

-C4SSC6

-c,c,s, - C,Co ]

-C4CSC~ - S4S6

-S4 CS C6

-S4CSC6

=

-CSc 6

+ C4C6

SSS6

(5)

o and the gripper-normal vector (orientation vector) as the function (6)

Find expressions for velocities of changing both the position vector and orientation vector:

x = R/(q)ilI +D2(q)q/l ,

(7)

;, = RIl(q)q1l + D2(q)iIJ ,

(8)

where

=-C1S23Y - C1C2~SC6 ' d 21 = C~23Y - Sl( S4 CSC6 + C4S6 ) d 23 = -Sl S23Y - SlC2~SC6' d 23 = -C23Y + S2~5C6 '

d 13

(9)

For manipulator "PUMA-560", with kinematics, parameters and position function defined in (Fu, et al., 1987) the following equation holds: (10)

Introducing designations :

where:

Dr(q,q-) = D1 (q)q2'

_ [-Sl X - c 1( d2

~ -

ClS2~SC6 '

+ d6 s4SS )

Dl1 (q,q) =D 2 (q)TqI'

-C1 X

define derivatives for

o

i

= Sin(ql) , S4 = Sin(Q4) , Cl = COs(Q1) , C2 =cos( Q2) ,CS =cos( Qs ) , X =a 2c 2 + a 3c 23 + d 4s23 + d6C23C4SS + d 6s23CS '

Sl

X

and ;, :

= Rr(q)iir + Rl1(q)qI + Dr(q,q) ,

;; =

Rl1 (q)iil1 + Rl1 (q)ql1 + Dl1 (q,q),

(12) (13)

Similar to the known systems with variable structure (Utkin, 1974) the expressions for sliding surfaces look as follows:

J

(14) Let us compute derivatives

Ttl

and

Ttll

and put the

expressions (7), (8), (12), (13) into them obtaining as the result

625

"1 = ~Xin +Xin - ~RI(q)ql - ~DI(q,q)-RI(ij)ql - DI(q,q) - RI (q)ql '

one can obtain a sliding mode on the surfaces

=0, TIll =0, in case when calculated matrices Ni' 4 ,Ci ,Hi and calculated vectors Q are cho-

TII

(15)

sen in the appropriate way:

Ni = M~~Ji(q)R;I(q)P; ,

(18) (19)

Using in (15) and (16) values ii[ and iiII taken from (I), and introducing the control function

4 = M~~Ji(q)R;l(q), Qi = Li [P;Di (q,q) + Di(q,q)], C i = M~:[ Ji(q )p; - Ms iH';-I K~~ ] +

"i = Hif3 i + LiX[ + Nix[ - Ciqi - Qi ,(17)

+M1i[Ji(q)~-l(q)~(q)- Bi(q,q)] ,

where i=/,11,

Hi = M~~Ji(q)~-l(q)Msi' where

Mi

(20)

(21) (22)

,i-l,2 and - diagonal coefficient matri-

ces. The structure of a control system for six-link manipulator designed on the base of synthesised controllers is given on Fig. 1.

Fig 1. The structure of a multilink control system with variable structure for a six-link manipulator.

626

existence of sliding mode on the both surfaces of the structure changing. The obtained results remain adequate when used in the main feedback circuit of direct kinematics corresponding both to the ideal kinematics scheme and the scheme which takes into account the mechanism errors.

Vectors of orientation of the gripper and vectors of its position in Cartesian space are given in it independently. There exist in controllers for the bottom and top three-link parts cross-<:ouples on the joint co-ordinates and joint velocities, and control vectors for them are calculated on separate microprocessors with restricted exchange between them. Gravitation vector and approximate character of the manipulator model were not taken into account when looking for the control function for the given system. One can use additional PI-regulators to compensate that (Lukyanets, et al., 1991). Having chosen Ni,L; ,Q,Ci,Hi , we obtain from (15) and (16):

=

:>. 0

,.......---~------------,

=

Ttl -MA~/' TtII -MII J3 II · It follows from here that under the suggested method for control of six-link manipulator there exists a sliding mode on the switching hypersur:face

[Tt/ , fllI

r

and the sys-

tem is stable.

3. EXPERIMENTAL RESEARCH OVER A CONTROL SYSTEM FOR A TWO-LINK MANIPULATOR

...•

The objective of experimental research over the control system over a ~link manipulator in Cartesian space is checking of existence of sliding mode, stability of the system, evaluation of its accuracy. For conducting natural testing of the control system for a robot of the type "PUMA 560" there were used its second and third degrees of its freedom located in the horizontal plane. It is necessary to have numerical values of initial driving torque' s, n~ load velocities and inertia moments. There was used in the experiment the manipulator PUMA-560, (PM01) produced by the firm NOKIA (Finland) with the control system "NoaM-36", developed in Minsk (/I=O.438m, IrO .433lm). That is wby parameters identification for a control object has been fulfilled firstly.

-
..

- --

..

- ... - -

......•. •_.. . -

- -.

. _. .. .. . ..

. .... _..

". -

Li--..----- - -- ----- . -.-.... - --i-.-.---- --- - -.. - - - . , . - ----

..-- - '--"-i

~ -o:3: ~_I.:~~:;;:~=~:E~=1

2 -1- n[J--:. ·~--- -- -- --~ --·- ---~--------1 . : ,

-0 . ""'D . .. hnn ~ "';

o

1SO

t.1I:

Fig.2. Changing co-ordinates and signals of the control system for a two-link manipulator under step input When obtaining this conclusion it was taken into account the deviation of joint rotation axis's from the normal positioning in regard to desired planes of rotation up to 2 grades, without changing by that the position function F and matrix R. Position control in response to sinusoidal input, i.e. the gripper moving along a circle of radius O.lm with velocities O.64mIs and 0 .05m1s is shown on Fig.3 . As can be seen from there, the error of response does not exceed O.5mm. The accuracy obtained by the suggested control algorithm more then in 1.5 times is higher than in case of the conventional control of the same manipulator, when for the velocity 0.52m1s the error equals 0.78mm (Lukyanets, et al., 1991)

As a result, the following values were obtained for ~load angular velocities and initial driving torque' s: W] . ]= 8.7 rad/s, W].3= 4.5rad/s,

M SI.) =255[1, M SJ,3 =1l0U.

"

By analysing the

curve of electric motor starting acceleration under nominal voltage UN there were found motor mechanical time constants Tml , Tm2 under orientation of the third degree of freedom correspondingly under q2 = 0 and q2 = 1f / 2 , and then correspondent inertia moments. Fig.2 illustrates basic signals changing by step input. The position accuracy equals by that 0.1 mm. The discontinuous nature of control signals confirms the

627

Dubowsky S. and D.T. DesForses (1979). The application of model referenced adaptive control to robot manipulators. Trans. ASME J. Dynam. Syst., Measur., Contr. voL101, pp. 193 - 200. Durrant-Shyte H. (1985). Practical adaptive control of actuated spatial mechanisms. IEEE Int. Con! Robot. Automat., pp. 650. Fu K.S., RC. Gonzales and C.S.G. Lee (1987).

Robotics: Control, Sensing, Vision and Intelligence. McGraw-Hill, New-York, pp. 5~2. Khalil W. and C. Chevallereau (1987). An efficient

""'5. ,,' :>1 OQ .

l00.~·

'" "

Fig.3. Changing co-ordinates and signals of the control system for a ~link manipulator under try-out a circle (v...... = O.64m/s).

4. CONCLUSION Manipulator control in Cartesian space enables not to solve inverse kinematics problem and to increase accuracy of control response to programmed trajectories, especially if to take into account deviations of manipulator parameters from the prescribed values in the direct kinematics or if to use the gripper position sensor.

algorythm for the dynamic control of robots in the Cartesian space. Proc. 26th IEEE Con! on Dec. and Contr., vol 1. pp. 582 - 588. Lim K. Y. and M Eslami (1987). Robust adaptive controller designs for robot manipulator systems. IEEE J. RobotiCS Automat., vol RA-3. pp. 54 -66. Lukvanets O.S., L.l Matioukhina and AS. Mikha- lyev (1991). Synthesis if tw~links manipulator control with programmed gripper position Avtomatika i vychislitelnaya feedback. tekhnilm, is.20, pp. 29-35 (in Russian). MiddletonRH. and G.S. Goodvin (1986). Adaptive computed torque control for rigid-link manipulators. 25th Con! on Decision and Control (Athens, Greece). pp. 68 - 73. Sadegh N. and R Horowitz (1987). Stability analysis of an adaptive controller for robotic manipulators. IEEE Int. Con! on Robotics and Automation (Raleigh, NC), pp 45-51 Sadegh N. and R. Horowitz (1990). Stability and robustness analysis of a class of adaptive controllers for robotic manipulators. The International J. of Robotics Research., vol. 9, No. 3. pp. 74-80 Utkin V.I. (1974). Sliding modes and their applica-

tion in variable structure systems. Moscow: Nauka (in Russian), pp 65-85. Young K.D. (1988). A variable structure model following control design for robotic application. IEEE J. Robot. Automat., voL 4, pp. 556 - 561.

ACKNOWLEDGEMENT The work was fulfilled at the chair of cybernetics of the Belarus State University, the chair head - DSc, Prof. AMichalyev.

REFERENCES Chem-Yuan K. and W. Shay-Ping (1991). Robust position control of robotic manipulator in Cartesian coordinates. IEEE Trans. Rob. and Autom. 7. No. 5. pp. 653 - 659. Craid J.J., P. Hsu. and S.S. Satry (1986). Adaptive control of mechanical manipulators. IEEE Int.

Con! on Robotics and Automation. (San Francisco, C4), pp 418-432. DeCarlo RA, S.H. Zak. and G. P. Matthews (1988). Variable structure control of nonlinear multivariable systems. Proc. of the IEEE., voL 76, N3, pp. 212 - 232

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