Optimal Adjustment of Feedback Gains of Point-to-Point Control Systems of Manipulators Dynamics

OPTIt\IAL ROBOTIC: C:O\:TROL

Copyright © IFAC Robot Control 'HS), Barceluna, Spain, 1985

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OPTIMAL ADJUSTMENT OF FEEDBACK GAINS OF POINT-TO-POINT CONTROL SYSTEMS OF MANIPULATORS DYNAMICS P. Marinov and P. Kiriazov IlI.Ililllll'

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Abstract. On the basis of real manipulator dynamics a direct metNod for feedback point -to-point control synthesis is proposed. This problem is very important especial l y in the case of higher operation speeds and ha vier manipulation l oads . The control synthesis algorithm consists in an iterative adjustment of the feedback gains in proportional plus derivative (PD) controllers of the joint actuators. Control laws are in accordance with minimum time - or energy l oss -cri ter i on . The values of the feedback gains are obtained from the exact solution of the corresponding two - point boundary - value problem. The method can be applied on the manipulator itself for final adjustment of the control parameters . As a demonstrative example , a dynamic model of a direct current dr i ven manipulator with cylindrical coordinates is taken into consideration. Keywords . Robots; dynamic response ; two - point boundary- va lue problem; con trol system synthesis; optimal control . IN TRODUCTION

derivative (PD ) controllers of the joint

In most tech n ological processes, manipu l a -

actuators. Control laws are in accordance

tors are mainly used as positioning devi -

with minimum time- or energy loss - crite-

ces . The path between operation points is

rion. With given initial and final states ,

not specified and depends on the control

the va lues of the feedback gains are obtai -

algorithm and hardware. In this case , it is

ned resolving the corresponding two - point

important to design an efficient controller

boundary - value problem (TPBVP ). The exact

which requires less time and energy, provi -

solution of this prob lem provides a desired

des smooth positioning and desired accuracy .

position ing accuracy and improved dynamic pe rf ormance .

This prob l em was the subject of some recent in vest igations by Marinov and Kiriazov

The algorithm of the proposed method is

(1983, 1984) where suboptimal open -l oop

illust rated on a dynamic model of a direct

control algorithms are proposed with the

current driven manipulator with cylindrical

foregoing goa l. With the concept of avera -

coordinates .

ging the manipu lator dynamics at each sampling interval, a feedback controller STATEMENT OF THE PROBLEM

with a weighted minimum time-fuel criterion is developed by Kim and Shin (1983). In fast simultaneous coupled jOint motions and

Using the Lagrangian formulation, the mathe-

hav ier loads, this approach leads to over-

matical mode l of manipulators dynamics in

shoots and settling time prolongation.

coupled joint operations can be written as follov.'s

In the present work, a method for feedback

M ( q )~

contro l synthesis on the basis of real ma -

+

N ( q , ~)

=T

where M(q ) is an nxn inertia matrix , N (q , q )

nipulator dynamics in pOint - to - point moti -

is a nxl vector which represents centrifu -

ons is proposed . The algorithm of this

gal , Coriolis , gravitation and friction

method consists in an iterati ve adjustment

forces , T is a generalized force / torque

of the feedback gains in proportional plus 117

ll H

P.

~I arin(}\'

q are vecto rs of n - joint

vector, q , q ,

a nd 1' . Kiria zoy

The n- dimensiona l vector F depends on the

coordinates , ve l ocities and accelerations ,

v ector of feedback ga i ns ki ( if there is no

respectively.

chatterin g rela y operation ). So , in o r der

The magnitudes of driving forces or t or-

one has t o solve the fol l owing shoot i ng

ques are depending o n the control inputs :

equation :

t o satisfy another final condition (4) 1 '

T

i

Ti (qi , qi ' u i )

l

:::: 1, . .. , n

(2)

Boundary conditions : q (t

o

0

q , q (t

)

o

f

o -

)

f q , q (tf)

)

q

f

(7)

This equat i on is solvable using the well in itial state (3)

q (t

F( k )

o-

fina l state (4 )

known grad ient' s , bisection ' s o r other search methods ( see Stoe r and Bulirsch , 1980 ), where convergence conditions are g i ven , too . Ha v ing obtained the exact values in the off -l ine case , the proposed it era -

We make use of a r e l ay se r vo system with ( PO) contro ll ers that has high po tential for improving the manipula t or pe rformance ( time or energy consumption ) :

tive proced ure can be perfo r med on the ma nipulator its e l f f o r final adjustme nt of these values. Such a self -l earning proce d ur e is needed since it is practically im-

(5)

poss ibl e the dynamic model ( 1

~

2 ) to pre -

sent the real man i pu l ator dynamics. The magnitude funct i on U for each joint is i composed of two alternatively changing con trol laws (when c r ossing t he switching line in the phase p l ane (q. , q. )): Ua (acce l era d

l

l

IMPROVEMENT OF MANIPU LATOR PERFORMANCE

l

tion regime) and U (dece l eration r egime ). i For the sake of a simp l e r explanation o f the method , we first assume bang - bang type of th ese c ontro l l aws in accordance with mi nimum ti me - cr i terion :

In the case of relativ e ly high l eve l s of t h e magnitudes

u~ , u~ , bang - b ang control

laws will not provide a go od dynamic beha viour of the manipula tor in the sw it ch i ng times . In the prese nc e of backlashes

(6)

such

co ntr ol strategy may give ris e to u nd esi rable vib rations and jerk s.

The problem is to find those va lues of the feedback gains k (1

~

which solve the TPBVP i 6 ), i . e . without joint motion oscilla -

tions (Fig . l a ) o r chattering relay opera ti on (Fig. lb ).

In order to improve the pe rf ormance capa bility of manipulators in point - to - point op erat i ons , we propose an extension of the relay servo system with (PO ) controllers . Instead o f us i ng on - off con tr ol laws , we can apply continuous magni t ude control func -

CONTROL SYNTHESIS ALGOR ITH M

tions as is shown on Fig . 2 ( such control laws correspond to Pontryagin Maximum pr in -

Control synthesis algorithm which exactly sol ve s the TPBVP (1

~

6 ), consists in per -

forming severa l test movements from the g i ven initial state to some termina l states ,

ciple for the linear second order p lant with ene r gy l oss crite ri on ), where r

is the i d istanc e between the current phase point (q i , qi ) and switch i ng line

converging to the r equired one . In the off li ne sence , each such movement means that the system of differential equations ( 1 ) and ( 2 ) with initial values ( 3 ) and control la ws ( 5 , 6) with some app r ox i mative feed back ga ins k

is int egrated unt i l satisfyi ing the final conditions for the velo f cities ( 4 ) 2 : q i(tf) = 0 ( t = maxtf ,

ri =

. I 2 - 1/ 2 - qi ) - qi .( k i + 1) I ki (qif

(8 )

Besides the i mprovement of the dynamic be hav i our o f manipu lators in the

s~itching

times , suc h control functions provide sub stantial energy - loss minimi zat ion ( se e Ma rino v and Kiri azov , 1984 , IFAC Cong ress).

i = 1, ... , n )- - each jOint mo tion stops af f with some termina l va lue

With fixed va l ues of the slopes in s uch

o f the corresponding coordinate: Fi =qi ( tf ).

rithm can be appl i ed , in the same way , to

ter the moment t

l

cont r o l functions , t he above propo sed algo -

li t)

Poillt -I< I-P' lillt (:0111 rol S\Stl'lll.'

obtain the corresponding feedback gains.

corresponding TPBVP . An illustrative

Making use of the parametric optimization

example for bang - bang control of a dynamic

procedure (developed in the above sited

mode l of a direct current driven manipula -

work ) with respect to these slopes and ini-

tor with cylindrical coordinates is pre -

tial times, a satisfactory suboptimal solu-

sented.

tion can be achieved . The method is simp l e for on -l ine implementation as an adaptive se l flearning NUMERICAL EXAMPLE

proce -

dure . Such a procedure is very use ful for fina l adjustment of the feedback gains

To be more illustrati ve , the proposed

since the exact identificati on of manipula -

method is applied to a dynamic model of ma -

tors parameters is i n general imposible

nipulator with cy l indrica l coordinates . In -

and compuation errors are unavoidable .

cluding direct current actuators and ta king some specified numerical values of the model parameters , the eguations of the two

REFERENCES

coupled joint motions are written as

f

/1 t /

5 . 0 (g 2 + 1 . 0 ) ~ er ( 0.4+0 .1 7 ( 20.0+10.0g2+5.0 (g2+ 1. O) ) and

i:j 1 = (u 1- 6 . 8q 1 - 0 . 34 ( 1 0 . Og

q2 = (u - 6 . 8q2 + 0 . 17 ( 1 O. Og2 + 5 . 0 (g2 + 1 .0 ) ) q~ ) /2 . 95, 2 a d a d =u =10 V (9) where u =u =15V u 1max 1max ' 2max 2max ' o f 0 f g 1 =0 . 0 rad, g 1 =1 . 0 rad , g2=0 .0 m, g2= 1. 0 m, desired accuracy - 0 . 5% . Bang - bang contro l is accepted for this example . The exact solution of this TPBVP is obtained with the following values of the feed back gains : k1 (t

f

1

= - 4.8 s

- 1

, k2 = - 6 . 2 s

- 1

f

= 1.4 s, t2 = 1.2 s I.

In o rder to show the sensitivity with respect to the gain values , an approximative oscillatory solution with k =k =- 6.8 s - 1 1 2 has been obtained. Besides the peak over shoots ( 4.4% ; 3 . 9 %, respectively ), the mo tion time prolongation to achieve the desi red accuracy is 43 %. The corresponding joint motions are depicted on Fig . 3 and 4 .

CONCL USIONS A method for

opti~al

adjustment of feedback

gains in point - to - point control systems of ~an i pulators

real

is proposed. On the basis of

~anipulator

dynamics , a control synthe -

sis algorithm is developed for the relay ser vo system with ( PO ) controllers of the joint actuators . Control laws are in accor dance with minimum time - or energy loss - cri terion. The va l ues of the feedback gains are obtained from the exact solution of the

Kiriazov , P., and P . Marinov . (1 983 ). Control synthesis of manipulator dyna mics in handling operations . Theor . App l. Mech ., Pub l. House Bulg~ad . Sci. , Year 14, 2 , 15-20 Marinov , P. and P. Kiriazov. (1 984 ). Syn thesis of time - optimal contro l for man i pul ator dynamics . Teor . App l. Mech ., Publ . House Bulg. Acad. Sci ., Year 15, ~, 13 -1 9 Marinov, P ., and P . Kiriazov . (1984). A di rect method for optima l contro l synthesis of manipulator pO int-to - point motion . Prepr. of 9th World congress. IFAC. Vol . IX, Co l loguia 14.2, 09.2, Budapest , Hungary, July 2 6 , 1984, 219 -222 Stoer , J. , and R. Bulirsch . ( 1980 ). Introduction to numerical ana l ysis . Sprin ger - Verlag. Kiriazov , P ., and P . Marinov . (1 984 ). A method for time-optimal control of dy namica ll y constrained manipulators . Prepr . of 5th CISM- IFToMM symposium on theory and practice of robots an man i pulators . June 26 - 29 , 1984 Ud i ne , Italy, 131 -1 38 Kim, Byung Kook and KangG . Shin . ( 1983 ) . Suboptimal control of industrial ma nipulators with a weighted minimum time - fuel criterion. Proc. 22nd IEEE COC , San Antonio , TX. , Dec . 1983, 1199- 1204

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1. 6

[ rad s

Cl l q.

slope= -k

1

1 . -1

i 1. :

1. '1

/ '

0 . ': 0

q.

0.6

1

0 . -1

Fig .l a. The joint motion oscilla ions

o. :

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.I /

v

-'.-

0

0.2

-0-

exact appr.

ql

o. 4

slope= - k i

Fig . 3 . The motion of joint (1)

Fig. lb. q [ ~

The cha tt ering relay opera ti on

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' -

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0.2 r

i

[)

0.2

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0 .6

0.8

Cl

Fig . 2 . The continuous mag nit ude contro l func tion s

Fi g . -l. The motion of joint (2)