Identification of Geometric Parameters of Robots

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IDENTIFICATION OF GEOMETRIC PARAMETERS OF ROBOTS

w.

Khalil and M. Gautier

,-"h"ml,,;"'· ,/".-\ 111,,"/111;'1/11' ,/,. .\'II/1ln . ('.4 -I.S]' /:'.\'.\.\/. ·1-IIJi] .\'11/111', C,.,/,·.\'. F rol/u '

Abstract . The positioning accuracy of the end effector depends on the accuracy of the geometric parameters of e a ch link . This paper presents a general method for the estimation of th es!.' parameters . The method is based on the Denav i t and Hartenberg notations a nd on the differential r ela tionships of the homogenous transformation mat r ices. The use of the deviation between the geometric parameters set up by the des i gner and that of the actual robot in the compe nsation of the position and the orientatioo errors of tile end effpctor is g i ven .

identification

Keywords . Robots ; parameter es timation

modelling .

error compensation

na!.

INTROD UCT ION

let i - IT. be the ma t rix which def i nes tce coordi -

The position and t he orientation of the end effec to r of a robot is depending on the joints variables

1

nate system of link (i) wi t h respect to the coo r di nate system of link (i - I) . From Fig.l , i t can be proved tha t

and on some geometric constants . A geometric cons tant is ei t her a distance between two axes or a

twist angle . On a real robot these parameters do not agree with the values s e t up by the designer because of the fab r ication tol e rance or after the co ll ision nf the r obot with rigid objects whi c h may occur during t he utili s ation of the robot . These deviations errors will affect the a ccu racy of the robot and make it impossib l e that a given rob o t can replace another one (of the same type) without reprogramming most of the positions used in the task .

i -I T.=Rot(Z,e . )Trans(Z, r . )Tr ans(X , d. )Rot(X , a. ) ( I ) 1

1

1

1

1

where : Rot(u , w) is a 4x4 mat r ix indicating the ro t ation by an angle W around the vector ~ ; Tra n s(u, ~ )

is a 4x4 mat r ix indica ti ng t he t ransl ation by a distance ~ along the vec t or u .

From( I), we get The aim of this paper is to prese nt a method to estima t e the r eal geometric parameters of a robot, and to illustrate how to use their deviation from the designed parameters to improve th e accurac y of the robo t • The geometric mode l of a robot is characterized a non linear model in the geometric parameters. the id entification procedure a lin ea rized mode l been used. This model is obtained bv th e use of

case. - cosa . sin e. s i na.sin e . d.cos e. 1

i- I

by In has the

1

1

1

1

1

1

0

sina.

COSCt. 1

r.

0

0

0

I

1

(2)

1

with re spec t to a reference coordinate system is

,~

fT

p,eome tric parame ters o f robot s , tllP literature con-

o

(3)

=

defines th e coo rdinate system 0 with r espec t

t o a fixed ref e rence coordinate system , it can be

tains very little r efe r ences treating the genera l problem (Sugimoto, 1 98~ ; \,'u . 19 8:.) .

spec ifi ed bv 4 co nstants (Oo,ro , do"o) ' The eleOlcnts of the matrix" are functions of at most ~(n+l) oarameters, for n=6, there are 28 pa r ameters . If j~int ( i) is prismatic th e coordina te sys te m 1 can be made such that the parameter d i

SYSTE~

e quals zero . The I!atrix fT

The system to be considered i :; an open -chain mecha-

nism consisting of n+1 rigid links . The" are numbe r ed such that link 0 i s t he base , ,ch ile link n is the end effector . link i and link i - I are connected

fT

bv joint number i . A coo rdinat e system (xi'Yi,zi)

is assigned fixed with resp ect t o link i . The axis of joint i is along the z;_1 axis . The Denavit and

n =

\-~ \~no

Hartenberg (D&H)(1955) pa ramt>t Ars ( \ ,ri , di , " i)' Fig . I, are used to assign t he co ordinate svst e ms. The joint variable qi is equal to \ if joint i is if ( i) is translatio -

.,-

~: -3 ·

11

given bv the followin? relation

In spite of th e importance o f the identification of

i

1

The des c ription of the end effector of the robot

ferential relationships . Th e so luti on of the model is then carried out hy the l east squares t echn iqu es .

rotational and i s equal to r

1

T.= 1

homogenous transformati on matrix 4x4 and on the dif-

DESCRIPTIO\ OF THE A\D \OTATIO\S

111

cosa.cos6. - sina .cos 8 . d.sin e .

sin e .

I

n n -n

can be r epresen ted b,' the f ollo" ing a

0

P -n J....

0

I

P

:~:- -~ I

1 (~

)

28

\\'. Khalil and \1. Galltier

where P

is the vector specifying the position of

-n

the origin of the coordinate system n (the end effector), with resnect to the fixed reference coordinate system, A represents the matrix or director cosinus of then coordinate system (n) with respect to the reference frame, it is of dimension 3xJ. Equation (3) is non linear in the geometric parameters, it can be rewritten as follows

" i-IT. ---1 3 r.

( 13)

1

~ TZ

represents a differential translation matrix

along the Z axis. 6

Q,

putting

can be obtained from (8) by TZ and d = ~. ( 14)

(5)

W = g(.!l.)

where

where 0 is a 3x3 zero matrix. (6) c)

To estimate B a linearized model will be derived

i-I ~

a

from (3) by the use of the differential relationships presented in the following section.

°i

i-I DIFFERENTIAL RELATIONSHIPS

~ RX

Given a transformation T the change dT can be represented in the following form by ignoring the higher order terms (Paul, 19B1) dT = ~ T (7)

Ti ~ RX

(15)

indicates a differential rotation matrix ~, ~

around the X axis, where by

o

6 = X

-0

thus

where ~ is a 4x4 matrix known as a differential transformation, in general, it can be calculated by the following relation

=10

in t RX are defined

~~ -\- ~-I Rot(Z, e i)Trans(Z,ri) ~ TXTrans(X,di)

(8)

Rot(X,oi) ( 16)

where

o=

( ox oy oz)

T

and

~ =

(dx dy dz)

T

~ TX 1S

are known as

the differential rotational and translational vectors, and 8 associated to the vector 0 is defined

a differential translation matrix along the

X axis, where the to 6

~

are equal to

TX vely, thus

by 0

-6

6

0

-~

6

0

-6

z y

6

z

y

~TX

y (9)

=

and

~

vectors corresponding

Q and ~ =

(1 0 O)T respecti-

10 - ~ -) -~~I

RELATIONSHIP BETWEEN KINEMATIC ERRORS AND CARTESIAN ERRORS

such that x V

=

( 10)

V

The derivative of W, given by (3), with respect to a parameter Bi will be presented in this section.

where x denotes vector product. DIFFERENTIAL CHANGES OF THE MATRIX i-IT.

1

i-1 Ti given by (1) is function of

( 8 i,ri' ~ i,di)'

The differential changes of i-IT. with respect to 1

the differential changes d8 ,dr , da i,dd can be i i i obtained bv the use of (1) and (7) as follows

It will be seen that the derivative of W could be obtained by premultiplying fTn by a differential transformation

matrix LB.' In the following sec1

seen that

~ B.

i-1 Ti

- RZ Rot(Z, Ci)Trans(Z,ri)Trans(X,d i )

3 :- i

Rot (X, ). i)

It will be

could be obtained by the use of the

1

elements of the matrices fTi' which will be calculated at any case in order to get

a) 3

~ Bi'

tion we present the calculation of

f

Tn'

The Calculation of Le i From (3) and (11) it can be proved that

(11)

wher e_ RZ represents a differential rotation matrix around the Z axis. putting 5 -

= -Z0 = (0

~ RZ

Cdn be obtained from (8) by

0

I)T, and d

= the

3 \,

a 9 1.

f

_ 3 i-1 Ti iT n Ti 1 ~ i

fT _ i 1

zero

i-IT ~ RZ

n

(b\· th e us e of ( 11))

ve c tor.

- RZ

O~

I

-

I

0

b) Similarly we can obtain that:

i-IT )fT ( fT _ f n i 1 -RZ ( 12)

( 17)

2~)

Geometric Parameters of Robots assuming E.i-l

o

~i-l

~i-l

o

o

dinates nf the end effector origin will be calculated by the use of the model. The value of the geometric parameters of the model will be adjusted such that the error between the measured positions and the model positions should be minimum.

~i-l 1

(J 8)

then i-\

=fT~~1 =10 ~r:.!O- r-!I:I

.!:i-l/

(19)

-n

fT

~,

to identif y

B we use thus a linearized model (Eykhoff, 1979). (20)

n

where

I-a-;~-~o-: !~-I_:_~i~I_1

lle i

--

(30) is non linear in the parameters

From (12), (17), (18), (19), we obtain ll e

The Identification Model From (3) the position of the origin of the end effector can be given by P = f(B) (30)

(21 )

This choice is justified because we can initialize the procedure by a value ~ from the robot industrial drawing which approches the real parameters of the robot. Thus from (30) we get probot

f(B ) + J dB

-n

-

(31 )

0

represents the differential changes of e ,' in

II

ei the base reference frame.

J dB

robot

En

-

where J

;;T

The Calculation of llr.

,

The derivative of W with respect to r

i

could be

J dB

d P

-n

af

found by the use of (3) and 64) as (22)

model p -n

(32)

(33)

By the use of sufficient number of points, say m, and from (33) we can constitute an equation as

(23) dB

where ll r i =

10 ~ - 0+ =~ ~1 ~

(24)

The Calculation of II "'i The derivative of W with respect to "'i could be

obtained by the use of (3) and (15) as

aW a "'i

dB can be found from (34) by the standard leastsquares techniques.The estimation problem is standard and will not be considered here. The Calculation of the Matrix J The matrix J is equal to

(25) J =

From (25) we obtain

aw

=

aa., (26)

where II

(27)

"'.,

(34)

I aa.!:n

eo

I

where J

ar 0 J

Bi

a -n p

a!'.n

r

(35)

J

o

can be obtained for Bi = e i , ri' a i' d i

by the use of the differential relationships defined previously as follows. The Calculation of J e. 1

The Calculation of ll d.

,

J e . can be calculated by the use of (20) and ( 2 1). 1

As in the previous section it can be proved that

aw

(28)

a;;-.

J

e1.

1

a i - 1 (.!'.n

where ll d j

=1-o-!-o-:i~1

(29)

drawing of the robot). The end effector origin will be put in some assigned points where the position coordinates with repsect to reference frame are accurately known. The positions sensors of the joints will be read out at each point and the coor-

(3 6)

.!:i-l)

The Calculation of J ri from (23) and (24) it can be proved that

=~

THE IDENTIFICATION OF THE GEOMETRIC PARAMETERS Introduction The identification of the geometric parameters B will be carried out by assuming a model initialized by the parameters ~ (to be got from the industrial

-

( 37 )

= ~i-l

If ther e are two axe s in parallel, for example -a.1 - 1• and

~i'

then the error of r

i

can be add e d to that

and the column -, a. can be eliminated.

:~()

\\". Khalil and 1\1. Galltier

The Calculation of J

that the differential changes of the end effector

ai

transformation matrix due to a change in a given

From (26) and (27) it can be proved that

parameter, can be obtained by premultiplying it by a differential transformation matrix. The elements of the differential transformation matrices could be obtained from the elements of the matrices

d~

P. )

s. (P

-n

1

-1

It is to be noted that a n cannot be identified by

fT .. The use of the estimated parameters in impro1

~n'

the measure of

To get a

an equation on the

orientation is necessary, or the measure of another point must be carried out.

ving tre accuracy is also presented. The given method has been tested numerically, a program is being written to get automatically the 6 matrices Bi by the use of the iterative symbolic techniques in

The Calculation of J

order t o minimize the number of operations requi-

i

di

From (28) and (29) we can obtain

~

s.

} d.

(39)

-1

red to get it. The efficiency of such technique has been demonstrated in so lvin g the problem of reducing the ca lculation complexity of the dynamic models of robots (Khalil, 1985).

1

Equations (36, 37 ,38, 39) constitute the relation s necessary to get the matrix J. It is to be noted that the vectors used in the calculation of J Bi will be deduced from the matrices fT. which will 1

be calculated at any case in order to get

~.

Thus

it is recommended to ca rry out the matrices multiplications of (3) from left to right to get at the same time the vectors ~i'~i '~i needed to calculate

the matrix J . APPLICATIONS In this section we present how to use the vector dB to improve the accuracy of the robot by solving the geometric inverse problem of the r obot by the u se of the actual geometric parameters. In fact the following relation is verified for all the parameters B

B.

Di

1

+ dB.

(40)

1.

REFERENCES Denavit, J.,and Hartenberg, R.S. (1955). A kinematic notation for lower-pair mechanism based on matrices. Journal of Applied Mechanism, ~, 215-221. Eykhoff, P . (1979). System identification. John Wiley. Khalil, H.Jand Kleinfinger, J.F. (1985) . Une modi'lisation performante pour la commande dynamique de robots. Rairo Automatique, System Analysis and Contro l, to appear. Paul, R. (19'B1). Robot Manipulators: Mathematics, Programming and Control. MIT Press. Sugimoto, K.,and Okada, T . (1984) . Compensation of positioning errors caused by geometric deviations in robot system. 2nd ISSR, Tokyo, 107-113. Wu,C.H. (1984). A kinematic CAD tool for the design and control of a robot manipulator.The InterHational Journal of Robotics Research-.Vol .3, No .l,5867.

This relation can be used for all the parameters if the inverse kinematic problem is solved by the use of the variational model of the robot and under the condition that no special values have been assumed in developing this model, that is to say that the inverse problem is based on the general values of the parameters. If the inverse problem is solved by the use of an inverse geometric model based on Ba, which is gene -

rally the case, only the following parameters could be fed back to the model by the use of (40). I) ai ' in fact dqi r epresents an off-set on the positions sensors values qi 2) the parameters of r

i

(i

t

or d

i

=

], ••• ,

r.

1

n).

which are not equal

to zero in B . -0

3) the parameters of the matrices

f

To and

n-I

Tn'

because they can be taken arbitrary in the inverse geometric model (Paul, 1931). As for the other parameters their effect on the position and orientation errors can be compensated by

the calculation of dq corresponding to the error dW defined bv the following equation dW

31,1

;~T

d'S

(41 )

where ~ contains only the parameters which have not been fed back to the model. The calculation of dW can be obtained by the use of (20) , (23), (25) ,(26). CO:\CL[SIO:\ This paper presents a method for the estimation of the Denavit and Hartenberg parameters of r obots. The homogenous transformation 4x4 proves to be an

efficient tool in dealing this problem . It is seen

Fig. I.

Denavit and Hartenberg parameters .