Automatica 36 (2000) 1755}1760

Technical Communique

More on the applications of the contraction mappingmethod in roboticsq

Eun-Seok Choi*, Byung-Ha AhnDepartment of Mechatronics, Kwangju Institute of Science and Technology (K-JIST), Kwangju, 500-712, South Korea

Received 23 April 1999; received in "nal form 11 February 2000

Abstract

The objective of this paper is two-fold. Firstly, we extend the applications of the control scheme, proposed by Ailon (1996,Automatica, 32(10), 1455}1461), which is based on the contraction mapping theorem, to a full model of a robot manipulator (Tomei,1991, IEEE Transactions on Automatic Control, 36(10), 1208}1213). Next, we present relaxed su$cient conditions on the gains involvedin the control scheme that ensure convergence of the system trajectory to the desired neighborhood of the equilibrium point.Simulation results demonstrate the contribution of this note. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Flexible-joint robot; Uncertainty; Contraction mapping; Output feedback; Set-point

1. Introduction and preliminaries

An iterative scheme based on the action of a PDcontroller for the set-point control problem of a #exible-joint robot model with unknown model terms andpayload has been demonstrated by DeLuca and Panzieri(1994). Latter, the application of the contraction map-ping theorem to the set-point control problem underconditions of system uncertainty has been establishedby Ailon (1995, 1996) for a control scheme that is basedon position signal measurements only. The model con-sidered in these references is based on the assumptionthat the kinetic energy of the rotor resulted from its ownrotation only, or, equivalently, the motion of the rotor isa pure rotation with respect to an inertial frame (Spong,1987).

However, in industrial robots the actuators play animportant role in the entire system dynamics, and inmany cases their contribution to the inertia matrix of thesystem should be considered. Hence, there remains anopen question as to under what conditions the control

qThis paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate EditorC.T. Abdallah under the direction of Editor P.M.J. Van den Hof.

*Corresponding author. Tel.: #82-62-970-2410; fax: #82-62-970-2384.

E-mail address: choies@geguri.kjist.ac.kr (E. -S. Choi).

scheme, proposed by Ailon (1996), can be applied to thefull model of a robot manipulator that has been suggestedby Tomei (1991). In this regard, our "rst step is to extendthe approach of Ailon (1996) to the model proposed byTomei (1991).

Next, regarding the conditions which ensure the con-vergence of the contraction-mapping-based algorithm(Ailon, 1996), we shall show how the relevant su$cientconditions can be relaxed and become more attractive forreal applications. Indeed, in view of the conditions estab-lished in the last reference the magnitudes of the control-ler gains increase with the norm of the elastic matrix.However, a high gain controller might be a source ofunwanted behavior, like a highly oscillatory response.Therefore, our motivation is to propose a su$cient con-dition that allows one to reduce the lower bound thatdominates the magnitude of the selected controller gainswhen the entries of the elastic matrix increase in size.

Let q1, q

23Rn be the vectors of the link and the motor

angles and de"ne qG[qT1, qT

2]T. The n-link yexible-joint

robot model including the e!ects of the actuator inertiaand the frictional forces is given by Tomei (1991):

D(q)qK#C(q,q5 )q5 #Kq#g(q)#f (q, q5 )"uf, (1)

where D(q)3R2nC2n is the inertia matrix of the entiresystem, structured as follows:

D(q)"D(q1)"C

D1(q

1) D

2(q

1)

DT2(q

1) D

3D, (2)

0005-1098/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 0 7 8 - 9

D1(q

1), D

2(q

1)3RnCn, and D

33RnCn is a constant

diagonal matrix depending on the actuators' inertias andgear ratios. The vector C(q, q5 )qR represents the Coriolisand centrifugal torques, and the elastic matrix K is

K"CK

%!K

%!K

%K

%D, (3)

where K%"diagMk

1,2, k

nN; k

i'0 is the elastic con-

stant of joint i.Referring to (1), the vector g(q) is given by

g(q)"L;(q)

Lq"C

g1(q

1)

0 D (4)

with g1(q

1)"L;(q

1)/Lq

1, ;(q

1) is the potential energy

due to the gravity "eld. The vector uf"[0, uT]T3R2n is

the applied torques. In view of Tomei's (1991) assump-tions, the vector f (q, q5 ) in (1) satis"es

f (q, 0)"0,

q5 Tf (q, q5 )50,

q5 Tf (q, q5 )"0Nf (q, q5 )"0. (5)

In this study, DDqDD denotes the Euclidean norm ofq, j

.!9(.*/)(H) is a maximal (minimal) eigenvalue of the

Hermitian matrix H and DDADD"Jj.!9

(ATA) is the in-duced norm of a matrix A.

In the course of this study, we assume knowledge ofa constant b'0 such that

b5KKLg

1(q

1)

Lq1KK, ∀q

13Rn. (6)

2. Main results

From the control scheme proposed by Ailon (1996), weadopt the following linear controller:

z5 "!S(z!q2)!Rz#v (7)

and

u"!S(q2!z), (8)

where the controller matrices S"ST,R"RT'0 are tobe determined, and v is an input vector.

Consider model (1) and controller (7)}(8) where v isa constant vector. De"ne the scalar function

Hf(q,q5 , z)G1

2Mq5 TD(q)q5 #qTKq#(q

2!z)TS(q

2!z)

#(z!R~1v)TR(z!R~1v)N#;(q), (9)

where K represents the augmented elastic matrix (3).Then, from the results obtained by Ailon and Lozano

(1996) one concludes that if

minMj.*/

(K%), j

.*/(S), j

.*/(R)N'3b, (10)

where b is given in (6), for any constant vector v the scalarfunction H

f(q, 0, z) has a unique global minimum de-

noted by xf"[qT, zT]T and the resulting equilibrium

point x"[qT, 0T, zT]T3R5n of the closed-loop system isglobally asymptotically stable. Conversely, it is easy tocheck that for any selected "xed q

1the equation

LHf/Lx

f"0, where H

fis given in (9), has a unique

solution [qT2, vT, zT]T and the following holds:

vG¸(q1)"(I

n#RS~1#RK~1

%)g

1(q

1)#Rq

1. (11)

Hence, according to Ailon (1996) (with K%

replacing K)¸ :RnPRn is a bijective map and we may write

q1"G(v)G¸~1(v). (12)

We now de"ne a map (Ailon, 1996) ¹ : RnPRn as fol-lows:

¹(v)"v!R(G(v)!q1d

), (13)

where q1d

is a given constant vector (the robot set-point).Our next step is to show that the main result, obtained byAilon (1996), can be extended to the model considered inthis paper, in which the e!ects of the actuator inertia aretaking into account.

Lemma 1. Consider the maps G(v) and ¹(v) in (12) and(13), respectively, and assume that j

.*/(K

e)'3b. Then

a pair MS,RN can be selected such that ¹(v) is a globalcontraction with a unique xxed point vH, i.e.

¹(vH)"vH, (14)

which implies

qH1"G(vH)"q

1d. (15)

Remark 1. The proof of the lemma is constructive andprovides a simple su$cient condition that ensures theconvergence of the sequence Mv

iN with

vi`1

G¹(vi)"v

i!R(G(v

i)!q

1d), i"0, 1,2 (16)

to a "xed point vH. In the beginning of the proof wefollow the proof of Lemma 3.2 by Ailon (1996). However,in the latter part we use a useful de"nition of some factorinvolved in the proof process that allows us to determinemild su$cient conditions that are suitable for real ap-plications, in particular because it is associated withreduced lower bounds on DDRDD and DDSDD with respect tothose obtained in the last reference.

Proof. To show that ¹(v) is a global contraction it issu$cient to show (Khalil, 1996, Chapter 2) that thereexists 0(g(1 such that

DD¹(vi)!¹(v

i`1)DD4gDDv

i!v

i`1DD, ∀v

i, v

i`13Rn. (17)

1756 E.-S. Choi, B.-H. Ahn / Automatica 36 (2000) 1755}1760

Suppose that the pairs Mqi1, v

iN and Mqi`1

1, v

i`1N satisfy

(11). Then

DDvi!v

i`1DD"DD(I

n#RS~1#RK~1

%)

]Mg1(qi

1)!g

1(qi`1

1)N

#R(qi1!qi`1

1)DD. (18)

Consider (10) and select S and R as follows:

R"rIn, S"R2, r'maxM3b,J3bN. (19)

Since S"R2 in (19) by the Schwarz inequality we havefrom (18)

DDvi!v

i`1DD5D DDR(qi

1!qi`1

1)DD!DD(I

n#R~1#RK~1

%)

]Mg1(qi

1)!g

1(qi`1

1)NDD D, (20)

and we de"ne m as

mGr#1

r#

r

j.*/

(K%)5DDI

n#R~1#RK~1

%DD. (21)

We can take an r such that in addition to (19) thefollowing relation holds:

r'2Ar#1

r#

r

j.*/

(K%)Bb"2mb. (22)

Then, from (20), (21), and (6) we have

D DDR(qi1!qi`1

1)DD!DD(I

n#R~1#RK~1

%)Mg

1(qi

1)

!g1(qi`1

1)NDDD5DrDDqi

1!qi`1

1DD!mbDDqi

1!qi`1

1DD D

"(r!mb)DDqi1!qi`1

1DD. (23)

Hence, (20) and (23) imply

1

r!mbDDv

i!v

i`1DD5DDqi

1!qi`1

1DD. (24)

Using (12) we may write G(vi)"qi

1for some v

i3Rn.

From (11) and (13) we have

DD¹(vi)!¹(v

i`1)DD"DD(I

n#R~1#RK~1

%)

]Mg1(qi

1)!g

1(qi`1

1)NDD. (25)

By (21), (6) and the mean-value theoremDD¹(v

i)!¹(v

i`1)DD4mbDDqi

1!qi`1

1DD. Hence, using the last

inequality and (24) we have

DD¹(vi)!¹(v

i`1)DD4

mbr!mb

DDvi!v

i`1DD, (26)

and "nally, following (22), we arrive at

0(gGmb

r!mb(1. (27)

Using j.*/

(K%)'3b it is straightforward to show that

any selected r that satis"es

r''G1

a(b#Jb2#2ab),

(28)

1

3(aG1!

2bj.*/

(K%)(1

satis"es (22) as well. Consequently, from (19) and (28) byselecting

R"rIn, S"R2, r'maxM3b,J3b,'N, (29)

we ensure that ¹(v) in (13) is a global contraction map-ping, and the sequence Mv

iN de"ned by (16) converges to

a unique "xed point vH that satis"es (14). Clearly (13) and(14) yield (15), and we complete the proof. h

Finally, by applying (29), the construction of a controlscheme based on Algorithm 3.2 established by Ailon(1996), follows immediately.

3. Simulation study

From (28) and (29) we see that as DDK%DD increases in size,

the resulting lower bound on r reduces. From the practi-cal point of view, this result is superior to the one deter-mining the lower bound on r in Ailon (1996) in whichr increases with DDK

%DD. Hence, the objective of this section

is to present a constructive comparison between theapplications of the relevant su$cient conditions obtainedin this study and in the previous reference mentionedabove. The simulation study emphasizes the contributionof the present results when DDK

%DD increases in size, i.e.,

when the e!ect of the joint #exibility becomes a moredominant factor in the system dynamics.

Example 1. To accomplish our objective we use the samemodel as demonstrated in Example 2 in Ailon (1996). Theselected set-point is q

1d"[1,0]. Using (6) we take

b"7.1. To be consistent with the su$cient conditionspresented in the last reference, let o"1.1 and r

1"11.

Then, using the inequality r'maxMr1,J3b, 6ob,

12oj

.*/(K)N"110 which dictates a lower bound to the

gain according to the last reference, and we de"neR"111I

2and S"R2. Applying Algorithm 3.2 in Ailon

(1996) we obtain the simulation results which all appeartogether in Fig. 1. Applying the conditions of this paper,

we select in the present case r'maxM3b,J3b, 'N"21.3according to (28) and (29), and implement the controllergain matrices R"22I

2and S"R2. The results are pre-

sented in Fig. 2.

Example 2. In order to show the e!ect of the controllergain matrices R and S for the system when the DDK

%DD

E.-S. Choi, B.-H. Ahn / Automatica 36 (2000) 1755}1760 1757

Fig. 1. Results obtained for K%"diag(200, 200). The controller design is based on the su$cient conditions presented in Ailon (1996).

Fig. 2. Results obtained for K%"diag(200, 200). The controller design is based on the su$cient conditions presented in this paper.

1758 E.-S. Choi, B.-H. Ahn / Automatica 36 (2000) 1755}1760

Fig. 3. Results obtained for K%"diag(500, 500). The controller design is based on the su$cient conditions presented in Ailon (1996).

Fig. 4. Results obtained for K%"diag(500, 500). The controller design is based on the su$cient conditions presented in this paper.

E.-S. Choi, B.-H. Ahn / Automatica 36 (2000) 1755}1760 1759

increases in size, we change K%

in Example 1 toK

%"diag(500, 500) keeping another robot parameters as

well as o and r1. Then, using r'maxMr

1,J3b,6ob,

12oj

.*/(K)N"275, we de"ne R"276I

2and S"R2

(when the DDK%DD increases, the controller gain matrices

R and S also increase compared with R and S obtainedby Algorithm 3.2 in Ailon (1996) Example 1. The simula-tion results obtained by these parameters are shown inFig. 3. Next, applying the conditions of this paper,

r'maxM3b,J3b,'N"21.3, we choose the controllergain matrices R"22I

2and S"R2. (Note that even

though the norm DDK%DD increases, due to the su$cient

condition presented in this study the controller gainmatrices R and S remain as in Example 1.) The simula-tion results implemented by these parameters are depic-ted in Fig. 4.

A comparison of the plots of the system responseshown in Figs. 1}4 reveals the contribution of the presentresults regarding the controller design.

4. Conclusions

In this paper we have extended the applications of thecontraction mapping approach to a full model of a robotmanipulator and proposed a new condition which deter-mines the values of the controller gains for guaranteeingglobal asymptotic stability of a desired equilibrium point.This condition is simple and useful for applications, inparticular since it establishes lower bounds on the de-

sired controller gains which decrease when the norm ofthe elastic matrix increases. Simulation results demon-strate the advantages of the suggested condition withrespect to the selected controller gains.

Acknowledgements

The authors are grateful to the reviewers for theirvaluable suggestions.

References

Ailon, A. (1995). Analysis and synthesis of an output feedback foran uncertain robot model with #exible joints. Proceedings of theIFAC conference on system structure and control, Nantes, France(pp. 370}375).

Ailon, A. (1996). Output controllers based on iterative schemes forset-point regulation of uncertain #exible-joint robot models.Automatica, 32(10), 1455}1461.

Ailon, A., & Lozano, R. (1996). Controller-observers for set-pointtracking of #exible-joint robots including Coriolis and centripetale!ects in motor dynamics. Automatica, 32(9), 1329}1331.

DeLuca, A., & Panzieri, S. (1994). An iterative scheme for learninggravity compensation in #exible robot arms. Automatica, 30(6),993}1002.

Khalil, H. K. (1996). Nonlinear systems (2nd ed.). Upper Saddle River,NJ: Prentice-Hall.

Spong, M. W. (1987). Modeling and control of elastic joint robots.ASME Journal of Dynamic Systems, Measurement, and Control, 109,310}319.

Tomei, P. (1991). A simple PD controller for robots with elastic joints.IEEE Transactions on Automatic Control, 36(10), 1208}1213.

1760 E.-S. Choi, B.-H. Ahn / Automatica 36 (2000) 1755}1760