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Modeling and adaptive control of robots interacting with environment
14th World Congress oflFAC
MODELTNG AND ADAPTIVE CONTROL OF ROBOTS TNTERACTlN ...
B-ld-05-1
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
MODELING AND ADAPTIVE CONTROL OF ROBOTS INTERACTING WITH ENVIRONMENT
Atanasko Tuneski*, Miomir Vukobratovic··
*The Faculty a/Mechanical Engineering, 91000 Skopje, Macedonia e-mail: [email protected] **RobotiC8 Laboratory, Mihallo Pupin Institute, 11000 Be/grade, Yugoslavia e-mail: [email protected]
Abstract: The adaptive control of multiple robot manipulators handling a dynamic object motion of which is constrained by the dynamic environment, when object and/or environment parameters are not known in advance, is proposed. It may be implemented when: (i) there is no good \Ulderstanding of all physical effects incorporated in the multiple robotslobject/enviroIWlent system; (ii) the parameters of the system are not precisely known; (iii) the system parameters do vary in a knOVYll. regions about their nominal values. Copyright © 1999 1FA C
Key words: adaptive control, robotic manipulators, stability analysis
force in the object, i.e., to regulate the forces which are applied by one manipulator against the others, through the object, that do not contribute to the object motion; (ii) possibility to minimize external requirements (like time or power requirements); (iii) the workspace of multiple robots is always larger than the workspace of only one. However, the addition of a second manipulator, or several manipulators, leads to a complex system since the motion of the multiple robot anns must be kinematicaIly and dynamically coordinated.
1. INTRODUcrrON
The problem addressed in this article is the synthesis of an adaptive control of multiple compliant manipulation on dynamical environment. Important advantages of multiple manipulator system are: (i) possibility to regulate the internal force in the object, Le., to regulate the forces which are applied by one manipulator against the others., through the obj ect, that do not contribute to the object motion; (ii) possibility to minimize external requirements (like time or power requirements); (ill) the workspace of multiple robots is always larger than the workspace of only one. Only a few papers consider the problem of multiple robot compliant manipulation on dynamic object (Yoshlkawa and Umeno, 1992, Luo et al., 1993, Yukawa et al., 1993), and several adaptive control methods for multiple manipulators in the contact tasks were proposed (Hu and Goldenberg, 1993, Su and Stepanenko, 1993, Yao et al., 1992a, 1993b).
The adaptive control law synthesized here has the following characteristics: (i) it may be implemented when there is uncertainty in the system parameters and they do vary in a known regions about their nominal values; (H) it is composed of an identification part (parameter update law), and a control law part; (ill) in deriving the adaptive control law, the inverse dynamics controller structure is adopted. Based on the Lyap\UloV stability theory a proof that the proposed adaptive controller is asymptotically stable is presented.
Important advantages of multiple manipulator system are: (i) possibility to regulate the internal
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Copyright 1999 IFAC
ISBN: 0 08 043248 4
MODELING AND ADAPTIVE CONTROL OF ROBOTS INTERACTIN ...
2. SYSTEM: MODEL AND TASK SETTING
The whole system consists of k robots with rigid joints, a dynamical tool or object, and a constraint dynamical environment. The coordinated robots have to simultaneously: (i) move the object along a predetennined known trajectory on the constraint eviromnent; (ii) exert a prespecified contact force on the environment; (ill) keep the value of the internal force between a certain minimum and maximum. In order to facilitate the formulation of the system dynamics, the following reasonable assumptions are made: (i) the dynamic object which is manipulated by the k -endeffectors has any size and/or geometrical shape. It has equivalent dynamic characteristics in all contact points with the robot end-effectors; (ii) the constraint environment is fixed; (iii) each robotic mechanism is non-redWldant and all coordinated robots have the same number ofjoints. The manipulated object has k connections with the multiple manipulators and one connection with the constraint environment. The manipulated object together with (k + 1) connections can be represented by a dynamic system of (k + 1) rigid bodies (Fig. 1), that is, each of the k connections between the manipulated object and the k manipulators can be represented as a local rigid body, and in the mass center of each local rigid body contact. gravitational, damping and elasticity forces act as external forces:
14th World Congress ofIFAC
system of (k + 1) bodies, and they are moving from the unloaded state. Using the Lagrange equations of motion, the mathematical model of multiple nonredundant rigid robot manipulators perfonning cooperative work on a single dynamical object of which the motion is constraIDed by a dynamical environment has been derived (Vukobratovic and Tuneski, 1997a). Here, the equations describing the multiple compliant manipulation model are briefly reported:
N(q)ij + n(q,q,Yo" ,to" )='Z"ER okx1
(2)
Yca =j(q )q+J(q)ij WOd (Yo.. )Yo,,+WOd (q,q,Yo••ro" )=F. Wca(Yca)Y.,.. +w"" (q,q.yOlI,Y"" )=F~ F. =G. (.u' .1'0" .io" 'yOd)= =(ST (Yo..)t [M(YOd )YOd +L{Yo.. ,i'o.. )]
(3)
(4) (5)
(6)
where
N(q )={H(q )+-J T (q )v... (g(q »).T(q»)eR"· x ".
q=[q; ,.... ,q;
r
ER ".X.
n(q,q,Y... ,Yo.)=C(q,q)q + G(q) + +JT(q)w.. (g(q»).i(q)q+JT(q)w.. (P{q1j(q)q,Yo.,Yo.}eR H(q}=diag{H. (q.1 .... ,H, (q, »)eR"·xn.
H,Xl
n, = tn=6k
'Z"=['Z"; •....., 'Z"; j
i~'
ER n• x!
». rr
C(q,q)q=kc!(ql'ti\)q\y ,.. , (Cl (q" ' q.. ER G(q )=[G]T (q\ ~ ....• G; (ql)j eR"·" J(q)=diag{J 1 (q.1 ... ,JIt (q" )}eR"·x". ; r ~T "H~~" f.T]T r FTI ,....,-FT]T f -L/l I. =Lk =- F ~E R".d
N xl •
Yoa e R 6 xl are the absolute coordinates of the mass center of the manipulated object for the loaded dynamic system state; Y,," = (y." ,.. E R6bl are the absolute coordinates of the mass centers of the local rigid bodies at the contacts between the robot end effectors and the object for the loaded dynamic system state; F. E R 6x\ are the generalized forces at the contact between the environment and the manipulated object; W OIr (Y0.. ) E Ro x6 is a matrix composed of the first six
.Y.J
'
kl
'-1
where m;,(i = 1,2, ...... , denotes the mass of the ith local rigid body at the contact between the i-th robot end-effector and the object (Fig. I). Two states of dynamic system of (k + 1) bodies may be defined: the unloaded stflte, that is. the state when no force system acts on the system of (k + I) elastically connected rigid bodies, and the loaded state, i.e. the state when a system of forces acts on the dynamic
ro-ws and the first six columns of the matrix WIZ (Y,,) representing the kinetic energy matrix; wo. (Y0.. , Yo" ) e Rr;;.\ is a colmnn matrix. representing the potential and dissipative energy matrix for the manipulated object; Wo. (Y.. )= W! (Y... )= (Wl .. (l'; .. ,w.. (Y.. ))e R,,·xlI, represents the kinetic energy matrix of the local rigid bodies at the contacts between the robots and the object; n=6 is the number of the degrees of freedom of the i-th robot mechanism; k is the number of the multiple robot manipulators; q/ eR" is the vector of the i-tb. robot joint coordinates;
l ...
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Copyright 1999 IF AC
ISBN: 008 0432484
MODELING AND ADAPTIVE CONTROL OF ROBOTS INTERACTIN ...
W~" (Yca,Y=)= (W ta (Y."
,Y;J...,
Wka
(Y Yu ))e Rn,xl
potential and dissipative energy matrix for the local rigid bodies at the contacts between the the robots and the object; =(FI~, eRfJ/,xl are the generalized forces at the contacts between the and the manipulated obj ect; manipulators Hi (q;) € R""" is the symmetric. bollllded, positive inertia matrix of the i - th robot manipulator;
Cj(q"qi)qi
... ,F:.r
e RIt"l is the vector of centrifugal and
Coriolis tenns; G j (q j) is the vector of gravitational terms; T; e Rn"l are the applied
3. ADAPTIVE CONTROL LAW SYNTHESIS·
ka ,
where w u, (1';,., Yii4J)e R 6xl , i = 1,2, .. , k represents the
Fca
14th World Congress ofIFAC
joint driving forces;
The deviation character of the real multiple robot motion and object motion from the programmed one may be described by:
.,,(t)=[171(t)1,,[ q(t)-qp~t) lER 6(hl}x1 1]2 (t)J Ya.(t)- Yo,,(t)J 171 (t )=(q(t}-q P(t )~R6kXl • 1Jz {t )=(Yo.. -y:a (t )~R6Xl
(9)
The family of fimctions representing the transient processes may be given by the vector differential equation (Ekalo and Vukobratovic. 1993): fi(t) = p(1J(t (10)
1r;(t»
p(1](t ),17(1»)=R 6(k+I}x1
J i (qi)e Rn"" is the generalized JacobiBn matrix;
where
f, = -Fi e RIt"l is the vector of forces and moments exerted by the i-tb robot end-effector on the object; G~ e Rn,,1 is nonlinear vector function, f.i' represents viscous friction, inertial, centrifugal, and elastic contributions; gravitational M(Yo..)e R'''''' is the non-singular inertia matrix representinj contribution of the environment; L(Yoa ,Yoa e Rn"l collects terms which represent viscous friction. Coriolis, centrifugal, gravitational S T (Yoa ) E R"M is and elastic contributions; continuous matrix.
continuous over all the set of its arguments. such that (10) has a trivial solution .,,(t} = o. The fimction P may be adopted in the form
The goal of the multiple robot acblptive control can be fonnulated in the following way: assuming that the constrained, closed set ~ eRr of possible values of the Wlknown multiple robots andIor object parameters has been given by
8(t}eV 1.1,
"tt~o
V 1.1 eR',
then, define the control law the control goals:
1"{t)
for t";? to to satisfy
' ....'"
Yo~ (t)=o
(7)
where the index p denotes nominal values of the corresponding vectors. The satisfaction of the control goals (7) also implies a satisfaction of the following control goals (Vukobratovic and Tuneski, 1997b):
lim(F~ (t) - F/ (t»=O ;
¥.here the matrices and
r
1
r
I
(11)
=diagh v l ' Yv 2 ••••• Y v(6 k+6) j
=diaglil p l ' Yp z ..... Y p(6tH) j have to be chosen
such that !he eigenvalues matrix
AI'
Az ••••• , AI2.t+12 of !he (12)
have negative real parts. In this way an asymptotic stability of the solution of the equation (11) is achieved. The following control law is proposed:
r(t }=N(q~q P +p' (1], 7j )}-li(q,q. Yo.. , t o,,)= =N(q~qp +r;'fJI +r;ilt}tli(q,q,Yo" ,Yoa )
(8)
t ....""
In addition to the control goals (7). (8) it is also necessary to ensure satisfaction of the internal force constraint In the cooperative manipulation of a dynamical object, the internal force is defined as a part of the contact forces which is canceled withln the object and therefore does not influence the object motion. The desired value of the internal force, the maximum internal force. and the minimum internal force are given by the task planner.
(13)
and the parameter adaptation law: (14)
where: p' (17, iJ}:R(6k}x1 is a vector fi.mction composed of the first 6k rows of the vector function p(1]. 17}eR 6(k+l),<1; 8=(e I ,. ••••e r) is the r-dimensional vector of the Wlknown multiple robots and object parameters; K=diag{kb ... ,k,], ki > 0, i=i,2,., r is (rxr) positive definite adaptation gain matrix; =diag[-Yvl'-YV2' ...• -Yv(6k)],and
r;
r;
t4
lim(Fc (t) - F.p (t»)=O
p(1](t),17(t») ;-rl 17(t)-rz 1J(t)
8=K
~(q(t) - qp (t»-o lim(Yoa (t) -
is a vector fimction,
=diag(-Y pp-yP2 ••.••-yp(6kI]
matrices, (see (11»);
are
lP(q,q,q'YOa,Yoa ):R(6A:)xr is
known matrix of fimctions of
8=(0-0 ~R'
r
is
constant diagonal
the
a
(q, q. q, Y to..);
parameter
Oa ,
error
vector,
8;;:::(6 1 , •••• , 6r eR' are the estimates of the parameters. such that a key feature of rigid robot dynamics, which is linearity of manipulator parameters, may be used to write the error equation in the form
631
Copyright 1999 IF AC
ISBN: 008 0432484
MODELING AND ADAPTIVE CONTROL OF ROBOTS INTERACTIN ...
iil+r; rlJ+r;iJI =N- 1(q iN(q )ij+n(q,ti,Yo",to,,)~ =l\rl(q }lI>(q,q,qY()" ,1'0,,)9
4. CONCLUSION
N(q)==N(q)-N{q), and ii(q,q, YOa ' Yoa )=n(q,q,Yaa ,Yoa )- n(q,q, Yua ,Yu,,) (:') is the estimate od·);
represent errors in the multiple robots dynamic model (2) used in the synthesis of the control law (13) arising from the errors in the multiple robots and/or object parameters, and
iil +1'; 111 +1'; iJl =N-1(qlN(q)q+ii(q,ti, Y tOol)] OG '
is the error equation obtained by equating (2) and (13);
11' =~iJl +If/TIJ ~ q=diag(~ 1>1; 2 ••••• ,~ 61: 11; i >0, and If/=diag(4 J'~ 2>•••• ,~ 6k I >0. i = 1,2, .... ,6k,
14
are positive constants chosen such that the transfer fimction
11
.
(15)
iV-1
is strictly positive real (SPR) function. The condition that the transfer function (15) is strictly positive real ensures the possibility 10 implement the Popov-KahnanYakubovitch lemma The following theorem may be established:
THEOREM Let us suppose that: (i) we have a perfect stnJctural model of the cooperative manipulation represented by the equations (2)-(6); (U) the constrained, closed set Vt/ eR' of possible values of the multiple robots and/or object parameters has been
O{t }eV IJ, "
excitation condition
for all to' where U
=(N €pr ' r is the number of the -I
unknown system parameters, a,p, and p are all positive,I, is the (rxr) unit matrix; (iv) the estimates of the multiple robots and/or object parameters lie within sufficiently small region such that the matrix remains positive definite and invertible
Bi(_)~Oi ~e;(IIWC»)' where the lower bound
N (q )
(};(mia)
14th World Congress ofIFAC
(i.e. and
the upper bol.Uld B;(mox} are a priori known. Then, if the control law (13) and the parameter adaptation law (14) are introduced into the multiple robots dynamic model (2), (i) the control goals (7) and (8) are satisfied; (ii) the errors of the parameter estimation will converge 10 zero; (ill) it is possible to satisfY the force constraints.
The proof of the theorem is given in Appendix A.
Adaptive control of multiple autonomous nonredundant robot manipulators with rigid joints handling one dynamic object motion of which is constrained by a dynamic environment is synthesized.. The manipulator dynamics, the object dynamics and the environment dynamics are taken into consideration, together with the material constraints imposed on the system (contact conditions between multiple robots and object and between object and environment), and program constraints (specified internal forces, contact forces, object motion). The object which is held by the k robot end-effectors makes a frictional point contact on the constraint environment. The robot and/or object parameters are assumed to be unknown and they are identified in the control synthesis process . The proposed adaptive control law ensures movement of the object along a predetennined known trajectory on the constraint environment, while simultaneously exerting a prespecified contact force unto the environment. The main contribution of this paper is that the adaptive coordinated control of multiple robot manipulators handling a dynamic object motion of which is constrained by the dynamic environment, when object and/or environment parameters are not kno'Wn in advance, is proposed. Based on Lyapunov stability theory a proof that the proposed adaptive controller is asymptotically stable is presented. The convergence of the unknown system parameters to their true values has been proved. The problem addressed in this paper may be further investigated in the following ways: (i) it is of practical interest to investigate the conditions for practical stability of the multiple robots around the desired position and force trajectories. The practical stability is defined by specifying the finite regions around the desired position and force trajectories within which multiple robots actual position and velocity, as well as contact forces, have to be during the task execution. In the constrained motion control tasks the practical stability is often more appropriate than the classical asymptotic/exponential stability since when the model and parameter uncertainties do not expire with time (although being constrained), the practical stability enables to consider effects of these uncertainties which cannot guarantee asymptotic/exponential stability of the system at all, but can fulfill practical stability conditions. So in the constrained motion tasks the practical stability requirements are more relaxed and more appropriate from a practical point of view than the asymptotic/exponential stability conditions.
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Copyright 1999 IF AC
ISBN: 008 0432484
MODELING AND ADAPTIVE CONTROL OF ROBOTS INTERACTIN ...
(ii)
(iii)
by taking into accoWlt some effects which have not been considered here (for example, joint flexibility, various environment models, elasticity of the multiple robots links, sensor noise); by experimental verification of the proposed adaptive controller, which will demonstrate the possibility for its implementation in the practical multiple robots contact tasks.
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Yukawa, T., M Uchiyama and G. Obinata (1993). Stability and Handling Characteristics of the Control System for Dual-Arm. Manipulators to Handle Flexible Objects, In: Proceedings of lEEElRSJ Coriference on Intelligent Robots and Systems, 11611164. APPENDIX A Proof of the Theorem. By equating (2) and (13), we obtain
RERERENCES Anderson B.D.O. (1977). Exponential Stability of Linear Equations Arising in Adaptive Identification, IEEE Trans. on Automatic Control, AC-ll, 83-88. Ekalo, Y. and M. Vukobratovic (1993). Robust and Adaptive PositionIForce Stabilization of Robotic Manipulators in Contact Tasks, Robotica, 11. 373386. Hu, Y. and A. Goldenberg (1993). An Adaptive Approach to Motion and Force Control of Multiple Coordinated Robot Arms", ASME Journal of Dynamic Systems, Measurement and Control, 115, 60-69. Luo, Z., K. Ito and M. Ito (1993a). Multiple Robot Manipulators: Cooperative Compliant Manipulation on Dynamical Environments, In: Proceedings of IEEElRSJ Conference on Intelligent Robots and Systems, 1927-1934. Luo, Z. and M. Ito (1993b). Control Design of Robot for Compliant Manipulation on Dynamical Environments, IEEE Transactions on Robotics and Automation, Vo1.9, No. 3,286-296. Suo C.Y. and Y. Stepanenko (1993). Adaptive Sliding Mode Coordinated Control Of Multiple Robot Arms Handling One Constrained Object, In: Proceedings ofAmerican Control Conference, 1406-1413. Vukobratovic, M. and A. Tlll1eski (1997a). Mathematical Model of Multiple Manipulators: Cooperative Compliant Manipulation on D)namical Enviromnent, Journal of Mechanism and Machine Theory, 33, 1211-1239. Vukobratovic, M. and A. TWleski (1 998b). Contribution to the Control of Multiple Compliant Manipulation on Dynamic Enviromnent, In: Third ECPD Int. Con/ on Advanced Robotics, Intelligent Automation and Active Systems, Bremen, September, 233-239. Yao, B., W.B. Gao, S.P.Chan and M. Cheng (1992a). VSC Coordinated Control of Two Manipulator Anns in the Presence of Enviromnental Constraints, IEEE Transactions on Automatic Control, 37, 1806-1812. Yao, B. and M. Tomizuka (1993b). Adaptive Coordinated Control of Multiple Manipulators Handling a Constrained Object, In: Proceedings of IEEE International Conference on Robotics and Automation, 624-629. Yoshikawa, '!". and A. Umeno (1992). Dynamic Hybrid Control of Manipulators Considering Object Dynamics, In: Proceedings of the Intern. Symp. Measurement and Control in Robotics, 597-604.
ii\+r;'lt +r;i]\ =iV-·(qXN(qR+n(q,q,yo•• Yo,,)~
=N- (q)cD(q.q,ii,y~",Yo.)9 1
(0) denotes the estimate of ( 0) ; N(q}=N(q}-N(q). $(q,ti,tJ'Yoa,Yo..)eR"'x, .
v.here
O=(O-O)e:R' . n(q,q,Yoa ,Yoa )=n(q,q,Yo" ,Y
Od ) -
n(q,q,Yoa,Yoa )
For the i-th element of the deviation vector '7. eR 6kx • , (9), an error equation may be written as:
~ 1i -y vi ~li -Ypi 'Ill =(.iV-'
v.here
r:
=diag[-YvP-Yv2"'.,-'r'V(61:)]
r; =diag[- Yp .. - yP2 •..• '-Y p (6k)]'
(iV-' rrJO)i
(AI) and
(13»,
(see
means the i-tb element of the (6k x 1)
vector (iV-1 ~). TIle stare-space representation of the error equation (AI) is given by:
xj=A,xt+Bt(.iV-1$O)j
(A2)
,,: =CiX j
,.,; is the i-th element of the vector (see (15)), and Xi
=[". ~.
r
". =O;i]. +'1'711
is the state vector. If the
positive constants ; =diag(~ P~ 1 , •••• ,~ H
",=diag(~ .. tP 2 ,.···,tP 6k chosen such that 11*
1~
>0. and tjI i >0, i = 1.2, ... ,6 k. are
1
i
,;r,. +1f/71.
iir-·Cf)iJ ii. +r; 71. + r: 7]. is strictly positive real (SPR) fimction. Then, by the Kalman-Yakubovitch-Popov lemma the symmetric, positive definite matrices PIER M and QjER M do exist such that
Ai P;+P;A;=-Qi
(A3)
p,Bj=cT
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Copyright 1999 IF AC
ISBN: 008 0432484
MODELING AND ADAPTIVE CONTROL OF ROBOTS INTERACTIN ...
vmere the matrices A j , Bi t and Cj are the matrices of the state-space realization (A2) of the error equation (Al). The error equation of the whole system of k multiple manipulators in the state-space form is given by:
X=AX+BN- I 4>O
(A4)
14th World Congress ofIFAC
.. Vc =V(Xo,Bo)- JXTQXdt
Since Vc and V( X o' ( 0 ) are finite terms, it may be
stated that 00
!XTQXdt
71· =Cx
o
A=blockdiag(At ,A2 •••• ,Ad )eR (2"k).(2nk) B=blockdiag(B1 ,B2, ... ,B"k )eR(2JJk)x.("k)
has a finite value, too. Since X T QX is positive, tmiformly continuous, and has a finite integral, using the well-known mathematical analysis results (Rudin, 1976), it may be stated that limXTQX=O (AB)
C=blockdiag(Cl' C 2 ," .,C"k ):R(Mi: }c(''''') X -I ~Xl
X 2 •· ••••• X"k
'(A12)
o
)T E R211kxI
'~oo
which leads to the conclusion that the state vector X= ( XI x ._ •••• x ) T E R(2"" )x(J)
IT we choose a symmetric positive definite matrix Q
Q=diag(QI ,Q2 ,... ,Q"" )ER (1""').(211.)
CA5)
2
where Xi ~['T/I
and solve the Lyapunovequation
ATp+PA=-Q
r,
nl
rh will asymptotically converge to zero vector as t ---+ 00 :
(A6)
the turique symmetric positive definite maIrix P D D P LR(2nl).«2 ..k) P =diag(..... 1' .....2'· .. ' ..l J<=
(A7)
maybedetennined. The follo\Ving LyaptUlOv :fi.mction candidate is chosen
v(X,O)=XT PX +OT K-1(j where K
(AS)
:;;;:diag[kl'kp......,kJ k j >0, i=1.2, ...,r , is
(rxr) positive definite adaptation gain matrix. The time derivative of (A4) is
v(x,e)
along the trajectories
i.e. the control goals (7) are satisfied. The satisfaction of the control goals (7) also implies a satisfaction of the control goals (8), (Vukobratovic and Tuneski., 1997a). The maximwn and minimum value of the internal force are given by the task plamter. So, if the control goals (8) are satisfied, "We may state that the intema1 force constraints are also satisfied. Now the parameter error convergence will be proved. According to (14) and (A4) the state-space representation of the entire system. may be written as
V{x.O}=- XT QX +20T[cI>T N-1'T/' +K-'{J] (A9) The equation (A9) suggests that, if the parameter update law is chosen to be
e=Kfb Tiv-IT]. then, since the parameter vector 8 is constant, the second tenn in (A9) equals zero, and
V(X,O)=_XTQX
e=8 , (AlO)
=(.iV- tPy .
J where U The asymptotic stability of (AI 1) has been already studied in the literatw"e. Anderson, 1977, has hown that (All) is uniformly asymptotically stable if: (i) the linear system (A, B, C) satisfies the strictly positive real condition (SPR); (ii) U satisfies the persistent excitation condition
oc I ,:S
'o"'P rut . .. 1':' 1':' ~ ~TrI • •• VI' --in \",<1'1 I J' w"q"q,. 0.' o.}J w"ql',q~,I."",l.o.P'-t' , I,
which is nonpositive because positive definite matrix (A5). Since
x,8,fI-
J ,
and
W
Q is chosen to be a
are bounded,:from (A4) it
X
is botmded as -well. Thus X is V(X,8). From (AS) and (AIO) it follows that
may be stated that
for all to. where
U =(iv-'
tPr .r
is the nwnber of the
unknown system parameters, a,p, and p positive, I, is the (rxr) Wlit matrix.
are all
The proof of the theorem is completed.
miformly continuous, and so is
limV(X,8)=V r.... c:o ~
(All)
does exist, and
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Copyright 1999 IF AC
ISBN: 008 0432484
MODELTNG AND ADAPTIVE CONTROL OF ROBOTS TNTERACTlN ...
B-ld-05-1
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
MODELING AND ADAPTIVE CONTROL OF ROBOTS INTERACTING WITH ENVIRONMENT
Atanasko Tuneski*, Miomir Vukobratovic··
*The Faculty a/Mechanical Engineering, 91000 Skopje, Macedonia e-mail: [email protected] **RobotiC8 Laboratory, Mihallo Pupin Institute, 11000 Be/grade, Yugoslavia e-mail: [email protected]
Abstract: The adaptive control of multiple robot manipulators handling a dynamic object motion of which is constrained by the dynamic environment, when object and/or environment parameters are not known in advance, is proposed. It may be implemented when: (i) there is no good \Ulderstanding of all physical effects incorporated in the multiple robotslobject/enviroIWlent system; (ii) the parameters of the system are not precisely known; (iii) the system parameters do vary in a knOVYll. regions about their nominal values. Copyright © 1999 1FA C
Key words: adaptive control, robotic manipulators, stability analysis
force in the object, i.e., to regulate the forces which are applied by one manipulator against the others, through the object, that do not contribute to the object motion; (ii) possibility to minimize external requirements (like time or power requirements); (iii) the workspace of multiple robots is always larger than the workspace of only one. However, the addition of a second manipulator, or several manipulators, leads to a complex system since the motion of the multiple robot anns must be kinematicaIly and dynamically coordinated.
1. INTRODUcrrON
The problem addressed in this article is the synthesis of an adaptive control of multiple compliant manipulation on dynamical environment. Important advantages of multiple manipulator system are: (i) possibility to regulate the internal force in the object, Le., to regulate the forces which are applied by one manipulator against the others., through the obj ect, that do not contribute to the object motion; (ii) possibility to minimize external requirements (like time or power requirements); (ill) the workspace of multiple robots is always larger than the workspace of only one. Only a few papers consider the problem of multiple robot compliant manipulation on dynamic object (Yoshlkawa and Umeno, 1992, Luo et al., 1993, Yukawa et al., 1993), and several adaptive control methods for multiple manipulators in the contact tasks were proposed (Hu and Goldenberg, 1993, Su and Stepanenko, 1993, Yao et al., 1992a, 1993b).
The adaptive control law synthesized here has the following characteristics: (i) it may be implemented when there is uncertainty in the system parameters and they do vary in a known regions about their nominal values; (H) it is composed of an identification part (parameter update law), and a control law part; (ill) in deriving the adaptive control law, the inverse dynamics controller structure is adopted. Based on the Lyap\UloV stability theory a proof that the proposed adaptive controller is asymptotically stable is presented.
Important advantages of multiple manipulator system are: (i) possibility to regulate the internal
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2. SYSTEM: MODEL AND TASK SETTING
The whole system consists of k robots with rigid joints, a dynamical tool or object, and a constraint dynamical environment. The coordinated robots have to simultaneously: (i) move the object along a predetennined known trajectory on the constraint eviromnent; (ii) exert a prespecified contact force on the environment; (ill) keep the value of the internal force between a certain minimum and maximum. In order to facilitate the formulation of the system dynamics, the following reasonable assumptions are made: (i) the dynamic object which is manipulated by the k -endeffectors has any size and/or geometrical shape. It has equivalent dynamic characteristics in all contact points with the robot end-effectors; (ii) the constraint environment is fixed; (iii) each robotic mechanism is non-redWldant and all coordinated robots have the same number ofjoints. The manipulated object has k connections with the multiple manipulators and one connection with the constraint environment. The manipulated object together with (k + 1) connections can be represented by a dynamic system of (k + 1) rigid bodies (Fig. 1), that is, each of the k connections between the manipulated object and the k manipulators can be represented as a local rigid body, and in the mass center of each local rigid body contact. gravitational, damping and elasticity forces act as external forces:
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system of (k + 1) bodies, and they are moving from the unloaded state. Using the Lagrange equations of motion, the mathematical model of multiple nonredundant rigid robot manipulators perfonning cooperative work on a single dynamical object of which the motion is constraIDed by a dynamical environment has been derived (Vukobratovic and Tuneski, 1997a). Here, the equations describing the multiple compliant manipulation model are briefly reported:
N(q)ij + n(q,q,Yo" ,to" )='Z"ER okx1
(2)
Yca =j(q )q+J(q)ij WOd (Yo.. )Yo,,+WOd (q,q,Yo••ro" )=F. Wca(Yca)Y.,.. +w"" (q,q.yOlI,Y"" )=F~ F. =G. (.u' .1'0" .io" 'yOd)= =(ST (Yo..)t [M(YOd )YOd +L{Yo.. ,i'o.. )]
(3)
(4) (5)
(6)
where
N(q )={H(q )+-J T (q )v... (g(q »).T(q»)eR"· x ".
q=[q; ,.... ,q;
r
ER ".X.
n(q,q,Y... ,Yo.)=C(q,q)q + G(q) + +JT(q)w.. (g(q»).i(q)q+JT(q)w.. (P{q1j(q)q,Yo.,Yo.}eR H(q}=diag{H. (q.1 .... ,H, (q, »)eR"·xn.
H,Xl
n, = tn=6k
'Z"=['Z"; •....., 'Z"; j
i~'
ER n• x!
». rr
C(q,q)q=kc!(ql'ti\)q\y ,.. , (Cl (q" ' q.. ER G(q )=[G]T (q\ ~ ....• G; (ql)j eR"·" J(q)=diag{J 1 (q.1 ... ,JIt (q" )}eR"·x". ; r ~T "H~~" f.T]T r FTI ,....,-FT]T f -L/l I. =Lk =- F ~E R".d
N xl •
Yoa e R 6 xl are the absolute coordinates of the mass center of the manipulated object for the loaded dynamic system state; Y,," = (y." ,.. E R6bl are the absolute coordinates of the mass centers of the local rigid bodies at the contacts between the robot end effectors and the object for the loaded dynamic system state; F. E R 6x\ are the generalized forces at the contact between the environment and the manipulated object; W OIr (Y0.. ) E Ro x6 is a matrix composed of the first six
.Y.J
'
kl
'-1
where m;,(i = 1,2, ...... , denotes the mass of the ith local rigid body at the contact between the i-th robot end-effector and the object (Fig. I). Two states of dynamic system of (k + 1) bodies may be defined: the unloaded stflte, that is. the state when no force system acts on the system of (k + I) elastically connected rigid bodies, and the loaded state, i.e. the state when a system of forces acts on the dynamic
ro-ws and the first six columns of the matrix WIZ (Y,,) representing the kinetic energy matrix; wo. (Y0.. , Yo" ) e Rr;;.\ is a colmnn matrix. representing the potential and dissipative energy matrix for the manipulated object; Wo. (Y.. )= W! (Y... )= (Wl .. (l'; .. ,w.. (Y.. ))e R,,·xlI, represents the kinetic energy matrix of the local rigid bodies at the contacts between the robots and the object; n=6 is the number of the degrees of freedom of the i-th robot mechanism; k is the number of the multiple robot manipulators; q/ eR" is the vector of the i-tb. robot joint coordinates;
l ...
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W~" (Yca,Y=)= (W ta (Y."
,Y;J...,
Wka
(Y Yu ))e Rn,xl
potential and dissipative energy matrix for the local rigid bodies at the contacts between the the robots and the object; =(FI~, eRfJ/,xl are the generalized forces at the contacts between the and the manipulated obj ect; manipulators Hi (q;) € R""" is the symmetric. bollllded, positive inertia matrix of the i - th robot manipulator;
Cj(q"qi)qi
... ,F:.r
e RIt"l is the vector of centrifugal and
Coriolis tenns; G j (q j) is the vector of gravitational terms; T; e Rn"l are the applied
3. ADAPTIVE CONTROL LAW SYNTHESIS·
ka ,
where w u, (1';,., Yii4J)e R 6xl , i = 1,2, .. , k represents the
Fca
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joint driving forces;
The deviation character of the real multiple robot motion and object motion from the programmed one may be described by:
.,,(t)=[171(t)1,,[ q(t)-qp~t) lER 6(hl}x1 1]2 (t)J Ya.(t)- Yo,,(t)J 171 (t )=(q(t}-q P(t )~R6kXl • 1Jz {t )=(Yo.. -y:a (t )~R6Xl
(9)
The family of fimctions representing the transient processes may be given by the vector differential equation (Ekalo and Vukobratovic. 1993): fi(t) = p(1J(t (10)
1r;(t»
p(1](t ),17(1»)=R 6(k+I}x1
J i (qi)e Rn"" is the generalized JacobiBn matrix;
where
f, = -Fi e RIt"l is the vector of forces and moments exerted by the i-tb robot end-effector on the object; G~ e Rn,,1 is nonlinear vector function, f.i' represents viscous friction, inertial, centrifugal, and elastic contributions; gravitational M(Yo..)e R'''''' is the non-singular inertia matrix representinj contribution of the environment; L(Yoa ,Yoa e Rn"l collects terms which represent viscous friction. Coriolis, centrifugal, gravitational S T (Yoa ) E R"M is and elastic contributions; continuous matrix.
continuous over all the set of its arguments. such that (10) has a trivial solution .,,(t} = o. The fimction P may be adopted in the form
The goal of the multiple robot acblptive control can be fonnulated in the following way: assuming that the constrained, closed set ~ eRr of possible values of the Wlknown multiple robots andIor object parameters has been given by
8(t}eV 1.1,
"tt~o
V 1.1 eR',
then, define the control law the control goals:
1"{t)
for t";? to to satisfy
' ....'"
Yo~ (t)=o
(7)
where the index p denotes nominal values of the corresponding vectors. The satisfaction of the control goals (7) also implies a satisfaction of the following control goals (Vukobratovic and Tuneski, 1997b):
lim(F~ (t) - F/ (t»=O ;
¥.here the matrices and
r
1
r
I
(11)
=diagh v l ' Yv 2 ••••• Y v(6 k+6) j
=diaglil p l ' Yp z ..... Y p(6tH) j have to be chosen
such that !he eigenvalues matrix
AI'
Az ••••• , AI2.t+12 of !he (12)
have negative real parts. In this way an asymptotic stability of the solution of the equation (11) is achieved. The following control law is proposed:
r(t }=N(q~q P +p' (1], 7j )}-li(q,q. Yo.. , t o,,)= =N(q~qp +r;'fJI +r;ilt}tli(q,q,Yo" ,Yoa )
(8)
t ....""
In addition to the control goals (7). (8) it is also necessary to ensure satisfaction of the internal force constraint In the cooperative manipulation of a dynamical object, the internal force is defined as a part of the contact forces which is canceled withln the object and therefore does not influence the object motion. The desired value of the internal force, the maximum internal force. and the minimum internal force are given by the task planner.
(13)
and the parameter adaptation law: (14)
where: p' (17, iJ}:R(6k}x1 is a vector fi.mction composed of the first 6k rows of the vector function p(1]. 17}eR 6(k+l),<1; 8=(e I ,. ••••e r) is the r-dimensional vector of the Wlknown multiple robots and object parameters; K=diag{kb ... ,k,], ki > 0, i=i,2,., r is (rxr) positive definite adaptation gain matrix; =diag[-Yvl'-YV2' ...• -Yv(6k)],and
r;
r;
t4
lim(Fc (t) - F.p (t»)=O
p(1](t),17(t») ;-rl 17(t)-rz 1J(t)
8=K
~(q(t) - qp (t»-o lim(Yoa (t) -
is a vector fimction,
=diag(-Y pp-yP2 ••.••-yp(6kI]
matrices, (see (11»);
are
lP(q,q,q'YOa,Yoa ):R(6A:)xr is
known matrix of fimctions of
8=(0-0 ~R'
r
is
constant diagonal
the
a
(q, q. q, Y to..);
parameter
Oa ,
error
vector,
8;;:::(6 1 , •••• , 6r eR' are the estimates of the parameters. such that a key feature of rigid robot dynamics, which is linearity of manipulator parameters, may be used to write the error equation in the form
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iil+r; rlJ+r;iJI =N- 1(q iN(q )ij+n(q,ti,Yo",to,,)~ =l\rl(q }lI>(q,q,qY()" ,1'0,,)9
4. CONCLUSION
N(q)==N(q)-N{q), and ii(q,q, YOa ' Yoa )=n(q,q,Yaa ,Yoa )- n(q,q, Yua ,Yu,,) (:') is the estimate od·);
represent errors in the multiple robots dynamic model (2) used in the synthesis of the control law (13) arising from the errors in the multiple robots and/or object parameters, and
iil +1'; 111 +1'; iJl =N-1(qlN(q)q+ii(q,ti, Y tOol)] OG '
is the error equation obtained by equating (2) and (13);
11' =~iJl +If/TIJ ~ q=diag(~ 1>1; 2 ••••• ,~ 61: 11; i >0, and If/=diag(4 J'~ 2>•••• ,~ 6k I >0. i = 1,2, .... ,6k,
14
are positive constants chosen such that the transfer fimction
11
.
(15)
iV-1
is strictly positive real (SPR) function. The condition that the transfer function (15) is strictly positive real ensures the possibility 10 implement the Popov-KahnanYakubovitch lemma The following theorem may be established:
THEOREM Let us suppose that: (i) we have a perfect stnJctural model of the cooperative manipulation represented by the equations (2)-(6); (U) the constrained, closed set Vt/ eR' of possible values of the multiple robots and/or object parameters has been
O{t }eV IJ, "
excitation condition
for all to' where U
=(N €pr ' r is the number of the -I
unknown system parameters, a,p, and p are all positive,I, is the (rxr) unit matrix; (iv) the estimates of the multiple robots and/or object parameters lie within sufficiently small region such that the matrix remains positive definite and invertible
Bi(_)~Oi ~e;(IIWC»)' where the lower bound
N (q )
(};(mia)
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(i.e. and
the upper bol.Uld B;(mox} are a priori known. Then, if the control law (13) and the parameter adaptation law (14) are introduced into the multiple robots dynamic model (2), (i) the control goals (7) and (8) are satisfied; (ii) the errors of the parameter estimation will converge 10 zero; (ill) it is possible to satisfY the force constraints.
The proof of the theorem is given in Appendix A.
Adaptive control of multiple autonomous nonredundant robot manipulators with rigid joints handling one dynamic object motion of which is constrained by a dynamic environment is synthesized.. The manipulator dynamics, the object dynamics and the environment dynamics are taken into consideration, together with the material constraints imposed on the system (contact conditions between multiple robots and object and between object and environment), and program constraints (specified internal forces, contact forces, object motion). The object which is held by the k robot end-effectors makes a frictional point contact on the constraint environment. The robot and/or object parameters are assumed to be unknown and they are identified in the control synthesis process . The proposed adaptive control law ensures movement of the object along a predetennined known trajectory on the constraint environment, while simultaneously exerting a prespecified contact force unto the environment. The main contribution of this paper is that the adaptive coordinated control of multiple robot manipulators handling a dynamic object motion of which is constrained by the dynamic environment, when object and/or environment parameters are not kno'Wn in advance, is proposed. Based on Lyapunov stability theory a proof that the proposed adaptive controller is asymptotically stable is presented. The convergence of the unknown system parameters to their true values has been proved. The problem addressed in this paper may be further investigated in the following ways: (i) it is of practical interest to investigate the conditions for practical stability of the multiple robots around the desired position and force trajectories. The practical stability is defined by specifying the finite regions around the desired position and force trajectories within which multiple robots actual position and velocity, as well as contact forces, have to be during the task execution. In the constrained motion control tasks the practical stability is often more appropriate than the classical asymptotic/exponential stability since when the model and parameter uncertainties do not expire with time (although being constrained), the practical stability enables to consider effects of these uncertainties which cannot guarantee asymptotic/exponential stability of the system at all, but can fulfill practical stability conditions. So in the constrained motion tasks the practical stability requirements are more relaxed and more appropriate from a practical point of view than the asymptotic/exponential stability conditions.
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(ii)
(iii)
by taking into accoWlt some effects which have not been considered here (for example, joint flexibility, various environment models, elasticity of the multiple robots links, sensor noise); by experimental verification of the proposed adaptive controller, which will demonstrate the possibility for its implementation in the practical multiple robots contact tasks.
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Yukawa, T., M Uchiyama and G. Obinata (1993). Stability and Handling Characteristics of the Control System for Dual-Arm. Manipulators to Handle Flexible Objects, In: Proceedings of lEEElRSJ Coriference on Intelligent Robots and Systems, 11611164. APPENDIX A Proof of the Theorem. By equating (2) and (13), we obtain
RERERENCES Anderson B.D.O. (1977). Exponential Stability of Linear Equations Arising in Adaptive Identification, IEEE Trans. on Automatic Control, AC-ll, 83-88. Ekalo, Y. and M. Vukobratovic (1993). Robust and Adaptive PositionIForce Stabilization of Robotic Manipulators in Contact Tasks, Robotica, 11. 373386. Hu, Y. and A. Goldenberg (1993). An Adaptive Approach to Motion and Force Control of Multiple Coordinated Robot Arms", ASME Journal of Dynamic Systems, Measurement and Control, 115, 60-69. Luo, Z., K. Ito and M. Ito (1993a). Multiple Robot Manipulators: Cooperative Compliant Manipulation on Dynamical Environments, In: Proceedings of IEEElRSJ Conference on Intelligent Robots and Systems, 1927-1934. Luo, Z. and M. Ito (1993b). Control Design of Robot for Compliant Manipulation on Dynamical Environments, IEEE Transactions on Robotics and Automation, Vo1.9, No. 3,286-296. Suo C.Y. and Y. Stepanenko (1993). Adaptive Sliding Mode Coordinated Control Of Multiple Robot Arms Handling One Constrained Object, In: Proceedings ofAmerican Control Conference, 1406-1413. Vukobratovic, M. and A. Tlll1eski (1997a). Mathematical Model of Multiple Manipulators: Cooperative Compliant Manipulation on D)namical Enviromnent, Journal of Mechanism and Machine Theory, 33, 1211-1239. Vukobratovic, M. and A. TWleski (1 998b). Contribution to the Control of Multiple Compliant Manipulation on Dynamic Enviromnent, In: Third ECPD Int. Con/ on Advanced Robotics, Intelligent Automation and Active Systems, Bremen, September, 233-239. Yao, B., W.B. Gao, S.P.Chan and M. Cheng (1992a). VSC Coordinated Control of Two Manipulator Anns in the Presence of Enviromnental Constraints, IEEE Transactions on Automatic Control, 37, 1806-1812. Yao, B. and M. Tomizuka (1993b). Adaptive Coordinated Control of Multiple Manipulators Handling a Constrained Object, In: Proceedings of IEEE International Conference on Robotics and Automation, 624-629. Yoshikawa, '!". and A. Umeno (1992). Dynamic Hybrid Control of Manipulators Considering Object Dynamics, In: Proceedings of the Intern. Symp. Measurement and Control in Robotics, 597-604.
ii\+r;'lt +r;i]\ =iV-·(qXN(qR+n(q,q,yo•• Yo,,)~
=N- (q)cD(q.q,ii,y~",Yo.)9 1
(0) denotes the estimate of ( 0) ; N(q}=N(q}-N(q). $(q,ti,tJ'Yoa,Yo..)eR"'x, .
v.here
O=(O-O)e:R' . n(q,q,Yoa ,Yoa )=n(q,q,Yo" ,Y
Od ) -
n(q,q,Yoa,Yoa )
For the i-th element of the deviation vector '7. eR 6kx • , (9), an error equation may be written as:
~ 1i -y vi ~li -Ypi 'Ill =(.iV-'
v.here
r:
=diag[-YvP-Yv2"'.,-'r'V(61:)]
r; =diag[- Yp .. - yP2 •..• '-Y p (6k)]'
(iV-' rrJO)i
(AI) and
(13»,
(see
means the i-tb element of the (6k x 1)
vector (iV-1 ~). TIle stare-space representation of the error equation (AI) is given by:
xj=A,xt+Bt(.iV-1$O)j
(A2)
,,: =CiX j
,.,; is the i-th element of the vector (see (15)), and Xi
=[". ~.
r
". =O;i]. +'1'711
is the state vector. If the
positive constants ; =diag(~ P~ 1 , •••• ,~ H
",=diag(~ .. tP 2 ,.···,tP 6k chosen such that 11*
1~
>0. and tjI i >0, i = 1.2, ... ,6 k. are
1
i
,;r,. +1f/71.
iir-·Cf)iJ ii. +r; 71. + r: 7]. is strictly positive real (SPR) fimction. Then, by the Kalman-Yakubovitch-Popov lemma the symmetric, positive definite matrices PIER M and QjER M do exist such that
Ai P;+P;A;=-Qi
(A3)
p,Bj=cT
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vmere the matrices A j , Bi t and Cj are the matrices of the state-space realization (A2) of the error equation (Al). The error equation of the whole system of k multiple manipulators in the state-space form is given by:
X=AX+BN- I 4>O
(A4)
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.. Vc =V(Xo,Bo)- JXTQXdt
Since Vc and V( X o' ( 0 ) are finite terms, it may be
stated that 00
!XTQXdt
71· =Cx
o
A=blockdiag(At ,A2 •••• ,Ad )eR (2"k).(2nk) B=blockdiag(B1 ,B2, ... ,B"k )eR(2JJk)x.("k)
has a finite value, too. Since X T QX is positive, tmiformly continuous, and has a finite integral, using the well-known mathematical analysis results (Rudin, 1976), it may be stated that limXTQX=O (AB)
C=blockdiag(Cl' C 2 ," .,C"k ):R(Mi: }c(''''') X -I ~Xl
X 2 •· ••••• X"k
'(A12)
o
)T E R211kxI
'~oo
which leads to the conclusion that the state vector X= ( XI x ._ •••• x ) T E R(2"" )x(J)
IT we choose a symmetric positive definite matrix Q
Q=diag(QI ,Q2 ,... ,Q"" )ER (1""').(211.)
CA5)
2
where Xi ~['T/I
and solve the Lyapunovequation
ATp+PA=-Q
r,
nl
rh will asymptotically converge to zero vector as t ---+ 00 :
(A6)
the turique symmetric positive definite maIrix P D D P LR(2nl).«2 ..k) P =diag(..... 1' .....2'· .. ' ..l J<=
(A7)
maybedetennined. The follo\Ving LyaptUlOv :fi.mction candidate is chosen
v(X,O)=XT PX +OT K-1(j where K
(AS)
:;;;:diag[kl'kp......,kJ k j >0, i=1.2, ...,r , is
(rxr) positive definite adaptation gain matrix. The time derivative of (A4) is
v(x,e)
along the trajectories
i.e. the control goals (7) are satisfied. The satisfaction of the control goals (7) also implies a satisfaction of the control goals (8), (Vukobratovic and Tuneski., 1997a). The maximwn and minimum value of the internal force are given by the task plamter. So, if the control goals (8) are satisfied, "We may state that the intema1 force constraints are also satisfied. Now the parameter error convergence will be proved. According to (14) and (A4) the state-space representation of the entire system. may be written as
V{x.O}=- XT QX +20T[cI>T N-1'T/' +K-'{J] (A9) The equation (A9) suggests that, if the parameter update law is chosen to be
e=Kfb Tiv-IT]. then, since the parameter vector 8 is constant, the second tenn in (A9) equals zero, and
V(X,O)=_XTQX
e=8 , (AlO)
=(.iV- tPy .
J where U The asymptotic stability of (AI 1) has been already studied in the literatw"e. Anderson, 1977, has hown that (All) is uniformly asymptotically stable if: (i) the linear system (A, B, C) satisfies the strictly positive real condition (SPR); (ii) U satisfies the persistent excitation condition
oc I ,:S
'o"'P rut . .. 1':' 1':' ~ ~TrI • •• VI' --in \",<1'1 I J' w"q"q,. 0.' o.}J w"ql',q~,I."",l.o.P'-t' , I,
which is nonpositive because positive definite matrix (A5). Since
x,8,fI-
J ,
and
W
Q is chosen to be a
are bounded,:from (A4) it
X
is botmded as -well. Thus X is V(X,8). From (AS) and (AIO) it follows that
may be stated that
for all to. where
U =(iv-'
tPr .r
is the nwnber of the
unknown system parameters, a,p, and p positive, I, is the (rxr) Wlit matrix.
are all
The proof of the theorem is completed.
miformly continuous, and so is
limV(X,8)=V r.... c:o ~
(All)
does exist, and
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