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Analysis and Synthesis of an Output Feedback for an Uncertain Robot Model with Flexible Joints*
Copyright © IFAC System Structure and Control, Nantes, France, 1995
ANALYSIS AND SYNTHESIS OF AN OUTPUT FEEDBACK FOR AN UNCERTAIN ROBOT MODEL WITH FLEXIBLE JOINTS· A. AILON Department of Electrical and Computer Engineering, Ben Gurion University of the Negev, Beer Sheva 84105, Israel Abstract. This study considers the set-point regulation problem of a flexible-joint robot with uncertain parameters and unknown gravity forces when only position measurements are available for the controller. An essential tool in the present approach is the contraction mapping theorem. An analytical solution to the underlying control problem is presented. Some practical issues associated with the implementation of the controller-observer are considered.
Resume. Nous considerons le probleme du controle point-a-point d 'un robot a jointures flexibles avec parametres incertains et forces de gravite inconnues, quand le controleur n'a que des donnees de position. L'outil essentiel dans notre approche est le theoreme des applications contractantes. Nous presentons une solution analytique a notre probleme de controle. Nous considerons aussi des questions pratiques associees a la realisation du controleur-observateur. Key Words- Flexible-joint robot ; controller-observer; point-to-point control; contraction mapping theorem.
1.
bitrarily small neighborhood of the operating point or, under mild physical conditions , global asymptotic stability. An analytical approach associated with the underlying control problem is presented. Some practical issues concerning the application of the controller are considered.
INTRODUCTION
In the majority of current robot applications, the basic set-point regulation problem has been solved using a proportional plus derivative (PD) controller with gravity compensation. For example: Takegaki and Arimoto (1982) and Tomei (1991) consider stabilizing controllers for a rigid robot model , while Tomei (1991 ) presents controllers for stabilizing a robot with elastic joints. The schemes presented in this reference is based on the exact knowledge of the gravity vector . In contrast , De Luca and Panzieri (1992) demonstrates an iterative scheme for generating gravity compensation at the desired set-point. However, the resulting control law is based on the action of a PD controller , and requires noisy measurements of joint velocity. The solution to the set-point regulation problem when only the output signals are available for the controller, has been solved by Berghuis and Nijmeijer (1993) for the rigid robot case and by Ailon and Ortega (1993) and Kelly et al. (1994) for flexiblejoint robots. We consider here the set-point regulation problem of a flexible-joint robot whose model contains uncertain parameters and unknown gravity forces, and only a set of possible values of these terms is known . The objective is to design a controller which implements only position measurements on the motor side, and yet ensures either globally ultimate boundedness such that every system response converges to an ar-
2. PRELIMINARY RESULTS From Spong and Vidyasagar (1989 ) the model of a flexible-joint robot is given by :
D(qdih
+ C(ql, qdql + g(qd = K(q2 Jij2 + K(q2 - qd = u,
ql) ,
( 1)
where ql , q2 E lRn represent the link and the motor angles, D(qd > 0 is the robot 's inertia matrix, J > 0 is the actuator inertias matrix, C (ql , ql)ql represents the Coriolis and centrifugal forces , g( ql) is the gravitational force vector, K > 0 is the joint stiffness coefficients diagonal matrix, and u E lRn is the applied torques. The Euclidean norm of x is denoted by IIxll and IIAlli is the induced norm of the matrix A with IIAII? == Amax(AT A) , where Amax(AT A) is the largest eigenvalue of (AT A) . The state space representation of (1 ) is given by Xl = X2 X2 = D(xd- l ( -C(Xl ' X2)X2 - g(x d - K(Xl - X3)) X3 = X4 X4 = J-l( -K(X3 - xt} + u), (2) where Xl == ql , X2 ==
·This work was supported in part by the joint FrenchIsraeli collaboration project " Arc en ciel" 1994 under contract FN 49035 .
335
The suggested n-dimensional linear controllerobserver is given by
z= -S(z u
X3) - Rz + v,
for all x r . But from (13), (14) is satisfied if
SO ] > f3 [In In]. o R In In Since there exists a permutation matrix P such that [
(4)
= -S(X3 - z) ,
(5)
=
=
where the constant matrices S SI' , R RT > 0 and the vector v E iRn , are to be determined. Remarks. The controller-observer (4) is different in its structure from the ones given in Ailon and Ortega (1993) and Kelly et al. (1994) . Note that the vector v (which appears in the first reference but not in the second) is essential for the approach below and the application of the contraction mapping theorem. We shall establishe the main results first for the case of a rigid robot, and then extend it to the flexible-joint robot model. For a rigid robot the model (2)-(3) reduces to Xl = X2 X2 = D(Xl )-l( -C(Xl ' X2)X2 - g(xd
Y=
p
where Ur == where
u
)jAmax(Urunl = 2, a;:{ 1:>:,=0> 0 if
Xl ,
xd - Rz + v,
~
(8)
Consider (6) , (8)-(9) and define a scalar function Hr
Hr(Xl , X2, z) = Hx{ D(Xl)X2 + (Xl - zf S(Xl - z) +(z - R-lv)T R( z - R-lv)] + Ug(xd (10) Evaluation of oHrlox r I:>:,=z,=o;x r == [xI. zTY yields
where it is assumed that there exists f3
+(z - R-lv)T R(z - R-lv)]
:>:lE~"
[ In In
= [ SO] 0 R
+
[In In
S
K+S
-S
(13)
-S]} [In +R 0
In] d'zag. In
(19)
-K
0 with
1:>:,=0>
(18) 0 im-
From (18)-(19) we have
OXl
0] {[ S +-S~
In
1:>:,=:>:.=0=
For x, == [xf,xI,zTY, oH,lox, plies
(12)
IIOg(xdll>O.
Following Sylvester 's law of inertia, °o'~' :>:, iff
+ Ug(xt).
K(Xl - X3) + g(Xl) = 0 -K(Xl - X3) + S(X3 - z) = 0 -S(X3 - z) + R(z - R-lv) = O. '
v)
xr
(11)
] ~
+ Rz -
H,(xl, X2, X3, X4, z) = Hx{ D(Xl)X2 + J X4 +(Xl - x3f K(Xl - X3) + (X3 - z)T S(X3 - z)
From (10) and (11) we have
-S
Xl)
o.
(17) Invoking LaSalle 's invariance theorem, we conclude that the system is globally asymptotically stable. For the flexible-joint robot (2) with the controller (4)-( 5) we define the following scalar function
(9)
S+R
(16)
Along the system 's trajectory we have
Vr= - (S(z - xd + Rz - v)T (S(z -
S(Xl - z) + g(Xl) = 0 -S(Xl - z) + R(z - R-lv) = O.
(15)
v;. (Xl , X2, z) = Hr(Xl , X2 , z)- fIr . (7)
= -S(Xl - z) .
f3=sup
[~ ~],
If {S, R} satisfies (15) then, for a given vector v the scalar function Hr(Xl , O,Z) has a unique global minimum point Xr = [xi, .iTy . Let the minimal value of Hr be denoted by fIr and define a Lyapunov candidate function
and the controller (4)-(5) becomes
z= -S(z -
diag .{Ur};
min{Amin(S) , Amin(R)} > 2f3.
(6)
+ u),
[~: ~:] pT = block -
Now, using previous arguments one concludes that the Hermitian matrix o;~' 1:>:,=:>:.=0 is positive definite for all iff the matrix
0
x,
[ InIn
In] In
In
{~} 0:>:1 >0
[Ko o
(14) 336
0 In In
0 0 In 0 0 S 0 0 R
[~ 0:>:; 1:>:,=:>:.=0 ]
In 0
0 +
[ InIn In
In In In
In In In
In In 0
In] In In
d'zag . {~} 0:>:1 '
From (24) and the fact that for each v E ~n (19) has (by recalling (20)), a unique solution the map Lr : ~n -+ ~n is bijective. In particular we may write
is positive definite. But again it is easy to see that there exists a permutation matrix Q such that
In In In
In In In
1
QT
where UJ ='=
= block - diag. {UJ};
1 1 1 1 1 1 [ 1 1 1
(25)
1
Let Xld be a given constant vector and define a map Tr : ~n --+ ~n by
,
(26) Using Theorem 2.1 we have the following result. Lemma 3.1. Consider the map Tr(v) defined by (24)-(26), and assume that g(x) satisfies (13). Then a pair of matrices {S, R} can be determined such that Tr is a global contraction with a unique fixed point
(21)
.
Suppose the triplet {K, 5, R} satisfies (21). Then, for a given v the function HJ(Xl, 0, X3, 0, z) has a unique global minimum. Let if J be the minimal value of H J and define a Lyapunov candidate function
VJ(Xl, X2, X3, X4, z) = HJ(Xl: X2, X3, X4, z)-
(27)
which implies (28)
if J
. (22) The derivative of VJ(Xl,X2,X3,X4,Z) along the trajectories is now given by .
.
v , J.e.,
Proof Let the pair {S, R} be selected according to the following rule: R ='= rIn ; r > 2{3, S ='= R2.
(29)
T
VJ=-(S(z-x3)+Rz-v) (S(z- x3)+Rz-v)
Since Lr in (24) is bijective, we may write using (25) Gr(vt) = xL and Gr (V2) = XI. Let f(xi,xI) ='= {w I w = Oxt + (1 - O)xI, 0 < 0 < I} be a line segment connecting xt to xi. By the mean value theorem
~O.
(23) Invoking LaSalle's invariance theorem, we conclude that the system is globally asymptotically stable. Finally, the following Theorem (Vidyasagar (1993)), is an essential tool along this study: Theorem 2.1. (Global Contraction Mapping) . Let (X,II .II) be a Banach space, and let T : X --+ X. Suppose there exists a fixed constant 1] < 1 such that 11
Tx - Ty II~ 1]
11 x - y
g(xD - g(xi) = IT(xt - xi); 1
IT ='=
1]n
1-1]
(30)
From (24)-(26), and (29) we have
1I,'l/x,yE X.
xnll ~ --IITxo -
8) xi) dO.
I
Tr(vt) - Tr (V2) = (In + R-l)g(xD + Rxt -R(xt - Xld) - «(In + R- l )g(xi) + Rxi -R(XI - Xld)) = (In + R- l )(g(xD - g(xi)),
Under these conditions, there exists exactly one x· E X such that Tx· = x·. For each Xo EX, the sequence {x n } in X defined by Xn+l = TX n converges to X·. Moreover,
IIx· -
J I;- (Oxt + (1 o
(~1)
where, by definition xi = Gr(Vi). Hence, by (30)
xoll·
3. BASIC ANALYTICAL RESULTS 3.1. Case a: Rigid Robot. Consider (6), and assume the pair {S, R} satisfies (15). Then, from the first equation of (11) we have
Recalling (24) one obtains
Hence, for every Xl E ~n the unique vector v that satisfies (11) is given by
and using (30) we arrive at
V2 = (In + R- l )g(xD + Rxt -«(In + R-l)g(xi) + RxI) = (In + R-l)(g(xD - g(xi)) + R(xt - xi) Vl -
(24) 337
point will be represented by (~n
By Schwarz inequality and its direct consequence
r+1
(35)
Select arbitrarily p > 1 and r > max{rl,2p.8} . Then, by (13) and (29) the right hand side of (34) satisfies IIIR(xi - xi)II - lI(In + R- I )II(xi - xnlll = IIR(xt - xi)l l - 11 (In + R-I)II(xt - xnll ~ rllxi - xiii - p.8l1xi - xiiI,
motion
II x~ - xi
=;>
comp .
_
T. ( ) motion I comp . Vl = r Vo ==> Xl =;> T.r (Vn-l ) motion n ==> Xl· · ··
In addition, (36), (39) and (41) ensure that:
11:511
VI - V2 II .
(36)
IlxId - x~II
From (32) and (30) one also concludes that II Tr(vI) - Tr (V2)11:5 p.8llx~ - xiii·
:5
(r_Ip{J) Ilv· - vnll
'1"
:5 (l-'1r)(r-p{J) IITr(vo) - voll, (37)
where v· is the limit of the sequence tively Xld is the limit of {xi}. 00
Therefore , since Gr ( v) is bijective (i.e. IIxi -xiII i= o~ IIVI - v2 11i= 0), for every r > max{rl, 2p.8} the following holds
{Vi}
and respec-
3.2. Case b: Flexible-Joint Robot. Now consider (2) , and assume that the triplet {K , S, R} satisfies (21). From (19) we obtain
V == Lf(XI) = (In+RS-l+RK-I)g(xt}+Rxl (42)
and we clearly have 7]r
0 comp .
Vo ==> Xl
.. . =;> Vn -
and as a result of (33)-(34)
(r - p.8)
The vec-
tor xi determines Vi+! through Vi+! == Tr ( Vi) = Vi - R(xi - Xld). The computation of Vi is based on measurable variables and will be represented below by (c~} Using these notations we state the following Algorithm . Algorithm 3.1. Select constants p > 1 and r > max{ rI, 2p.8} where TI is any constant that satisfies (35), and determine according to (29) a pair {S, R} . Then, an arbitrary selection of a constant vector Vo E ~n initiates the following chain:
1I[(In + R-l)II + R](xl- xI)II ~ I IIR(xi - xI)ll- II(In + R-l)II(xl- xi)II I . (34) For every P > lone can select rl > 0 such that O<-r-
).
p.8a < 1. =. - r - PI-'
and by previous considerations the map L f : ~n is bijective. In particular we may write
(39)
( 43)
Since the space ~n with the Euclidean norm is a Banach space, following Theorem 2.1, Tr in (26) is a global contraction, and the sequence {v n } defined by (40)
Let XId be a given constant vector and define a map Tf : ~n -+ ~n as follows : (44)
converges to a unique fixed point v· which satisfies (27)-(28). The rate of convergence of the sequence {vn } to v· is determined by the parameter r, and there exists:
IIv· - vn ll:5 1 7]~ IITr(vo) - voll, - 7]r
~n -+
Lemma 3.2. Consider the map Tf (v) defined by (42)-(44) and assume that g(x) satisfies (13) and >'min(K) > 3.8. Then a pair of matrices {S, R} can be determined such that Tf is a global contraction with a unique fixed point v· , i. e. ,
(41)
(45)
where Vo E ~n is an arbitrary initial vector. 00 Let {vd be the sequence determined by (40). For any given initial position and for each Vi the closedloop system trajectory
which implies (46)
Proof. Let the pair {S, R} be selected according to the following rule R= r1n;r > 3.8 , 5 = R2 .
[xf ,o.zTr
with XI= xi . In the following algorithms the system's motion toward its equilibrium
From (21) L f in (42) is bijective. Let Gf From (42)-(44), and (29) we have 338
(47) (Vi)
= xi ·
which implies, using (50)-(51):
TJ(vd - TJ (V2) = (In + R- I + RK-I)g(xl) + Rx! -R(x! - XId) - «(In + R- I + RK-I)g(xt) +RxI - R(xI - XId)) =(In + R-I + RK-I )(g(xl) - g(xt)),
(r (48)
~,8)
11 x~ - xi II~II VI - V2 11 .
(55)
From (49) and (54) we have (56)
and using (30)
TJ(VI) - TJ(V2) = (In
Hence, by observing the last two equations and recalling that G J (v) is bijective we arrive at:
+ R- I + RK-I)II(x~ - xi). (49)
Recalling (42) one obtains It remains to show that
VI -
V2 = (In + R-I + RK- I )g(xi) + Rx! -«(In + R- I + RK-I)g(xi) + Rxi) = (In + R- I + RK-I)(g(xl) - g(xi)) + R(x! - xi).
TJJ
.~,8 ~.8
=r -
Clearly (58) holds if r
Again, using (30) we have
2r+2
r - ).min(K),8r
and hence
IIVI - v211 = 11[(In +R- I + RK-I)II + Rl(x~ - xi)lI· (50) By Schwarz inequality one gets
IH(In + R- I + RK-I)II + R](xt - xi)11 2: IIIR(xt - xnll-ll(In + R- I + RK-I)II(xi - xD111 .
(58)
> 2.8~ or (see (54)):
2
VI - V2 = (In +R- I +RK- I )II(x~ - xi)+R(x~ -xi),
1
< .
(59)
> -r-,8·
But recalling (35) and using the fact that ).min(K) > 3.8, (59) is satisfied ifr(l-~) > 2p,8 or r > 6p,8. With this result one concludes by Theorem 2.1 that TJ in (44) is a global contraction, and the sequence {v n } in ~n defined by
(51) Arbitrarily select P > 1, and define rl such that (35) holds, and let r > max{ rI, 6p,8, tP).min (K)}. Clearly r satisfies (47). Next we claim that
converges to a unique fixed point v· which satisfies (45). Hence by (44), (46) holds and there exists:
(52)
(61)
Indeed, since ).min (K) > 3,8 and r > one obtains by (35):
t P).min (K),
where Vo E ~n is an arbitrary initial vector. 00 Following Lemma 3.2 and using previous notations we have the following algorithm for the present case. Algorithm 3.2. Select constants P > 1 and r > max{rl' 6p,8, tP).min(K)} where rl is any number that satisfies (35). Then determine according to (47) a pair {S, R}. Then, an arbitrary selection of a constant vector Vo E ~n initiates the following chain:
2r 2 ,8 > (r + l).min(K),8 => r2().min(K) -,8) > (r + l).min(K).8, and the last inequality yields
r>
r2
+ r>.min (K) + ).min (K),8 r).min(K
).
(53)
The last inequality together with the relation (recall that by definition K and R are diagonal matrices)
motion
0 comp.
T (
)
motipn
I comp.
Vo ==> XI ~ VI = J Vo ~ XI ~ comp. ) motion n ... ~ Vn = TJ(Vn-1 ==> Xl·· ··
In addition , (54), (55), (58) and (61) ensure that (54) assert the claim. Therefore, if r is sufficiently large we have
IIXld -
Xn 11 ~ (r_l~.6) 1)"
IIv· -
~ (1-1)J)(r-~.6) IITJ(va) -
IIIR(xt - xDII-II(In + R- 1 + RK-l )II(xi - xD111 = IIR(xi - xi)II-II(In + R- I + RK- I )II(xt - xi)11 ~ rllx! - xi 11- ~,8llxi - XIII,
Vn 11
vall,
where V· is the limit of the sequence {Vi} and respectively XId is the limit of {xi}. 00 339
4.
say the decision maker. This action ensures asymptotic convergence to the desired target.
FINAL REMARKS CONCERNING THE PRACTICAL SOLUTION TO THE CONTROL PROBLEM
REFERENCES
Some remarks with regard to the applications of the previous analytical results to the framework established by synthesis are in order. (i)- In the case of a rigid robot the rate of convergence is determined by the parameter TJr which, as observed in (39) , depends on r. If, for a given p > 0, one takes a relatively large r , T}r becomes as small as desired and the robot converges rapidly to Xld . (ii)- While the rate of convergence of the rigid robot to the final destination can be tuned arbitrarily, this is not the case for a flexible-joint robot. In fact, the key parameter with respect to the rate of convergence in this case is T}J in (58) . Since ~ depends also on r (see (54)) there is a lower bound to TJJ . By a direct calculation it can be shown that
[1] Takegaki, M. and S. Arimoto (1981) . A new feedback method for dynamic control of manipulators, Trans . ASME J. on Dynamic Systems, Measurement, and Control 102, 119-125. [2] Tomei P. (1991) . Adaptive PD controller for robot manipulators, IEEE Trans. on Robotics and Automation, 7, 565-570 . [3] Tomei P. (1991) . A simple PD controller for robots with elastic joints, IEEE Trans. on Automatic Control, 36, 1208-1213. [4] De Luca A. and S. Panzieri (1992). An iterative scheme for learning gravity compensation in flexible robot arms, Cancun Workshop on Nonlinear Systems and Robotics, Cancun, Mexico. [5] Berghuis H. and H. Nijmeijer (1993) . Global regulation of robots using only position measurements, Systems and Control Letters, 21 , 289-293.
Following the assumption that .Amin(K) > 3{3 it can be seen that iffor example .Amin (K) ~ 3{3, then for a large r, TJJ ~ t. If .Amin (K) is relatively a large number, that lower bound becomes respectively small. (iii)- Since asymptotic convergence is associated with infinite-time process, the above algorithms cannot be implemented in a straight-forward manner in practical applications. However, this problem can be overcomed using the following approach . (Due to limitation in space we present here just the outlines of the relevant approach .) Let g > 0 be a (small as desired) constant. Since from (17) and (23) the functions Vr (.) and VJ (-) depend just on measurable signals , assuming some upper bound on the system initial 'energy' is known , it can be shown formally that, based on available accumulated data it is possible to determine when the system's state vector belongs to an g - neighborhood of an (unknown) equilibrium point. Now, whatever the initial condition is , it takes finite time for the system's trajectory to enter that neighborhood. Following this observation, the algorithm can be synthesized such that in each step of the process a vector Vn+l, rather than Vn+l = T(v n ) , is determined. Moreover , it can be shown that ifin each step 11 Vn+l -vn +111 is sufficiently small then, finally the system will converge to an arbitrarily small neighborhood of the desired equilibrium point. Furthermore, if g > 0 was selected sufficiently small, then , as soon as the system's trajectory enters into an g - neighborhood of the desired equilibrium point, the control strategy can be switched to a new mode, and a different controller-observer which is based on the principle of stability in the first approximation (Vidyasagar (1993» can be initiated by,
[6] Ailon A. and R. Ortega (1993). An observerbased set-point controller for robot manipulators with flexible joints, Systems and Control Letters, 21 , 329-335. [7] Kelly R., R. Ortega, A. Ailon, and A. Loria (1994). Global regulation of Flexible joints robots using approximate differentiation, IEEE Trans . on Automatic Control, 39, 1222-1224. [8] Spong M. and M. Vidyasagar (1989). Robot Dynamics and Control, John Wiley & Sons, New York . [9] Vidyasagr M. (1993). Nonlinear Systems Analysis, Prentice-Hall, New York .
340
ANALYSIS AND SYNTHESIS OF AN OUTPUT FEEDBACK FOR AN UNCERTAIN ROBOT MODEL WITH FLEXIBLE JOINTS· A. AILON Department of Electrical and Computer Engineering, Ben Gurion University of the Negev, Beer Sheva 84105, Israel Abstract. This study considers the set-point regulation problem of a flexible-joint robot with uncertain parameters and unknown gravity forces when only position measurements are available for the controller. An essential tool in the present approach is the contraction mapping theorem. An analytical solution to the underlying control problem is presented. Some practical issues associated with the implementation of the controller-observer are considered.
Resume. Nous considerons le probleme du controle point-a-point d 'un robot a jointures flexibles avec parametres incertains et forces de gravite inconnues, quand le controleur n'a que des donnees de position. L'outil essentiel dans notre approche est le theoreme des applications contractantes. Nous presentons une solution analytique a notre probleme de controle. Nous considerons aussi des questions pratiques associees a la realisation du controleur-observateur. Key Words- Flexible-joint robot ; controller-observer; point-to-point control; contraction mapping theorem.
1.
bitrarily small neighborhood of the operating point or, under mild physical conditions , global asymptotic stability. An analytical approach associated with the underlying control problem is presented. Some practical issues concerning the application of the controller are considered.
INTRODUCTION
In the majority of current robot applications, the basic set-point regulation problem has been solved using a proportional plus derivative (PD) controller with gravity compensation. For example: Takegaki and Arimoto (1982) and Tomei (1991) consider stabilizing controllers for a rigid robot model , while Tomei (1991 ) presents controllers for stabilizing a robot with elastic joints. The schemes presented in this reference is based on the exact knowledge of the gravity vector . In contrast , De Luca and Panzieri (1992) demonstrates an iterative scheme for generating gravity compensation at the desired set-point. However, the resulting control law is based on the action of a PD controller , and requires noisy measurements of joint velocity. The solution to the set-point regulation problem when only the output signals are available for the controller, has been solved by Berghuis and Nijmeijer (1993) for the rigid robot case and by Ailon and Ortega (1993) and Kelly et al. (1994) for flexiblejoint robots. We consider here the set-point regulation problem of a flexible-joint robot whose model contains uncertain parameters and unknown gravity forces, and only a set of possible values of these terms is known . The objective is to design a controller which implements only position measurements on the motor side, and yet ensures either globally ultimate boundedness such that every system response converges to an ar-
2. PRELIMINARY RESULTS From Spong and Vidyasagar (1989 ) the model of a flexible-joint robot is given by :
D(qdih
+ C(ql, qdql + g(qd = K(q2 Jij2 + K(q2 - qd = u,
ql) ,
( 1)
where ql , q2 E lRn represent the link and the motor angles, D(qd > 0 is the robot 's inertia matrix, J > 0 is the actuator inertias matrix, C (ql , ql)ql represents the Coriolis and centrifugal forces , g( ql) is the gravitational force vector, K > 0 is the joint stiffness coefficients diagonal matrix, and u E lRn is the applied torques. The Euclidean norm of x is denoted by IIxll and IIAlli is the induced norm of the matrix A with IIAII? == Amax(AT A) , where Amax(AT A) is the largest eigenvalue of (AT A) . The state space representation of (1 ) is given by Xl = X2 X2 = D(xd- l ( -C(Xl ' X2)X2 - g(x d - K(Xl - X3)) X3 = X4 X4 = J-l( -K(X3 - xt} + u), (2) where Xl == ql , X2 ==
·This work was supported in part by the joint FrenchIsraeli collaboration project " Arc en ciel" 1994 under contract FN 49035 .
335
The suggested n-dimensional linear controllerobserver is given by
z= -S(z u
X3) - Rz + v,
for all x r . But from (13), (14) is satisfied if
SO ] > f3 [In In]. o R In In Since there exists a permutation matrix P such that [
(4)
= -S(X3 - z) ,
(5)
=
=
where the constant matrices S SI' , R RT > 0 and the vector v E iRn , are to be determined. Remarks. The controller-observer (4) is different in its structure from the ones given in Ailon and Ortega (1993) and Kelly et al. (1994) . Note that the vector v (which appears in the first reference but not in the second) is essential for the approach below and the application of the contraction mapping theorem. We shall establishe the main results first for the case of a rigid robot, and then extend it to the flexible-joint robot model. For a rigid robot the model (2)-(3) reduces to Xl = X2 X2 = D(Xl )-l( -C(Xl ' X2)X2 - g(xd
Y=
p
where Ur == where
u
)jAmax(Urunl = 2, a;:{ 1:>:,=0> 0 if
Xl ,
xd - Rz + v,
~
(8)
Consider (6) , (8)-(9) and define a scalar function Hr
Hr(Xl , X2, z) = Hx{ D(Xl)X2 + (Xl - zf S(Xl - z) +(z - R-lv)T R( z - R-lv)] + Ug(xd (10) Evaluation of oHrlox r I:>:,=z,=o;x r == [xI. zTY yields
where it is assumed that there exists f3
+(z - R-lv)T R(z - R-lv)]
:>:lE~"
[ In In
= [ SO] 0 R
+
[In In
S
K+S
-S
(13)
-S]} [In +R 0
In] d'zag. In
(19)
-K
0 with
1:>:,=0>
(18) 0 im-
From (18)-(19) we have
OXl
0] {[ S +-S~
In
1:>:,=:>:.=0=
For x, == [xf,xI,zTY, oH,lox, plies
(12)
IIOg(xdll>O.
Following Sylvester 's law of inertia, °o'~' :>:, iff
+ Ug(xt).
K(Xl - X3) + g(Xl) = 0 -K(Xl - X3) + S(X3 - z) = 0 -S(X3 - z) + R(z - R-lv) = O. '
v)
xr
(11)
] ~
+ Rz -
H,(xl, X2, X3, X4, z) = Hx{ D(Xl)X2 + J X4 +(Xl - x3f K(Xl - X3) + (X3 - z)T S(X3 - z)
From (10) and (11) we have
-S
Xl)
o.
(17) Invoking LaSalle 's invariance theorem, we conclude that the system is globally asymptotically stable. For the flexible-joint robot (2) with the controller (4)-( 5) we define the following scalar function
(9)
S+R
(16)
Along the system 's trajectory we have
Vr= - (S(z - xd + Rz - v)T (S(z -
S(Xl - z) + g(Xl) = 0 -S(Xl - z) + R(z - R-lv) = O.
(15)
v;. (Xl , X2, z) = Hr(Xl , X2 , z)- fIr . (7)
= -S(Xl - z) .
f3=sup
[~ ~],
If {S, R} satisfies (15) then, for a given vector v the scalar function Hr(Xl , O,Z) has a unique global minimum point Xr = [xi, .iTy . Let the minimal value of Hr be denoted by fIr and define a Lyapunov candidate function
and the controller (4)-(5) becomes
z= -S(z -
diag .{Ur};
min{Amin(S) , Amin(R)} > 2f3.
(6)
+ u),
[~: ~:] pT = block -
Now, using previous arguments one concludes that the Hermitian matrix o;~' 1:>:,=:>:.=0 is positive definite for all iff the matrix
0
x,
[ InIn
In] In
In
{~} 0:>:1 >0
[Ko o
(14) 336
0 In In
0 0 In 0 0 S 0 0 R
[~ 0:>:; 1:>:,=:>:.=0 ]
In 0
0 +
[ InIn In
In In In
In In In
In In 0
In] In In
d'zag . {~} 0:>:1 '
From (24) and the fact that for each v E ~n (19) has (by recalling (20)), a unique solution the map Lr : ~n -+ ~n is bijective. In particular we may write
is positive definite. But again it is easy to see that there exists a permutation matrix Q such that
In In In
In In In
1
QT
where UJ ='=
= block - diag. {UJ};
1 1 1 1 1 1 [ 1 1 1
(25)
1
Let Xld be a given constant vector and define a map Tr : ~n --+ ~n by
,
(26) Using Theorem 2.1 we have the following result. Lemma 3.1. Consider the map Tr(v) defined by (24)-(26), and assume that g(x) satisfies (13). Then a pair of matrices {S, R} can be determined such that Tr is a global contraction with a unique fixed point
(21)
.
Suppose the triplet {K, 5, R} satisfies (21). Then, for a given v the function HJ(Xl, 0, X3, 0, z) has a unique global minimum. Let if J be the minimal value of H J and define a Lyapunov candidate function
VJ(Xl, X2, X3, X4, z) = HJ(Xl: X2, X3, X4, z)-
(27)
which implies (28)
if J
. (22) The derivative of VJ(Xl,X2,X3,X4,Z) along the trajectories is now given by .
.
v , J.e.,
Proof Let the pair {S, R} be selected according to the following rule: R ='= rIn ; r > 2{3, S ='= R2.
(29)
T
VJ=-(S(z-x3)+Rz-v) (S(z- x3)+Rz-v)
Since Lr in (24) is bijective, we may write using (25) Gr(vt) = xL and Gr (V2) = XI. Let f(xi,xI) ='= {w I w = Oxt + (1 - O)xI, 0 < 0 < I} be a line segment connecting xt to xi. By the mean value theorem
~O.
(23) Invoking LaSalle's invariance theorem, we conclude that the system is globally asymptotically stable. Finally, the following Theorem (Vidyasagar (1993)), is an essential tool along this study: Theorem 2.1. (Global Contraction Mapping) . Let (X,II .II) be a Banach space, and let T : X --+ X. Suppose there exists a fixed constant 1] < 1 such that 11
Tx - Ty II~ 1]
11 x - y
g(xD - g(xi) = IT(xt - xi); 1
IT ='=
1]n
1-1]
(30)
From (24)-(26), and (29) we have
1I,'l/x,yE X.
xnll ~ --IITxo -
8) xi) dO.
I
Tr(vt) - Tr (V2) = (In + R-l)g(xD + Rxt -R(xt - Xld) - «(In + R- l )g(xi) + Rxi -R(XI - Xld)) = (In + R- l )(g(xD - g(xi)),
Under these conditions, there exists exactly one x· E X such that Tx· = x·. For each Xo EX, the sequence {x n } in X defined by Xn+l = TX n converges to X·. Moreover,
IIx· -
J I;- (Oxt + (1 o
(~1)
where, by definition xi = Gr(Vi). Hence, by (30)
xoll·
3. BASIC ANALYTICAL RESULTS 3.1. Case a: Rigid Robot. Consider (6), and assume the pair {S, R} satisfies (15). Then, from the first equation of (11) we have
Recalling (24) one obtains
Hence, for every Xl E ~n the unique vector v that satisfies (11) is given by
and using (30) we arrive at
V2 = (In + R- l )g(xD + Rxt -«(In + R-l)g(xi) + RxI) = (In + R-l)(g(xD - g(xi)) + R(xt - xi) Vl -
(24) 337
point will be represented by (~n
By Schwarz inequality and its direct consequence
r+1
(35)
Select arbitrarily p > 1 and r > max{rl,2p.8} . Then, by (13) and (29) the right hand side of (34) satisfies IIIR(xi - xi)II - lI(In + R- I )II(xi - xnlll = IIR(xt - xi)l l - 11 (In + R-I)II(xt - xnll ~ rllxi - xiii - p.8l1xi - xiiI,
motion
II x~ - xi
=;>
comp .
_
T. ( ) motion I comp . Vl = r Vo ==> Xl =;> T.r (Vn-l ) motion n ==> Xl· · ··
In addition, (36), (39) and (41) ensure that:
11:511
VI - V2 II .
(36)
IlxId - x~II
From (32) and (30) one also concludes that II Tr(vI) - Tr (V2)11:5 p.8llx~ - xiii·
:5
(r_Ip{J) Ilv· - vnll
'1"
:5 (l-'1r)(r-p{J) IITr(vo) - voll, (37)
where v· is the limit of the sequence tively Xld is the limit of {xi}. 00
Therefore , since Gr ( v) is bijective (i.e. IIxi -xiII i= o~ IIVI - v2 11i= 0), for every r > max{rl, 2p.8} the following holds
{Vi}
and respec-
3.2. Case b: Flexible-Joint Robot. Now consider (2) , and assume that the triplet {K , S, R} satisfies (21). From (19) we obtain
V == Lf(XI) = (In+RS-l+RK-I)g(xt}+Rxl (42)
and we clearly have 7]r
0 comp .
Vo ==> Xl
.. . =;> Vn -
and as a result of (33)-(34)
(r - p.8)
The vec-
tor xi determines Vi+! through Vi+! == Tr ( Vi) = Vi - R(xi - Xld). The computation of Vi is based on measurable variables and will be represented below by (c~} Using these notations we state the following Algorithm . Algorithm 3.1. Select constants p > 1 and r > max{ rI, 2p.8} where TI is any constant that satisfies (35), and determine according to (29) a pair {S, R} . Then, an arbitrary selection of a constant vector Vo E ~n initiates the following chain:
1I[(In + R-l)II + R](xl- xI)II ~ I IIR(xi - xI)ll- II(In + R-l)II(xl- xi)II I . (34) For every P > lone can select rl > 0 such that O<-r-
).
p.8a < 1. =. - r - PI-'
and by previous considerations the map L f : ~n is bijective. In particular we may write
(39)
( 43)
Since the space ~n with the Euclidean norm is a Banach space, following Theorem 2.1, Tr in (26) is a global contraction, and the sequence {v n } defined by (40)
Let XId be a given constant vector and define a map Tf : ~n -+ ~n as follows : (44)
converges to a unique fixed point v· which satisfies (27)-(28). The rate of convergence of the sequence {vn } to v· is determined by the parameter r, and there exists:
IIv· - vn ll:5 1 7]~ IITr(vo) - voll, - 7]r
~n -+
Lemma 3.2. Consider the map Tf (v) defined by (42)-(44) and assume that g(x) satisfies (13) and >'min(K) > 3.8. Then a pair of matrices {S, R} can be determined such that Tf is a global contraction with a unique fixed point v· , i. e. ,
(41)
(45)
where Vo E ~n is an arbitrary initial vector. 00 Let {vd be the sequence determined by (40). For any given initial position and for each Vi the closedloop system trajectory
which implies (46)
Proof. Let the pair {S, R} be selected according to the following rule R= r1n;r > 3.8 , 5 = R2 .
[xf ,o.zTr
with XI= xi . In the following algorithms the system's motion toward its equilibrium
From (21) L f in (42) is bijective. Let Gf From (42)-(44), and (29) we have 338
(47) (Vi)
= xi ·
which implies, using (50)-(51):
TJ(vd - TJ (V2) = (In + R- I + RK-I)g(xl) + Rx! -R(x! - XId) - «(In + R- I + RK-I)g(xt) +RxI - R(xI - XId)) =(In + R-I + RK-I )(g(xl) - g(xt)),
(r (48)
~,8)
11 x~ - xi II~II VI - V2 11 .
(55)
From (49) and (54) we have (56)
and using (30)
TJ(VI) - TJ(V2) = (In
Hence, by observing the last two equations and recalling that G J (v) is bijective we arrive at:
+ R- I + RK-I)II(x~ - xi). (49)
Recalling (42) one obtains It remains to show that
VI -
V2 = (In + R-I + RK- I )g(xi) + Rx! -«(In + R- I + RK-I)g(xi) + Rxi) = (In + R- I + RK-I)(g(xl) - g(xi)) + R(x! - xi).
TJJ
.~,8 ~.8
=r -
Clearly (58) holds if r
Again, using (30) we have
2r+2
r - ).min(K),8r
and hence
IIVI - v211 = 11[(In +R- I + RK-I)II + Rl(x~ - xi)lI· (50) By Schwarz inequality one gets
IH(In + R- I + RK-I)II + R](xt - xi)11 2: IIIR(xt - xnll-ll(In + R- I + RK-I)II(xi - xD111 .
(58)
> 2.8~ or (see (54)):
2
VI - V2 = (In +R- I +RK- I )II(x~ - xi)+R(x~ -xi),
1
< .
(59)
> -r-,8·
But recalling (35) and using the fact that ).min(K) > 3.8, (59) is satisfied ifr(l-~) > 2p,8 or r > 6p,8. With this result one concludes by Theorem 2.1 that TJ in (44) is a global contraction, and the sequence {v n } in ~n defined by
(51) Arbitrarily select P > 1, and define rl such that (35) holds, and let r > max{ rI, 6p,8, tP).min (K)}. Clearly r satisfies (47). Next we claim that
converges to a unique fixed point v· which satisfies (45). Hence by (44), (46) holds and there exists:
(52)
(61)
Indeed, since ).min (K) > 3,8 and r > one obtains by (35):
t P).min (K),
where Vo E ~n is an arbitrary initial vector. 00 Following Lemma 3.2 and using previous notations we have the following algorithm for the present case. Algorithm 3.2. Select constants P > 1 and r > max{rl' 6p,8, tP).min(K)} where rl is any number that satisfies (35). Then determine according to (47) a pair {S, R}. Then, an arbitrary selection of a constant vector Vo E ~n initiates the following chain:
2r 2 ,8 > (r + l).min(K),8 => r2().min(K) -,8) > (r + l).min(K).8, and the last inequality yields
r>
r2
+ r>.min (K) + ).min (K),8 r).min(K
).
(53)
The last inequality together with the relation (recall that by definition K and R are diagonal matrices)
motion
0 comp.
T (
)
motipn
I comp.
Vo ==> XI ~ VI = J Vo ~ XI ~ comp. ) motion n ... ~ Vn = TJ(Vn-1 ==> Xl·· ··
In addition , (54), (55), (58) and (61) ensure that (54) assert the claim. Therefore, if r is sufficiently large we have
IIXld -
Xn 11 ~ (r_l~.6) 1)"
IIv· -
~ (1-1)J)(r-~.6) IITJ(va) -
IIIR(xt - xDII-II(In + R- 1 + RK-l )II(xi - xD111 = IIR(xi - xi)II-II(In + R- I + RK- I )II(xt - xi)11 ~ rllx! - xi 11- ~,8llxi - XIII,
Vn 11
vall,
where V· is the limit of the sequence {Vi} and respectively XId is the limit of {xi}. 00 339
4.
say the decision maker. This action ensures asymptotic convergence to the desired target.
FINAL REMARKS CONCERNING THE PRACTICAL SOLUTION TO THE CONTROL PROBLEM
REFERENCES
Some remarks with regard to the applications of the previous analytical results to the framework established by synthesis are in order. (i)- In the case of a rigid robot the rate of convergence is determined by the parameter TJr which, as observed in (39) , depends on r. If, for a given p > 0, one takes a relatively large r , T}r becomes as small as desired and the robot converges rapidly to Xld . (ii)- While the rate of convergence of the rigid robot to the final destination can be tuned arbitrarily, this is not the case for a flexible-joint robot. In fact, the key parameter with respect to the rate of convergence in this case is T}J in (58) . Since ~ depends also on r (see (54)) there is a lower bound to TJJ . By a direct calculation it can be shown that
[1] Takegaki, M. and S. Arimoto (1981) . A new feedback method for dynamic control of manipulators, Trans . ASME J. on Dynamic Systems, Measurement, and Control 102, 119-125. [2] Tomei P. (1991) . Adaptive PD controller for robot manipulators, IEEE Trans. on Robotics and Automation, 7, 565-570 . [3] Tomei P. (1991) . A simple PD controller for robots with elastic joints, IEEE Trans. on Automatic Control, 36, 1208-1213. [4] De Luca A. and S. Panzieri (1992). An iterative scheme for learning gravity compensation in flexible robot arms, Cancun Workshop on Nonlinear Systems and Robotics, Cancun, Mexico. [5] Berghuis H. and H. Nijmeijer (1993) . Global regulation of robots using only position measurements, Systems and Control Letters, 21 , 289-293.
Following the assumption that .Amin(K) > 3{3 it can be seen that iffor example .Amin (K) ~ 3{3, then for a large r, TJJ ~ t. If .Amin (K) is relatively a large number, that lower bound becomes respectively small. (iii)- Since asymptotic convergence is associated with infinite-time process, the above algorithms cannot be implemented in a straight-forward manner in practical applications. However, this problem can be overcomed using the following approach . (Due to limitation in space we present here just the outlines of the relevant approach .) Let g > 0 be a (small as desired) constant. Since from (17) and (23) the functions Vr (.) and VJ (-) depend just on measurable signals , assuming some upper bound on the system initial 'energy' is known , it can be shown formally that, based on available accumulated data it is possible to determine when the system's state vector belongs to an g - neighborhood of an (unknown) equilibrium point. Now, whatever the initial condition is , it takes finite time for the system's trajectory to enter that neighborhood. Following this observation, the algorithm can be synthesized such that in each step of the process a vector Vn+l, rather than Vn+l = T(v n ) , is determined. Moreover , it can be shown that ifin each step 11 Vn+l -vn +111 is sufficiently small then, finally the system will converge to an arbitrarily small neighborhood of the desired equilibrium point. Furthermore, if g > 0 was selected sufficiently small, then , as soon as the system's trajectory enters into an g - neighborhood of the desired equilibrium point, the control strategy can be switched to a new mode, and a different controller-observer which is based on the principle of stability in the first approximation (Vidyasagar (1993» can be initiated by,
[6] Ailon A. and R. Ortega (1993). An observerbased set-point controller for robot manipulators with flexible joints, Systems and Control Letters, 21 , 329-335. [7] Kelly R., R. Ortega, A. Ailon, and A. Loria (1994). Global regulation of Flexible joints robots using approximate differentiation, IEEE Trans . on Automatic Control, 39, 1222-1224. [8] Spong M. and M. Vidyasagar (1989). Robot Dynamics and Control, John Wiley & Sons, New York . [9] Vidyasagr M. (1993). Nonlinear Systems Analysis, Prentice-Hall, New York .
340