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Control and Stability of a Torsional Elastic Robot Arm
Journal of Mathematical Analysis and Applications 243, 140᎐162 Ž2000. doi:10.1006rjmaa.1999.6666, available online at http:rrwww.idealibrary.com on
Control and Stability of a Torsional Elastic Robot Arm Xuezhang Hou and Sze-Kai Tsui Department of Mathematics and Statistics, Oakland Uni¨ ersity, Rochester, Michigan 48309-4485 E-mail: [email protected] Submitted by Joseph A. Ball Received October 18, 1999
A mathematical model for a long elastic torsional robot beam is determined. The above system can be expressed as an evolution equation on a Hilbert space. Furthermore the system is shown to be stable and controllable, and we design a controller for the system. 䊚 2000 Academic Press Key Words: Euler᎐Bernoulli model; torsional vibration; internal viscous damping; nonlinear control systems; controllability; stability; differential operators; compact resolvents; self-adjoint operators; spectrum of a closed operator.
1. INTRODUCTION With the rapid development of robotics in engineering, the coupled bending and torsional vibrations of elastomers appear frequently in application. In this case, the motion of the system is governed by some coupled ordinary differential equations and partial differential equations. In this paper, we shall discuss the control problem for a long and thin flexible robot arm. We first describe the flexible robot system as an evolution equation in an appropriate Hilbert space, and then apply functional analysis, spectral theory of linear operators and semigroup theory of linear operators to investigate stability. Finally, we design a controller so that the considered system is exponentially stable under this control, and the tip of the arm of the robot can reach any designated point. Related works on the control of beams can be found in w2, 3, 13, 15x. The mathematical model in our paper is taken from w13x, but our approach is different from theirs. We are concerned with more theoretical issues such as controllability and stability. This paper is a continuation of our earlier work w15x. The traditional approaches to prove the existence of solution of the original 140 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
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systems with nonlinear boundary conditions wsee Eqs. Ž2.1. ᎐ Ž2.7.x will run into problems with the turning angle being a function of time, t, and the transverse displacement, y Ž t, x .. The issue for us is to determine the controllability and the stability of the robot. Presently, we are working on a simulation of such a closed loop feedback control.
2. THE MODEL OF A BEAM WITH A TIP BODY Consider a long and thin flexible beam which is rotated by a motor in a horizontal plane. The beam is clamped on a vertical shaft of the motor at one end and has a tip body rigidly attached at the free end as shown in Fig. 1. The beam is of length l and with a uniform mass density per unit length, uniform flexural rigidity EI, and uniform torsional rigidity GJ. Let X 0 , Y0 , Z0 be the inertial Cartesian coordinate axes, where X 0 , Y0 axes span a horizontal plane, and Z0 axis is the axle of rotation of the motor. Let X 1 , Y1 , Z1 with Z1 s Z0 denote coordinate axes rotating with the motor and Ž t . be the angle of rotation of the motor. Let Q be the mass center of the rigid tip body, and P be the intersection of the beam tip’s tangent with a perpendicular plan passing through the Q. Let C denote the distance between the beam’s tip point and P, and C is assumed to be small. It is also assumed that P and Q never coincide and lie on the same vertical line in the equilibrium state. Let e be the distance between P and Q. We take another coordinate axes, X 2 , Y2 , Z2 attached to the tip body, where X 2 is the beam’s tip tangent and is obtained by rotation X 1 axis by 1 due to the bending of the beam. During the motion the tip body oscillates about a shear-center axis PX 2 like a pendulum. Let ⌽ be the
FIG. 1. Bending and torsion of a flexible beam with a tip body.
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HOU AND TSUI
angle of rotation of the tip body about PX 2 . The axes Y2 , Z2 also oscillates together with the tip body. Since the tip body is a rigid body, it is characterized by mass m, and two moments of inertia J 0 and JE , where J 0 is 0 with respect to the line passing through Q and parallel to the axis PZ2 and JE is with respect to the line passing through Q and parallel to the axis PX 2 . Now let y Ž t, x . and Ž t, x . be the transverse displacement of the beam in the rotating frame X 1 , Y1 and the angle of twist of the beam, respectively, at position x, 0 - x - l, and at time t. For the transverse vibration we use the Euler᎐Bernoulli model with internal viscous damping of the Voigt type w13x
¡⭸ y Ž t , x . EI ⭸ y Ž t , x . EI ⭸ ~ ⭸ t q 2 ␦ ⭸ t⭸ x q ¢y Ž t , 0. s y⬘Ž t , 0. s 0, 2
5
2
4
4
yŽ t, x.
⭸ x4
s yx ¨Ž t . ,
Ž 2.1.
where ␦ ) 0 is a small damping constant of the beam material. The initial conditions are due to the fact that the beam is clamped at x s 0. We assume that the beam material is isotropic and the internal damping constant for the torsional vibration is equal to that of the transverse vibration. Therefore, the torsional vibration is governed by
¡⭸ Ž t , x . GJ ⭸ Ž t , x . GJ ⭸ Ž t , x . ~ ⭸ t y 2 ␦ k ⭈ ⭸ t⭸ x y k ⭈ ⭸ x s 0, ¢ Ž t , 0. s 0, 2
3
2
2
2
2
2
2
Ž 2.2.
where k 2 is the polar momentum of inertia mass for per length of beam. Obviously Ž t, l . s ⌽ Ž t ., y x Ž t, l . s 1Ž t .. Neglecting some nonlinear small quantities, we obtain the total kinetic energy of end body by T s 12 JE ˙ Ž t , l .
2
q 12 J 0 ˙Ž t . q ˙ y⬘ Ž t , l .
2
2
q 12 m Ž l q c . ˙Ž t . q ˙ y Ž t , l . q cy⬘ ˙ Ž t , l . q e ˙ Ž t , l . , where ‘‘the overdot’’ denotes the time derivative, and ‘‘prime’’ denotes the spatial derivative. We choose y Ž t, l ., y⬘Ž t, l ., Ž t, l . as the generalized coordinates, and f 1 , f 2 , and f 3 as the corresponding generalized forces
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
143
defined by f 1 s EIy Ž t , l . q 2 ␦ EIy ˙ Ž t, l. , f 2 s yEIy⬙ Ž t , l . y 2 ␦ EIy⬙ ˙ Ž t, l. , f 3 s yGI ⬘ Ž t , l . y 2 ␦ GI ˙⬘Ž t , l . . From the second class Lagrange’s equation we have d
⭸T
ž /
dt ⭸ q˙i
y
⭸T ⭸ qi
s fi
Ž i s 1, 2, 3 . ,
where q1 s y Ž t, l ., q2 s y⬘Ž t, l ., q3 s Ž t, l .. We can derive the following boundary equations of coupled bending and torsional vibrations of flexible beam as follows. m Ž l q c. Ž t . q ¨ y Ž t , l . q cy⬘ ¨ Ž t , l . q e ¨ Ž t , l . s EIy Ž t , l . q 2 ␦ EIy ˙ Ž t, l. ,
Ž 2.3.
mc Ž l q c . ¨Ž t . q ¨ y Ž t , l . q cy⬘ ¨ Ž t , l . q e ¨ Ž t , l . q J0 ¨⬘ Ž t . q ¨y⬘ Ž t , l . s yEIy⬙ Ž t , l . y 2 ␦ EIy⬙ ˙ Ž t, l. ,
Ž 2.4.
me Ž l q c . ¨Ž t . q ¨ y Ž t , l . q cy⬘ ¨ Ž t , l . q e ¨ Ž t , l . q JE ¨ Ž t , l . s yGJ ⬘ Ž t , l . y 2 ␦ GJ ˙⬘Ž t , l . .
Ž 2.5.
The rigid turning angle Ž t . of the beam is described by
½
Jm ¨Ž t . q ˙Ž t . s c Ž t . q EIy⬙ Ž t , 0 . ,
Ž 0 . s Ž1. ,
˙Ž 0 . s Ž2. ,
Ž 2.6.
where Jm is the inertia moment of the electrical motor, is the viscousfriction coefficient, EIy⬙ Ž t, 0. is the bending moment of the flexible beam, and c Ž t . is the torque of the motor. Unlike in Eqs. Ž2.3., Ž2.4., and Ž2.5., there is no damping term for the angle of turning Ž t . at the shaft end in Ž2.6., for it is negligible in comparison with the damping at the tip body. The motion differential equations of the robot system is described by Ž2.1. ᎐ Ž2.6.. The turning angle Ž t, y Ž t .. depends on the time variable, t, and the bending moment, y⬙ Ž t, 0..
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We shall choose the space H s L2 Ž0, l . = L2 Ž0, l . = R 3 as a state space, which is a Hilbert space equipped with the inner product defined as 5
l
² u, ¨ :H s
u1 Ž x . ¨ 1 Ž x . q k 2 u 2 Ž x . ¨ 2 Ž x . dx q
H0
Ý u i¨ i , is3
where u s Ž u1 , u 2 , . . . , u 5 .T , ¨ s Ž ¨ 1 , ¨ 2 , . . . , ¨ 5 .T , u, ¨ g H , and Ž ⭈⭈⭈ .T means the transpose of Ž ⭈⭈⭈ .. Let V be a subspace of H defined by T
V s u s Ž u1 , u 2 , . . . , u 5 . N u1 Ž x . g H 2 Ž 0, l . , u 2 Ž x . g H 1 Ž 0, l . , u 3 s u1 Ž l . , u 4 s uX1 Ž l . , u 5 s u 2 Ž l . , u1 Ž 0 . s 0, uX1 Ž 0 . s 0, u 2 Ž 0 . s 0 4 , where H m Ž0, l . s f g L2 Ž0, l . s f ⬘, f ⬙, . . . , f Ž m. g L2 Ž0, l .4 is the mth degree Sobolev space, m s 1, 2. We now define the inner product on V by ² u, ¨ : V s
l
H0
uY1 Ž x . ¨ Y1 Ž x . q uX2 Ž x . ¨ X2 Ž x . dx q
5
Ý u i¨ i . is3
It is easy to see that V with the inner product ² ⭈ , ⭈ : V is a Hilbert space. Define an operator n: H ª H as 1 nu s 0 0
0 1
u,
ŽugH.,
M
where m mc Ms me
mc J 0 q mc 2
me mce
mce
JE q me 2
.
It is obvious that n and M are symmetric positive operators. Due to the positivity of the operator n, we can define another inner product as ² u, ¨ :H ⬘ s ² n u, ¨ :H s
l
H0
u1 Ž x . ¨ 1 Ž x . q k 2 u 2 Ž x . ¨ 2 Ž x . dx T
q Ž u 3 , u 4 , u5 . M Ž ¨ 3 , ¨ 4 , ¨ 5 . .
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
145
We denote the space Ž H , ² ⭈ , ⭈ :H ⬘ . by H ⬘. It is apparent that there are two constants c1 and c2 such that c1 5 u 5 H F 5 u 5 H ⬘ F c 2 5 u 5 H . Thus, H ⬘ is also a Hilbert space. Furthermore, we define the operator B: DŽ B . ª H by Bu s diag
ž
EI d 4
dx
4
,y
GJ d 2
k dx 2
2
, yEI
d3 dx
3
, EI
d dx
, GJ
d dx
/
u,
u g DŽ B. .
Here DŽ B . s u s Ž u1 , u 2 , . . . , u 5 .T N u g V, uY1 g H 2 Ž0, l ., uX2 g H 1 Ž0, l .4 is the domain of B. In the systems Ž2.1. ᎐ Ž2.6., the turn angle is related to time t and the bending vibration displacement y of the beam, i.e., s Ž t, y .. If we introduce the notation ⍀ s yŽ x, 0, mŽ l q c ., J 0 q mcŽ l q c ., meŽ l q c ..T , and then the system Ž2.1. ᎐ Ž2.5. can be described as the following second order homogeneous evolution equation: nu ¨Ž t . q 2 ␦ Bu˙Ž t . q Bu Ž t . s ⍀ ¨Ž t , y Ž t . . .
Ž 2.7.
Let A s ny1 B, DŽ A. s DŽ B .. Then Ž2.7. becomes u ¨Ž t . q 2 ␦ Au˙Ž t . q Au Ž t . s ¨Ž t , y Ž t . . ny1 Ž ⍀ . .
Ž 2.8.
The corresponding second order homogeneous evolution equation is as follows: u ¨Ž t . q 2 ␦ Au˙Ž t . q Au Ž t . s 0. Set © u s Ž uŽ1. , uŽ2. .T , uŽ1. s uŽ t ., uŽ2. s ©
du dt
Ž 2.9.
0 I x Ž . Ž . , A s w yA y2 ␦ A , D A s D A
=DŽ A., and F Ž t, © u. s Ž0, ny1 Ž ⍀ .¨Ž t, y Ž t ...T . Then Ž2.8. becomes © du Ž t.
dt
©
s A© uŽ t . q F Ž t , © u. .
Ž 2.10.
The corresponding homogeneous evolution equation is © du Ž t.
dt
s A© uŽ t . .
Ž 2.11.
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HOU AND TSUI
3. THE SPECTRAL PROPERTIES OF A AND A THEOREM 3.1. The operator A is a densely defined, self-adjoint, positi¨ e definite operator on V. Proof. Since H 2 , H 1 are dense in L2 , it follows that DŽ A. is dense in V and subsequently A is densely defined. See Theorem 1 in w13x for A being self-adjoint and positive definite operator on V. THEOREM 3.2. The in¨ erse of A exists and it is compact. Proof. See Theorem 2 in w13x. THEOREM 3.3. The spectrum Ž A. of A consists of only countable eigen¨ alues n4 with finite multiplicity, so that 0 - 1 - 2 - ⭈⭈⭈ - n ⭈⭈⭈ and n ª ⬁Ž n ª ⬁.. Proof. It follows from Theorem 3.2 that Ay1 can be diagonalized as
1y1 y1 2
0 ..
Ay1 s
.
y1 n 0
..
.
with 1 G 2 G 3 . . . and lim nª⬁ n s 0. Thus
1 2 As 0
0 ..
.
.
n
..
.
Let orthogonal unital eigenvectors of A corresponding to the eigenvalue n be n j where Ž j s 1, 2, . . . , n k ; n k is finite. such that A n j s n n j , 5 n j 5 H ⬘ s 1. It is known that k , . . . , k 4⬁ks1 form an orthonormal basis 1 nk for H ⬘.
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
147
Next, we shall discuss the spectral properties of the main operator A in the evolution equation Ž2.10.. Let’s consider a dense subspace E s DŽ A1r2 . = H ⬘ with a new inner product defined by ²© u, © ¨ :E s ² A1r2 uŽ1. , A1r2 ¨ Ž1. :H ⬘ q ² uŽ2. , ¨ Ž2. :H ⬘ ,
Ž 3.1.
where © u s Ž uŽ1. , uŽ2. .T , © ¨ s Ž ¨ Ž1. , ¨ Ž2. .T g E LEMMA 3.4. The space E with the inner product defined in Ž3.1. is a Hilbert space. Proof. Let © u n4 be a Cauchy sequence in E. For any positive number ⑀ , there exists a positive integer n 0 such that 5© un y© u m 5 - ⑀ whenever n, m G n 0 . Therefore, we have 5 u n Ž2. y u m Ž2. 5 - ⑀ , 5 A1r2 Ž u n Ž1. y u m Ž1. 5 H ⬘ for n, m G n 0 . Since H ⬘ is a Hilbert space, there exists u*, ¨ * in H ⬘ such that u n Ž2. ª u*, A1r2 u n Ž1. ª ¨ *. Note that AyŽ1 r2. exists and hence u n Ž1. ª AyŽ1 r2. ¨ *. By the closedness of A1r2 , we have AyŽ1r2. ¨ * g DŽ AŽ1r2. . and A1r2 Ž AyŽ1r2. ¨ *. s ¨ *. Thus, © u n converges to © u s Ž AyŽ1r2. ¨ *, u*.T . It is easy to see that E has an orthonormal basis consisting of the following vectors:
kn 0 0 k , ,..., , k nk k1 0
k1
½ž / ž / ž / ž /5 0
⬁
. ks 1
THEOREM 3.5. Denote the spectrum of A by Ž A ., the point spectrum of A by p Ž A ., the resol¨ ent of A by Ž A ., then we ha¨ e the following results:
Ž 1 . Ž A . s p Ž A . j y
1
½ 5 2␦
,
⬁
p Ž A . s k , k 4 ks1 ,
where
k s y␦ k q s
ª
'Ž ␦ .
2
y k ,
k
k s y␦ k y
k y␦ k y
'Ž ␦ . k
1 y␦ y '␦
2
s
2
y k
1 y2 ␦
as k ª ⬁
'Ž ␦ . k
2
y k
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HOU AND TSUI
and the eigen¨ ectors of A corresponding to k and k are, respecti¨ ely, as follows:
k j s
' q < < k
©
k j
1
©
k
2
ž / k k j
,
k j s
k j
1
©
' q < < k
2
k
ž / k k j
©
with 5 k j 5 E s 5 k j 5 E s 1. Ž2. If g Ž A ., then
Ž I y A.
y1 y1
s
y1
Ž 2 q 2 ␦ A q A . Ž q 2 ␦ A . Ž 2 q 2 ␦ A q A . y1 y1 yI q Ž 2 q 2 ␦ A q A . Ž 2 q 2 ␦ A . Ž 2 q 2 ␦ A q A .
.
Ž 3.2. ©
Proof. We first verify that k , k 4⬁ks1 ; Ž A . by checking that k j and © k j are eigenvectors of A corresponding to k and k , respectively. Since Ž . k ª ⬁ as k ª ⬁, and k ª y 21␦ as k ª ⬁; hence, y1 2 ␦ g A . Next, we show that Ž A . : k , k , y 21␦ 4⬁ks1. Suppose / k , k , y 21␦ and f Ž . s 2y1 q 2 ␦ q 1. Then, f Ž A. s 2 y1 A q 2 ␦ q I by the functional calculus. Since the extended spectrum e Ž A. of A is equal to Ž A. j ⬁4 s k 4⬁ks1 j ⬁4 . We have f Ž k . s 2y1 k q 2 ␦ q 1 / 0,
otherwise s k , or k
and lim f Ž k . s 2 ␦ q 1 / 0
kª⬁
for / y
1 2␦
.
It follows from spectral mapping theorem that 0 f f Ž e Ž A.. s Ž f Ž A.., and so 0 g Ž f Ž A... Note that
Ž 2 q 2 ␦ A q A .
y1
s Ay1 Ž 2Ay1 q 2 ␦ q I .
y1
s Ay1 f Ž A .
y1
.
Therefore Ž 2 q 2 ␦ A q A.y1 exists and it is a bounded linear operator on H⬘. If we denote the right-hand side of Ž3.2. by S, then it is easy to verify that S is the bounded linear operator on H and Ž I y A . S s IH , SŽ I y A . s I D Ž A . . Thus, Ž I y A .y1 s S, and the proof of the theorem is complete.
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CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
COROLLARY 3.6. The operator A is a closed linear operator. Proof. Since Ž I y A .y1 is a bounded linear operator on H for g Ž p A ., it follows that Ž I y A . s wŽ I y A .y1 xy1 is closed. Hence A is a closed operator.
4. EIGENVECTORS OF A FORM AN UNCONDITIONAL BASIS Ž1 r 2 . T HE©ORE M © 4.1. ©Let ␦ / y . Then eigen¨ ectors of A , k ⬁ Ž k , k ., . . . , Ž k , k .4ks1 , constitutes an unconditional basis of H ⬘ [ H ⬘. 1 1 n n
©
k
k
Proof. Since for arbitrary Ž w¨ . g H ⬘ [ H ⬘ we have ¨
žw/
s
⬁
nk
¨kj
Ý Ý ks1 js1
k j
0 q wk j kj
ž / ž / 0
⬁
s
nk
Ž1.© k j k j
ža
Ý Ý ks1 js1
©
q aŽ2. k j k j ,
/
Ž 4.1. where
' q < < , s Ž y ¨ y w . 2'Ž ␦ . y ' q < < . a s Ž ¨ yw . 2'Ž ␦ . y 2
2 k
aŽ1. kj
k kj
k
kj
2
k
2
2 k
Ž2. kj
k kj
k
kj
2
k
ª
Since Ž k j , tional basis.
k
ª
n k 4⬁ k j . js1 ks1
k
is an orthogonal set in H ⬘ [ H ⬘, it is an uncondi©
©
r2. Next, if ␦ s yŽ1 , then k s k , k j s k j . In this case the eigenveck tors do not constitute the unconditional basis of H ⬘ [ H ⬘ in general. However, we have the following result. r2. THEOREM 4.2. If ␦ s yŽ1 for some k 0 Ž for at most one k ., then the k © © 0 © © © © eigen¨ ectors of A, k 1, k 1, . . . , k n , k n 4k / k 0 j k 0 , . . . , k 0 4 together k
k
1
0 0 with Ž k ., . . . , Ž k 0 .4 , constitute the bases of H ⬘ [ H ⬘. nk
01
nk
0
0
r2. Proof. Since ␦ s yŽ1 , we see from Theorem 3.4 that k 0 s k 0 s k0 1r2 y k 0 , and the eigenvectors corresponding to k 0 and k 0 are
1
©
k0 s j
2k 0
'
q k 0
k 0
j
y1r2 k 0 k 0j
0
,
j s 1, 2, . . . , jn k . 0
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HOU AND TSUI
Then,
¨k0
k 0
j
0 k 0
ž / ž / 0
j
q wk 0
j
s
j
ž
1r2 k0 ¨ k0j
0 J bkŽ1.0 k0 j
q wk 0
j
/
0 k 0
ž / j
©
q bkŽ2.0 k 0
ž /
j
j
j
q ¨ k0
k 0
j
0 j
y1r2 k 0 k
0j
Ž j s 1, 2, . . . , jk . , 0
where bkŽ1.0 s 1r2 k 0 ¨ k 0 q wk 0 , j
j
bkŽ2.0 s ¨ k 0 j
j
'
2 k0
j
q k 0 .
For every Ž w¨ . g H ⬘ [ H ⬘ we have ⬁
¨
žw/ s Ý
nk
Ý
ks1 js1 nk0
s
Ý js1
k 0
0 q wk j kj
ž / ž / Ý Ýž ž / ¨kj
0 bkŽ1.0 k0 j
0
j
⬁
©
j
q bkŽ2.0 k 0 j
j
nk
©
©
Ž2. aŽ1. k j k j q ak j k j .
ks1 js1 k/k 0
/
5. STABILITY AND CONTROL OF THE SYSTEM In this section we shall investigate stability and control of the robot system by means of the theory of operator semigroups. We are able to prove that the real parts of the eigenvalues of A are upper bounded, and there exists a constant 1 ) 0 such that sup Re n : n g p Ž A . 4 s y 1
Ž 1 ) 0. .
Ž 5.1.
In fact, it is easy to see from Theorem 3.5 that Re k - 0, Re k - 0. Since lim k ª⬁ k s y 21␦ , lim k ª⬁ k s y⬁, it follows that lim k ª⬁ Re k s y 21␦ and lim k ª⬁ Re k s y⬁. Hence we have the following theorem. THEOREM 5.1. The operator A in Ž2.10. or Ž2.11. is the infinitesimal generator of a C0-semigroup T Ž t . on Hilbert space H ⬘ [ H ⬘, and there are constants M ) 0 and ) 0 such that 5 T Ž t .5 F Mey T Ž t G 0..
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
151
r2. Proof. Ž1. Assume ␦ / yŽ1 for any positive number k. We denote k the eigenvalues and their corresponding eigenvectors of A by n4 and © e n4 , respectively. Let s ysup Re : g Ž A .4 . We see from the discussions above and Theorem 3.5 that for any real number ) y , g Ž A .. For any © u g H ⬘ [ H ⬘, since © e n4 constitutes an unconditional basis of H ⬘ [ H ⬘ Žsee Theorem 4.2., it follows that © u s Ý⬁ns 1 a n© e n , and
Ž I y A.
⬁
y1©
us
1
Ý an y
ns1
Ž I y A.
y1
m
©
u s
s
⬁
©
en ,
n
1
en Ý an Ž y . m © ns1 n m
⬁
an
Ý ns1
1 Ž q . © m m en , Ž y n . Ž q .
Since y s sup n Re n4 - 0 and ) y , we see that <Ž q . m rŽ y n . m < F 1, and
Ž I y A.
m
u F
⬁
1
©
y1
Ž q .
m
en Ý an©
1
s
Ž q .
ns1
m
5© u5.
Thus,
Ž I y A.
y1
m
F
1
Ž q .
m
,
) y ,
m s 1, 2, 3, . . . .
By Theorem 1.5.3 in w11x, it follows that A is the infinitesimal generator of a C0-semigroup T Ž t . on H , and 5 T Ž t .5 F e t , letting M s 1. r2. Ž2. Assume ␦ s yŽ1 for some positive number k. We see from k © Theorem 4.3 that for any u g H ⬘ [ H ⬘
©
us
⬁
nk
Ý Ý ks1 js1 k/k 0
ž
©
©
nk0
Ž2. aŽ1. k j k j q ak j k j q
/
Ý js1
0 bkŽ1.0 k0 j
ž / j
©
q bkŽ2.0 k 0 . j
j
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HOU AND TSUI
Since ©
©
A k 0 s k 0 k 0 , j
0 A k0
ž /
s
j
s
s
ž
I y2 ␦ A
0 yA
k 0
j
y1r2 k 0 k 0j
'
2 k0
j s 1, 2, . . . , jk 0 ,
j
0 k 0
/ž
j
/
s
0 q y1r2 k0 k0
0 ž
k 0
j
y2 ␦ A k 0
0 s
j
k 0
j
y2 1r2 k0 k0j
0
/
j
0 © q k 0 k 0 q Ž y1r2 . . k 0 k0 j
ž / j
©
©
0
0 Hence, the space spanned by k 0 , . . . , k 0 4 j Ž k ., . . . , Ž k 01
nk
1
0
0n
.4 is an k0
invariant subspace of A and of 2 jk 0 dimensions denoted by Mk 0 . Thus we have Ž I y A .y1 Mk 0 ; Mk 0 for g pŽ A . by functional calculus. It follows from w13x that Ž A N M k . ; Ž A .. 0 We now arrange the vectors spanning Mk 0 as follows: 0 0 0 © © k 01 , k 01 , k 02 , k 02 , . . . , k 0 n
ž /
ž / ž /
k0
©
, k0 . nk
0
In this order, let ⺑s
y1r2 k0 0
'
2 k0
q k 0
k 0
0
then A N M k has the form 0
⺑ A N Mk s 0
0
0 ⺑
..
.
⺑
0
Ž there are
jk 0 ⺑’s in the diagonal . .
It is clear that A generates a C0-semigroup T1Ž t .Ž t G 0. in Mk 0 . Since Ž A N M k . ; Ž A . and Ž A N M k . s p Ž A N M k . it follows that there is a 0 0 0 t constant M1 ) 0 satisfying 5 T1Ž t .5 F M1© ey© , where© s© 1 in Ž5.1.. On the other hand, since the family k 1, k 1, . . . , k n , k n 4k / k 0 consists k k of the eigenvectors of A, the subspace Mk spanned by them is an invariant
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
153
subspace of A, and this family is just an unconditional basis of Mk . Thus, we know from the case Ž1. that A generates a C0-semigroup T2 Ž t ., t G 0 in Mk . Since g Ž A . we have
Ž I y A.
y1 © k j
s
1
©
y k
k j ,
Ž I y A.
y1 © k j
s
1
y k
©
k j ,
Ž j s 1, 2, . . . , n k ; k / k 0 . . Hence Ž I y A .y1 Mk ; Mk . Thus, we see from w13x that Ž A N M k . ; Ž A . and therefore there exists a constant M2 ) 0 satisfying 5 T2 Ž t . 5 F M2 ey t .
Ž 5.2.
Note that H s Mk 0 [ Mk , where [ expresses the orthogonal sum in Hilbert space H ⬘ [ H ⬘. Now, let T Ž t . s T1Ž t . [ T2 Ž t . Žobviously, T1Ž t . ⭈ T2 Ž t . s T2 Ž t . ⭈ T1Ž t . s 0.. We shall next prove that T Ž t . is just a semigroup generated by A. In fact, we need to verify the following Ži. ᎐ Živ.: Ži. T Ž0. s T1Ž0. [ T2 Ž0. s Im [ Im s IH ; k0 k Žii. T Ž t q s . s T1 Ž t q s . [ T2 Ž t q s . s T1 Ž t . T1 Ž s . [ T2 Ž t . T2 Ž s . s T1 Ž t . [ T2 Ž t . ⭈ T1 Ž s . [ T2 Ž s . s T Ž t . ⭈ T Ž s. Žiii.
Ž t , s G 0. ;
x g H , x s x k 0 [ x k , x k 0 g Mk 0 , x k g Mk , lim T Ž t . x s limq T1 Ž t . [ T2 Ž t .
tª0 q
tª0
Ž xk
0
[ xk .
s limq T1 Ž t . x k 0 [ T2 Ž t . x k tª0
s limq T1 Ž t . x k 0 [ lim T2 Ž t . x k tª0
s xk0 [ xk s x.
tª0
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HOU AND TSUI
Živ. For any x g DŽ A .. We have that x s x k [ x k , where x k g Mk , 0 0 0 x k g Mk , A x s AŽ xk0 [ xk . s A xk0 [ A xk s limq
T1 Ž t . x k 0 y x k 0 t
tª0
s limq
[ limq tª0
tª0
t
T1 Ž t . x k 0 [ T2 Ž t . x k y Ž x k 0 [ x k . t
tª0
s limq
T2 Ž x . x k y x k
T Ž x. y x t
.
Thus, T Ž t . is the co-semigroup on H generated by A. Taking M s max 1, M2 4 , then we have 5 T Ž t .5 F Mey t. This proves the Theorem 5.1. THEOREM 5.2. The first order homogeneous e¨ olution equation Ž2.11. has a unique solution © uŽ t .. Proof. Since A is the infinitesimal generator of a C0 semigroup T Ž t . Ž t G 0. Žsee Theorem 5.1.. According to the theory of the homogeneous evolution equation we know that the evolution equation Ž2.11. has a unique solution © uŽ t . s T Ž t .© u 0 , where © u 0 s Ž uŽ0., u ˙Ž0..T, uŽ0., u˙Ž0. are the initial values for system Ž2.1. ᎐ Ž2.5.. THEOREM 5.3. The solution uŽ t . of the second order e¨ olution equation Ž2.9. is asymptotically stable. Proof. We have known from Theorem 5.2 and Theorem 5.1 that the solution © uŽ t . of the evolution equation Ž2.11. satisfies the following inequality: 5© u Ž t . 5 E F 5© u 0 5 Mey t
Ž ) 0. .
©
Thus, lim t ª⬁ 5 uŽ t .5 E s 0. It follows from the definitions Ž3.5. and Ž3.4. that 5© u 5 2E G ² A1r2 uŽ1. , A1r2 uŽ1. :H ⬘ s ² AuŽ1. , uŽ1. :H ⬘ G C 5 uŽ1. 5 2H ⬘ . However, since uŽ1.Ž t . s uŽ t . is the solution of Ž2.9., it follows that 5 uŽ t . 5 H ⬘ F
1
'c
5© uŽ t . 5 E
and lim t ª⬁ 5 uŽ t .5 H ⬘ s 0. This implies that the solution uŽ t . of Ž2.9. is asymptotically stable.
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
155
THEOREM 5.4. Suppose for e¨ ery T ) 0, ¨ : w0, T x = L2 Ž0, l . ª L2 Ž0, l . is Lipschitz continuous Ž with constant N . in y on L2 Ž0, l ., then nonlinear e¨ olution equation Ž2.10. has a unique weak solution © u g C Žw0, T x; H .. Proof. In the evolution Ž2.10., it is clear that ©
F Ž t,© u . s Ž 0, ny1 ⍀ ¨Ž t , y . .
T
and ©
5 F Ž t ,© u . 5 E s 5 ny1 ⍀ ¨Ž t , y . 5 H ⬘ F 5 ny1 5 5 ⍀ ¨Ž t , y . 5 H ⬘ .
Ž 5.3.
It follows from the definition of ⍀ and Ž3.2. that 5 ⍀ ¨Ž t , y . 5 2H ⬘ s
l
H0 < x¨Ž t , y . <
2
dx q m m Ž l q c .
q Ž J 0 q mc 2 . J 0 q mc Ž l q c . q Ž JE q me 2 . me Ž l q c . q 2 mŽ l q c .
2
2
2
< ¨Ž t , y . < 2
< ¨Ž t , y . < 2
< ¨Ž t , y . < 2
J 0 q mc Ž l q c . mc < ¨Ž t , y . < 2
q 2 J 0 q mc Ž l q c . me Ž l q c . mce < ¨Ž t , y . < 2 q 2 me Ž l q c . m Ž l q c . me < ¨Ž t , y . < 2 .
Ž 5.4.
Note that every term on the right-hand side of Ž5.4. contains the factor < ¨Ž t, y .< and is independent of x, except for the first term. The first term becomes, for 0 F x F l, l
H0 < x¨Ž t , y . <
2
dx F
l 2 <
H0 l
¨Ž t , y . < 2 dx s l 3 < ¨Ž t , y . < 2 .
Ž 5.5.
Combining Ž5.4. and Ž5.5., we have 5 ⍀ ¨Ž t , y . 5 2H F a20 < ¨Ž t , y . < 2 F a20 N 2 5 y 5 2L2 Ž0 , l . ,
Ž 5.6.
where a02 is the sum of the coefficients from all terms in Ž5.4. and a0 ) 0. Hence it is easy to see that ©
5 F Ž t,© u . 5 E F 5 ny1 5 a0 5 ¨Ž t , y . 5 .
Ž 5.7.
By the assumption that ¨Ž t, y . is uniformly Žin t . Lipschitz continuous in y on L2 Ž0, l ., we have 5 ¨Ž t, y .5 F N 5 y 5 for some N. Note that y s u1 in u s Ž u1 , u 2 , u 3 , u 4 , u 5 . of systems Ž2.1. ᎐ Ž2.5., and 5 y 5 2 s 5 u1 5 2 F 1 5 u 5 2H ⬘.
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HOU AND TSUI
From Ž3.4. and Ž3.5. we get 5 u 5 2H ⬘ F 1c 5© u 5 E . Hence we have ©
5 F Ž t ,© u . 5 E F Ž a0 N 5 ny1 5r c . 5© u5 E .
'
Ž 5.8.
Since ¨Ž t, y . in t is © infinitely differentiable on w0, t x, it must be continuous on w0, t x, and then F Ž t, © u. in t is continuous on w0, t x. Thus, according to Theorem 1.2 we know that the nonlinear evolution equation Ž2.10. has a unique weak solution © u g C Žw0, t x, H ., and this completes the proof. In a similar manner, using Theorem 1.6 of w12x we get THEOREM 5.5. Let T ) 0, ¨: w0, T x = L2 Ž0, l . ª L2 Ž0, l . be continuously differentiable, then © u 0 s Ž uŽ0., u ˙Ž0..T g DŽ A ., and nonlinear e¨ olution equation Ž2.10. has a unique strong solution. In order to investigate the properties of the solution to Ž2.10., we denote C Žw0, q⬁.. s f : f is continuous on w0, ⬁. and 5 f 5 ⬁ s sup t G 0 < f Ž t .< - q⬁4 . It is clear that the space C Žw0, q⬁.. with norm 5 ⭈ 5 ⬁ is a Banach space. We define an operator on C Žw0, q⬁.. by Kg Ž t . s
t y Ž tys.
H0 e
g Ž s . ds,
g g C Ž 0, q⬁ . . ,
where can be found in Theorem 5.1. LEMMA 5.6. The operator K is a linear bounded operator on C Žw0, q⬁.. and 5 K 5 ⬁ F 1r . Proof. It is obvious that K is linear. Next we prove that K is bounded. In fact, for any y g C Žw0, q⬁.., t G 0 we have 5 Kg Ž t . 5 F
t y Ž tys. <
H0 e
F 5 g 5⬁
g Ž s . < ds
t y Ž tys.
H0 e
ds s 5 g 5 ⬁ Ž 1 y ey t . r F
1
5 g 5⬁ .
Therefore 5 Kg 5 ⬁ s max t G 0 < Kg Ž t .< F 1 5 g 5 ⬁ , that is, 5 K 5 ⬁ F 1r . THEOREM 5.7. Suppose ¨: w0, T x = L2 Ž0, l . ª L2 Ž0, l . is uniformly Lipshitz continuous in y on L2 Ž0, l . for any T ) 0 with a Lipschitz constant N - c ra0 M 5 ny1 5. Then the solution © uŽ t . to the nonlinear e¨ olution Ž2.10. decays exponentially so that the solution uŽ t . to the original system Ž2.1. ᎐ Ž2.5. is asymptotically stable in exponential form.
'
Proof. We have seen from Theorem 5.4 that the nonlinear evolution equation has a unique solution © uŽ t ., and the system Ž2.1. ᎐ Ž2.5. has a unique solution uŽ t .. Now, we decompose the solution uŽ t . as follows uŽ t . s Ž t . q Ž t . ,
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
157
where Ž t ., Ž t . satisfy the following equations, respectively,
½
¨Ž t . q 2 ␦ A ˙Ž t . q A Ž t . s 0 Ž 0. s u Ž 0.
and
½
¨ Ž t . q 2 ␦ A˙ Ž t . q A Ž t . s ny1 ⍀ ¨Ž t , y . Ž 0 . s 0.
In space H , the above equations become
¡d Ž t . ©
~
dt
©
s A Ž t .
¢ Ž 0. s u ©
© 0
Ž 5.9. T
s Ž u Ž 0. , u ˙Ž 0 . . ,
©
where s Ž 1 , 2 .T , 1 s , 2 s d rdt, and ©
½
d© Ž t . s A© Ž t . q F Ž t,© u. © Ž 0 . s 0,
Ž 5.10.
where © s Ž1 , 2 .T , 1 s , 2 s drdt, ©
T
F Ž t,© u . s Ž 0, ny1 ⍀ ¨Ž t , y . . . Thus, ©
©
uŽ t . s Ž t . q © Ž t. .
By the semigroup theory of linear operators, we know that ©
Ž t. s
©
t
H0 T Ž t y s . F Ž s, ©u Ž s . . ds.
It follows from Theorem 5.1 and Ž5.8. that t y Ž tys. © 5 5
5© Ž t . 5 E F Ž Ma0 N 5 ny1 5r c .
H0 e
F Ž Ma0 N 5 ny1 5r c .
H0 e
' '
t y Ž tys.
u
E
ds
5 E . ds. Ž 5.11. Ž 5 5 E q 5©
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HOU AND TSUI
The solution of Ž5.9. should satisfy Žsee Theorem 5.1 and Theorem 5.2. ©
©
5 Ž t . 5 E F Mey t 5 0 5 E
Ž t G 0. .
Ž 5.12.
Substitute Ž5.12. into Ž5.11., we have 5© Ž t . 5 E F Ž M 2 a0 N 5 ny1 5r c .
'
t ©
H0 5
q Ž Ma0 N 5 ny1 5r c .
'
5 y t 0 Ee
ds
t y Ž tys. © 5 5
H0 e
E
ds
' © q Ž Ma0 N 5 ny1 5r'c . K Ž 5© Ž t. 5 E.
s Ž M 2 a0 N 5 ny1 5r c . 5 0 5 t ⭈ ey t
and
ž I y Ž Ma
0
N 5 ny1 5r c . K 5© 5 E
'
©
F Ž M 2 a0 N 5 ny1 5r c . 5 0 5 E t ⭈ ey t .
'
Ž 5.13.
We observe Ž5.13. in space C Žw0, q⬁.., using Lemma 5.6 and obtain that 5 Ž Ma0 N 5 ny1 5r c . K 5
'
s Ž Ma0 N 5 ny1 5r c . 5 K 5 F Ma0 N 5 ny1 5r Ž c . .
'
'
But, N - c ra0 M 5 ny1 5, that is, Ma0 N 5 ny1 5r c - 1, and hence the inverse of the operator Ž I y Ž Ma0 N 5 ny1 5r c . K .y1 exists and it is bounded. Thus
'
'
'
I y Ž Ma0 N 5 ny1 5r c . K
'
y1
s
⬁
Ý Ž Ma0 N 5 ny1 5r'c .
n
K n.
ns0
Since K preserves the ordering on C Žw0, q⬁.x, so does w I y Ž Ma0 N 5 ny1 5r c . K xy1. Applying w I y Ž Ma0 N 5 ny1 5r c . K xy1 on both sides of Ž5.13., we get
'
'
5© 5 E F
⬁
Ý Ž Ma0 N 5 ny1 5r'c .
n
Kn
ns0
©
Ž M 2 a0 N 5 ny1 5r'c . 5 0 5 E t ⭈ ey t © s Ž M 2 a0 N 5 ny1 5r'c . 5 0 5 E =
=
⬁
Ý Ž Ma0 N 5 ny1 5r'c . ns0
n
K n Ž t ⭈ ey t . .
Ž 5.14.
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
159
From the known result below y t
K Ž te
t y Ž tys.
. sH e
y s
Ž se
t2
. ds s
2!
0
ey t ,
we obtain that K n Ž t ⭈ ey t . s
t nq 1
Ž n q 1. !
ey t .
Ž 5.15.
Substituting Ž5.15. into Ž5.14., we see that ©
5© 5 E F Ž M 2 a0 N 5 ny1 5r c . 5 0 5 E
'
=
⬁
n
t nq 1
Ý Ž Ma0 N 5 ny1 5r'c . Ž n q 1. ! ey t ns0 ©
s M 5 05 E
⬁
Ý Ž Ma0 N 5 ny1 5r'c .
nq 1
⭈
ns0
©
y1
s M 5 0 5 E eyŽ yM a 0 N 5 n
5 r c .t
'
t nq 1
Ž n q 1. !
ey t
.
Since N - c ra0 M 5 ny1 5, it follows that y Ma0 N 5 ny1 5r c ) 0. Let r s y Ma0 N 5 ny1 5r c , then r ) 0, and
'
'
'
©
5© 5 E F M 5 0 5 E eyr t
Ž r ) 0. .
Ž 5.16.
©
Combining Ž5.2. and Ž5.16., we obtain from © u s q© that ©
©
©
5© u 5 E F 5 5 E q 5© 5 E F M 5 0 5 E ey t q M 5 0 5 Z eyr t and ©
5© u 5 E F 2 M 5 0 5 E eyr t
Ž r ) 0, t G 0 . .
©
Note that in Ž5.9. 0 s © u 0 is the initial value of system Ž2.10., and it follows that 5© u Ž t . 5 E F 2 M 5© u 0 5 E eyr t
Ž r ) 0, t G 0 . .
Ž 5.17.
This implies that the solution © uŽ t . of the Ž2.10. is exponentially decay by the energy norm 5 ⭈ 5 E .
160
HOU AND TSUI
Recall the definition of norm, 5 ⭈ 5 E and Ž3.4., we have 5© u 5 2E s ² A1r2 uŽ1. , A1r2 uŽ1. :H ⬘ q ² uŽ2. , uŽ2. :H ⬘ s ² AuŽ1. , uŽ1. :H ⬘ q 5 uŽ2. 5 2H ⬘ G ² AuŽ1. , uŽ1. :H ⬘ G C 5 uŽ1. 5 2H ⬘ . However, uŽ1. s u is just the solution of the system Ž2.1. ᎐ Ž2.5.. Hence we have 5 uŽ t . 5 H ⬘ F 5 uŽ t . 5 H ⬘ F
2 M 5© u0 5 E
'c
1
'c
5© u5 E ,
eyr t
Ž r ) 0, t G 0 . ,
Ž 5.18.
which indicates that the solution uŽ t . of the original robot system Ž2.1. ᎐ Ž2.5. is exponentially decay. THEOREM 5.8.
If we design the controller
e Ž t . s yEIy⬙ Ž 0, t . y Ž Ž t . y 0 . ,
Ž 5.19.
where 0 g w0, 2 x, 0 F F 2r4 Jm . and Jm can be found in Ž2.6., for the system Ž2.1. ᎐ Ž2.6., then the bending ¨ ibration y Ž t, x . and torsional ¨ ibration Ž t, x . of the robot arm can be suppressed to be exponentially stable, and the elastic arm of the robot can be arri¨ ed at any designated position, that is, lim Ž t . s 0 .
tª⬁
Proof. Let e Ž t . s Ž t . y 0 . Substitute Ž5.19. into Eq. Ž2.6. to find Jm ¨e Ž t . q ˙e Ž t . q e Ž t . s 0. Solve the above ordinary differential equation with the initial condition that e Ž0. s Ž1. y 0 , ˙e Ž0. s Ž2. to obtain
e Ž t . s C1 eyp 1 t q C2 eyp 2 t ,
Ž 5.20.
where C1 s
p1 s
p 2 0 y p 2 Ž1. y Ž2.
q
p1 y p 2
'
2
y 4 Jm
2 Jm
,
C2 s
G 0,
p2 s
p1 Ž1. y p1 0 q Ž2. p1 y p 2
y
'
2
y 4 Jm
2 Jm
,
G 0.
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
161
We shall see from Ž5.20. that there exist two constants, C0 , p, such that < ¨e Ž t . < F C0 eyp t ,
< ˙e Ž t . < F C0 eyp t ,
< e Ž t . < F C0 eyp t . Ž 5.21.
Considering the first order evolution equation Ž2.10.. We have seen in Theorem 5.5 that for every © u 0 g DŽ A ., there is a unique strong solution given by ©
uŽ t . s T Ž t . u0 q
©
t
H0 T Ž t y s . F Ž s . ds,
Ž 5.22.
where A is the generator of a semigroup T Ž t .. Since 5© T Ž t .5 F Mey t Ž t G 0. © yp t Žsee Theorem 5.1., 5 F Ž t .5 F C0 K 0 e wsee Ž2.10.x, F Ž t . s Ž0, ny1 Ž ⍀ .¨.T , we can evaluate Ž5.22. in norm as 5© u Ž t . 5 E F M 5© u 0 5 ey t q
MC0 K 0
yp
Ž eyp t y ey t . .
Ž 5.23.
It follows from Ž5.23. that lim t ª⬁ 5© uŽ t .5 s 0, and uŽ t . s uŽ1.Ž t ., u ˙Ž t . s uŽ2.Ž t . decay exponentially. This implies that the robot system Ž2.1. ᎐ Ž2.6. is exponentially stable. In other words, the bending vibration y Ž t, x . and torsional vibration Ž t, x . are suppressed to be exponentially stable. Furthermore, in terms of relationship Ž t . s e Ž t . q 0 and Ž5.21., we see that lim Ž t . s 0 .
tª⬁
This means that the robot arm will eventually arrive at any designated position. The proof is complete.
REFERENCES 1. R. A. Adams, ‘‘Sobolev Space,’’ Academic Press, New York, 1975. 2. A. V. Balakrishnan, ‘‘Applied Functional Analysis,’’ 2nd ed., Springer-Verlag, New York, 1981. 3. G. Chen and J. Zhou, ‘‘Vibration and Damping in Distributed Systems,’’ Vol. 1, Chemical Rubber, Boca Raton, FL, 1993. 4. G. Chen and J. Zhou, ‘‘Vibration and Damping in Distributed Systems,’’ Vol. 2, Chemical Rubber, Boca Raton, FL, 1993. 5. R. W. Clough and J. Penzien, ‘‘Dynamics of Structures,’’ Wiley, New York, 1975. 6. C. A. Desoer, ‘‘Notes for a Second Course on Linear Systems,’’ Von Nostrand Reinhold, New York, 1970. 7. I. Erdelyi and R. Lange, ‘‘Spectral Decompositions on Banach Spaces,’’ Springer-Verlag, New York, 1977. 8. H. Goodstein, ‘‘Classical Mechanics,’’ 2nd. ed., Addison-Wesley, Reading, MA, 1980.
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9. T. Kato, ‘‘Perturbation Theory for Linear Operators,’’ Springer-Verlag, New York, 1966. 10. J. L. Lions and E. Magenes, ‘‘Non-homogeneous Boundary Value Problems and Applications 1,’’ Springer-Verlag, New York, 1972. 11. A. E. Taylor and D. C. Lay, ‘‘Introduction to Functional Analysis,’’ Wiley, New York, 1980. 12. A. Pazy, ‘‘Semigroups of Linear Operators and Applications to Partial Differential Equations,’’ Springer-Verlag, Berlin, 1983. 13. Y. Sakawa and Z. H. Lou, Modeling and control of coupled bending and torsional vibrations of flexible beams, IEEE Trans. Auto. Contr. 34 Ž1989., 970᎐977. 14. S. Timoshenko, D. H. Yong, and W. Weaver, ‘‘Vibration Problem in Engineering,’’ 4th ed., Wiley, New York, 1974. 15. S. K. Tsui and X. Z. Hou, A control theory for Cartesian flexible robot arms, J. Math. Anal. Appl. 225 Ž1998., 265᎐288.
Control and Stability of a Torsional Elastic Robot Arm Xuezhang Hou and Sze-Kai Tsui Department of Mathematics and Statistics, Oakland Uni¨ ersity, Rochester, Michigan 48309-4485 E-mail: [email protected] Submitted by Joseph A. Ball Received October 18, 1999
A mathematical model for a long elastic torsional robot beam is determined. The above system can be expressed as an evolution equation on a Hilbert space. Furthermore the system is shown to be stable and controllable, and we design a controller for the system. 䊚 2000 Academic Press Key Words: Euler᎐Bernoulli model; torsional vibration; internal viscous damping; nonlinear control systems; controllability; stability; differential operators; compact resolvents; self-adjoint operators; spectrum of a closed operator.
1. INTRODUCTION With the rapid development of robotics in engineering, the coupled bending and torsional vibrations of elastomers appear frequently in application. In this case, the motion of the system is governed by some coupled ordinary differential equations and partial differential equations. In this paper, we shall discuss the control problem for a long and thin flexible robot arm. We first describe the flexible robot system as an evolution equation in an appropriate Hilbert space, and then apply functional analysis, spectral theory of linear operators and semigroup theory of linear operators to investigate stability. Finally, we design a controller so that the considered system is exponentially stable under this control, and the tip of the arm of the robot can reach any designated point. Related works on the control of beams can be found in w2, 3, 13, 15x. The mathematical model in our paper is taken from w13x, but our approach is different from theirs. We are concerned with more theoretical issues such as controllability and stability. This paper is a continuation of our earlier work w15x. The traditional approaches to prove the existence of solution of the original 140 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
141
systems with nonlinear boundary conditions wsee Eqs. Ž2.1. ᎐ Ž2.7.x will run into problems with the turning angle being a function of time, t, and the transverse displacement, y Ž t, x .. The issue for us is to determine the controllability and the stability of the robot. Presently, we are working on a simulation of such a closed loop feedback control.
2. THE MODEL OF A BEAM WITH A TIP BODY Consider a long and thin flexible beam which is rotated by a motor in a horizontal plane. The beam is clamped on a vertical shaft of the motor at one end and has a tip body rigidly attached at the free end as shown in Fig. 1. The beam is of length l and with a uniform mass density per unit length, uniform flexural rigidity EI, and uniform torsional rigidity GJ. Let X 0 , Y0 , Z0 be the inertial Cartesian coordinate axes, where X 0 , Y0 axes span a horizontal plane, and Z0 axis is the axle of rotation of the motor. Let X 1 , Y1 , Z1 with Z1 s Z0 denote coordinate axes rotating with the motor and Ž t . be the angle of rotation of the motor. Let Q be the mass center of the rigid tip body, and P be the intersection of the beam tip’s tangent with a perpendicular plan passing through the Q. Let C denote the distance between the beam’s tip point and P, and C is assumed to be small. It is also assumed that P and Q never coincide and lie on the same vertical line in the equilibrium state. Let e be the distance between P and Q. We take another coordinate axes, X 2 , Y2 , Z2 attached to the tip body, where X 2 is the beam’s tip tangent and is obtained by rotation X 1 axis by 1 due to the bending of the beam. During the motion the tip body oscillates about a shear-center axis PX 2 like a pendulum. Let ⌽ be the
FIG. 1. Bending and torsion of a flexible beam with a tip body.
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HOU AND TSUI
angle of rotation of the tip body about PX 2 . The axes Y2 , Z2 also oscillates together with the tip body. Since the tip body is a rigid body, it is characterized by mass m, and two moments of inertia J 0 and JE , where J 0 is 0 with respect to the line passing through Q and parallel to the axis PZ2 and JE is with respect to the line passing through Q and parallel to the axis PX 2 . Now let y Ž t, x . and Ž t, x . be the transverse displacement of the beam in the rotating frame X 1 , Y1 and the angle of twist of the beam, respectively, at position x, 0 - x - l, and at time t. For the transverse vibration we use the Euler᎐Bernoulli model with internal viscous damping of the Voigt type w13x
¡⭸ y Ž t , x . EI ⭸ y Ž t , x . EI ⭸ ~ ⭸ t q 2 ␦ ⭸ t⭸ x q ¢y Ž t , 0. s y⬘Ž t , 0. s 0, 2
5
2
4
4
yŽ t, x.
⭸ x4
s yx ¨Ž t . ,
Ž 2.1.
where ␦ ) 0 is a small damping constant of the beam material. The initial conditions are due to the fact that the beam is clamped at x s 0. We assume that the beam material is isotropic and the internal damping constant for the torsional vibration is equal to that of the transverse vibration. Therefore, the torsional vibration is governed by
¡⭸ Ž t , x . GJ ⭸ Ž t , x . GJ ⭸ Ž t , x . ~ ⭸ t y 2 ␦ k ⭈ ⭸ t⭸ x y k ⭈ ⭸ x s 0, ¢ Ž t , 0. s 0, 2
3
2
2
2
2
2
2
Ž 2.2.
where k 2 is the polar momentum of inertia mass for per length of beam. Obviously Ž t, l . s ⌽ Ž t ., y x Ž t, l . s 1Ž t .. Neglecting some nonlinear small quantities, we obtain the total kinetic energy of end body by T s 12 JE ˙ Ž t , l .
2
q 12 J 0 ˙Ž t . q ˙ y⬘ Ž t , l .
2
2
q 12 m Ž l q c . ˙Ž t . q ˙ y Ž t , l . q cy⬘ ˙ Ž t , l . q e ˙ Ž t , l . , where ‘‘the overdot’’ denotes the time derivative, and ‘‘prime’’ denotes the spatial derivative. We choose y Ž t, l ., y⬘Ž t, l ., Ž t, l . as the generalized coordinates, and f 1 , f 2 , and f 3 as the corresponding generalized forces
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
143
defined by f 1 s EIy Ž t , l . q 2 ␦ EIy ˙ Ž t, l. , f 2 s yEIy⬙ Ž t , l . y 2 ␦ EIy⬙ ˙ Ž t, l. , f 3 s yGI ⬘ Ž t , l . y 2 ␦ GI ˙⬘Ž t , l . . From the second class Lagrange’s equation we have d
⭸T
ž /
dt ⭸ q˙i
y
⭸T ⭸ qi
s fi
Ž i s 1, 2, 3 . ,
where q1 s y Ž t, l ., q2 s y⬘Ž t, l ., q3 s Ž t, l .. We can derive the following boundary equations of coupled bending and torsional vibrations of flexible beam as follows. m Ž l q c. Ž t . q ¨ y Ž t , l . q cy⬘ ¨ Ž t , l . q e ¨ Ž t , l . s EIy Ž t , l . q 2 ␦ EIy ˙ Ž t, l. ,
Ž 2.3.
mc Ž l q c . ¨Ž t . q ¨ y Ž t , l . q cy⬘ ¨ Ž t , l . q e ¨ Ž t , l . q J0 ¨⬘ Ž t . q ¨y⬘ Ž t , l . s yEIy⬙ Ž t , l . y 2 ␦ EIy⬙ ˙ Ž t, l. ,
Ž 2.4.
me Ž l q c . ¨Ž t . q ¨ y Ž t , l . q cy⬘ ¨ Ž t , l . q e ¨ Ž t , l . q JE ¨ Ž t , l . s yGJ ⬘ Ž t , l . y 2 ␦ GJ ˙⬘Ž t , l . .
Ž 2.5.
The rigid turning angle Ž t . of the beam is described by
½
Jm ¨Ž t . q ˙Ž t . s c Ž t . q EIy⬙ Ž t , 0 . ,
Ž 0 . s Ž1. ,
˙Ž 0 . s Ž2. ,
Ž 2.6.
where Jm is the inertia moment of the electrical motor, is the viscousfriction coefficient, EIy⬙ Ž t, 0. is the bending moment of the flexible beam, and c Ž t . is the torque of the motor. Unlike in Eqs. Ž2.3., Ž2.4., and Ž2.5., there is no damping term for the angle of turning Ž t . at the shaft end in Ž2.6., for it is negligible in comparison with the damping at the tip body. The motion differential equations of the robot system is described by Ž2.1. ᎐ Ž2.6.. The turning angle Ž t, y Ž t .. depends on the time variable, t, and the bending moment, y⬙ Ž t, 0..
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HOU AND TSUI
We shall choose the space H s L2 Ž0, l . = L2 Ž0, l . = R 3 as a state space, which is a Hilbert space equipped with the inner product defined as 5
l
² u, ¨ :H s
u1 Ž x . ¨ 1 Ž x . q k 2 u 2 Ž x . ¨ 2 Ž x . dx q
H0
Ý u i¨ i , is3
where u s Ž u1 , u 2 , . . . , u 5 .T , ¨ s Ž ¨ 1 , ¨ 2 , . . . , ¨ 5 .T , u, ¨ g H , and Ž ⭈⭈⭈ .T means the transpose of Ž ⭈⭈⭈ .. Let V be a subspace of H defined by T
V s u s Ž u1 , u 2 , . . . , u 5 . N u1 Ž x . g H 2 Ž 0, l . , u 2 Ž x . g H 1 Ž 0, l . , u 3 s u1 Ž l . , u 4 s uX1 Ž l . , u 5 s u 2 Ž l . , u1 Ž 0 . s 0, uX1 Ž 0 . s 0, u 2 Ž 0 . s 0 4 , where H m Ž0, l . s f g L2 Ž0, l . s f ⬘, f ⬙, . . . , f Ž m. g L2 Ž0, l .4 is the mth degree Sobolev space, m s 1, 2. We now define the inner product on V by ² u, ¨ : V s
l
H0
uY1 Ž x . ¨ Y1 Ž x . q uX2 Ž x . ¨ X2 Ž x . dx q
5
Ý u i¨ i . is3
It is easy to see that V with the inner product ² ⭈ , ⭈ : V is a Hilbert space. Define an operator n: H ª H as 1 nu s 0 0
0 1
u,
ŽugH.,
M
where m mc Ms me
mc J 0 q mc 2
me mce
mce
JE q me 2
.
It is obvious that n and M are symmetric positive operators. Due to the positivity of the operator n, we can define another inner product as ² u, ¨ :H ⬘ s ² n u, ¨ :H s
l
H0
u1 Ž x . ¨ 1 Ž x . q k 2 u 2 Ž x . ¨ 2 Ž x . dx T
q Ž u 3 , u 4 , u5 . M Ž ¨ 3 , ¨ 4 , ¨ 5 . .
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
145
We denote the space Ž H , ² ⭈ , ⭈ :H ⬘ . by H ⬘. It is apparent that there are two constants c1 and c2 such that c1 5 u 5 H F 5 u 5 H ⬘ F c 2 5 u 5 H . Thus, H ⬘ is also a Hilbert space. Furthermore, we define the operator B: DŽ B . ª H by Bu s diag
ž
EI d 4
dx
4
,y
GJ d 2
k dx 2
2
, yEI
d3 dx
3
, EI
d dx
, GJ
d dx
/
u,
u g DŽ B. .
Here DŽ B . s u s Ž u1 , u 2 , . . . , u 5 .T N u g V, uY1 g H 2 Ž0, l ., uX2 g H 1 Ž0, l .4 is the domain of B. In the systems Ž2.1. ᎐ Ž2.6., the turn angle is related to time t and the bending vibration displacement y of the beam, i.e., s Ž t, y .. If we introduce the notation ⍀ s yŽ x, 0, mŽ l q c ., J 0 q mcŽ l q c ., meŽ l q c ..T , and then the system Ž2.1. ᎐ Ž2.5. can be described as the following second order homogeneous evolution equation: nu ¨Ž t . q 2 ␦ Bu˙Ž t . q Bu Ž t . s ⍀ ¨Ž t , y Ž t . . .
Ž 2.7.
Let A s ny1 B, DŽ A. s DŽ B .. Then Ž2.7. becomes u ¨Ž t . q 2 ␦ Au˙Ž t . q Au Ž t . s ¨Ž t , y Ž t . . ny1 Ž ⍀ . .
Ž 2.8.
The corresponding second order homogeneous evolution equation is as follows: u ¨Ž t . q 2 ␦ Au˙Ž t . q Au Ž t . s 0. Set © u s Ž uŽ1. , uŽ2. .T , uŽ1. s uŽ t ., uŽ2. s ©
du dt
Ž 2.9.
0 I x Ž . Ž . , A s w yA y2 ␦ A , D A s D A
=DŽ A., and F Ž t, © u. s Ž0, ny1 Ž ⍀ .¨Ž t, y Ž t ...T . Then Ž2.8. becomes © du Ž t.
dt
©
s A© uŽ t . q F Ž t , © u. .
Ž 2.10.
The corresponding homogeneous evolution equation is © du Ž t.
dt
s A© uŽ t . .
Ž 2.11.
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HOU AND TSUI
3. THE SPECTRAL PROPERTIES OF A AND A THEOREM 3.1. The operator A is a densely defined, self-adjoint, positi¨ e definite operator on V. Proof. Since H 2 , H 1 are dense in L2 , it follows that DŽ A. is dense in V and subsequently A is densely defined. See Theorem 1 in w13x for A being self-adjoint and positive definite operator on V. THEOREM 3.2. The in¨ erse of A exists and it is compact. Proof. See Theorem 2 in w13x. THEOREM 3.3. The spectrum Ž A. of A consists of only countable eigen¨ alues n4 with finite multiplicity, so that 0 - 1 - 2 - ⭈⭈⭈ - n ⭈⭈⭈ and n ª ⬁Ž n ª ⬁.. Proof. It follows from Theorem 3.2 that Ay1 can be diagonalized as
1y1 y1 2
0 ..
Ay1 s
.
y1 n 0
..
.
with 1 G 2 G 3 . . . and lim nª⬁ n s 0. Thus
1 2 As 0
0 ..
.
.
n
..
.
Let orthogonal unital eigenvectors of A corresponding to the eigenvalue n be n j where Ž j s 1, 2, . . . , n k ; n k is finite. such that A n j s n n j , 5 n j 5 H ⬘ s 1. It is known that k , . . . , k 4⬁ks1 form an orthonormal basis 1 nk for H ⬘.
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
147
Next, we shall discuss the spectral properties of the main operator A in the evolution equation Ž2.10.. Let’s consider a dense subspace E s DŽ A1r2 . = H ⬘ with a new inner product defined by ²© u, © ¨ :E s ² A1r2 uŽ1. , A1r2 ¨ Ž1. :H ⬘ q ² uŽ2. , ¨ Ž2. :H ⬘ ,
Ž 3.1.
where © u s Ž uŽ1. , uŽ2. .T , © ¨ s Ž ¨ Ž1. , ¨ Ž2. .T g E LEMMA 3.4. The space E with the inner product defined in Ž3.1. is a Hilbert space. Proof. Let © u n4 be a Cauchy sequence in E. For any positive number ⑀ , there exists a positive integer n 0 such that 5© un y© u m 5 - ⑀ whenever n, m G n 0 . Therefore, we have 5 u n Ž2. y u m Ž2. 5 - ⑀ , 5 A1r2 Ž u n Ž1. y u m Ž1. 5 H ⬘ for n, m G n 0 . Since H ⬘ is a Hilbert space, there exists u*, ¨ * in H ⬘ such that u n Ž2. ª u*, A1r2 u n Ž1. ª ¨ *. Note that AyŽ1 r2. exists and hence u n Ž1. ª AyŽ1 r2. ¨ *. By the closedness of A1r2 , we have AyŽ1r2. ¨ * g DŽ AŽ1r2. . and A1r2 Ž AyŽ1r2. ¨ *. s ¨ *. Thus, © u n converges to © u s Ž AyŽ1r2. ¨ *, u*.T . It is easy to see that E has an orthonormal basis consisting of the following vectors:
kn 0 0 k , ,..., , k nk k1 0
k1
½ž / ž / ž / ž /5 0
⬁
. ks 1
THEOREM 3.5. Denote the spectrum of A by Ž A ., the point spectrum of A by p Ž A ., the resol¨ ent of A by Ž A ., then we ha¨ e the following results:
Ž 1 . Ž A . s p Ž A . j y
1
½ 5 2␦
,
⬁
p Ž A . s k , k 4 ks1 ,
where
k s y␦ k q s
ª
'Ž ␦ .
2
y k ,
k
k s y␦ k y
k y␦ k y
'Ž ␦ . k
1 y␦ y '␦
2
s
2
y k
1 y2 ␦
as k ª ⬁
'Ž ␦ . k
2
y k
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HOU AND TSUI
and the eigen¨ ectors of A corresponding to k and k are, respecti¨ ely, as follows:
k j s
' q < < k
©
k j
1
©
k
2
ž / k k j
,
k j s
k j
1
©
' q < < k
2
k
ž / k k j
©
with 5 k j 5 E s 5 k j 5 E s 1. Ž2. If g Ž A ., then
Ž I y A.
y1 y1
s
y1
Ž 2 q 2 ␦ A q A . Ž q 2 ␦ A . Ž 2 q 2 ␦ A q A . y1 y1 yI q Ž 2 q 2 ␦ A q A . Ž 2 q 2 ␦ A . Ž 2 q 2 ␦ A q A .
.
Ž 3.2. ©
Proof. We first verify that k , k 4⬁ks1 ; Ž A . by checking that k j and © k j are eigenvectors of A corresponding to k and k , respectively. Since Ž . k ª ⬁ as k ª ⬁, and k ª y 21␦ as k ª ⬁; hence, y1 2 ␦ g A . Next, we show that Ž A . : k , k , y 21␦ 4⬁ks1. Suppose / k , k , y 21␦ and f Ž . s 2y1 q 2 ␦ q 1. Then, f Ž A. s 2 y1 A q 2 ␦ q I by the functional calculus. Since the extended spectrum e Ž A. of A is equal to Ž A. j ⬁4 s k 4⬁ks1 j ⬁4 . We have f Ž k . s 2y1 k q 2 ␦ q 1 / 0,
otherwise s k , or k
and lim f Ž k . s 2 ␦ q 1 / 0
kª⬁
for / y
1 2␦
.
It follows from spectral mapping theorem that 0 f f Ž e Ž A.. s Ž f Ž A.., and so 0 g Ž f Ž A... Note that
Ž 2 q 2 ␦ A q A .
y1
s Ay1 Ž 2Ay1 q 2 ␦ q I .
y1
s Ay1 f Ž A .
y1
.
Therefore Ž 2 q 2 ␦ A q A.y1 exists and it is a bounded linear operator on H⬘. If we denote the right-hand side of Ž3.2. by S, then it is easy to verify that S is the bounded linear operator on H and Ž I y A . S s IH , SŽ I y A . s I D Ž A . . Thus, Ž I y A .y1 s S, and the proof of the theorem is complete.
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CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
COROLLARY 3.6. The operator A is a closed linear operator. Proof. Since Ž I y A .y1 is a bounded linear operator on H for g Ž p A ., it follows that Ž I y A . s wŽ I y A .y1 xy1 is closed. Hence A is a closed operator.
4. EIGENVECTORS OF A FORM AN UNCONDITIONAL BASIS Ž1 r 2 . T HE©ORE M © 4.1. ©Let ␦ / y . Then eigen¨ ectors of A , k ⬁ Ž k , k ., . . . , Ž k , k .4ks1 , constitutes an unconditional basis of H ⬘ [ H ⬘. 1 1 n n
©
k
k
Proof. Since for arbitrary Ž w¨ . g H ⬘ [ H ⬘ we have ¨
žw/
s
⬁
nk
¨kj
Ý Ý ks1 js1
k j
0 q wk j kj
ž / ž / 0
⬁
s
nk
Ž1.© k j k j
ža
Ý Ý ks1 js1
©
q aŽ2. k j k j ,
/
Ž 4.1. where
' q < < , s Ž y ¨ y w . 2'Ž ␦ . y ' q < < . a s Ž ¨ yw . 2'Ž ␦ . y 2
2 k
aŽ1. kj
k kj
k
kj
2
k
2
2 k
Ž2. kj
k kj
k
kj
2
k
ª
Since Ž k j , tional basis.
k
ª
n k 4⬁ k j . js1 ks1
k
is an orthogonal set in H ⬘ [ H ⬘, it is an uncondi©
©
r2. Next, if ␦ s yŽ1 , then k s k , k j s k j . In this case the eigenveck tors do not constitute the unconditional basis of H ⬘ [ H ⬘ in general. However, we have the following result. r2. THEOREM 4.2. If ␦ s yŽ1 for some k 0 Ž for at most one k ., then the k © © 0 © © © © eigen¨ ectors of A, k 1, k 1, . . . , k n , k n 4k / k 0 j k 0 , . . . , k 0 4 together k
k
1
0 0 with Ž k ., . . . , Ž k 0 .4 , constitute the bases of H ⬘ [ H ⬘. nk
01
nk
0
0
r2. Proof. Since ␦ s yŽ1 , we see from Theorem 3.4 that k 0 s k 0 s k0 1r2 y k 0 , and the eigenvectors corresponding to k 0 and k 0 are
1
©
k0 s j
2k 0
'
q k 0
k 0
j
y1r2 k 0 k 0j
0
,
j s 1, 2, . . . , jn k . 0
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HOU AND TSUI
Then,
¨k0
k 0
j
0 k 0
ž / ž / 0
j
q wk 0
j
s
j
ž
1r2 k0 ¨ k0j
0 J bkŽ1.0 k0 j
q wk 0
j
/
0 k 0
ž / j
©
q bkŽ2.0 k 0
ž /
j
j
j
q ¨ k0
k 0
j
0 j
y1r2 k 0 k
0j
Ž j s 1, 2, . . . , jk . , 0
where bkŽ1.0 s 1r2 k 0 ¨ k 0 q wk 0 , j
j
bkŽ2.0 s ¨ k 0 j
j
'
2 k0
j
q k 0 .
For every Ž w¨ . g H ⬘ [ H ⬘ we have ⬁
¨
žw/ s Ý
nk
Ý
ks1 js1 nk0
s
Ý js1
k 0
0 q wk j kj
ž / ž / Ý Ýž ž / ¨kj
0 bkŽ1.0 k0 j
0
j
⬁
©
j
q bkŽ2.0 k 0 j
j
nk
©
©
Ž2. aŽ1. k j k j q ak j k j .
ks1 js1 k/k 0
/
5. STABILITY AND CONTROL OF THE SYSTEM In this section we shall investigate stability and control of the robot system by means of the theory of operator semigroups. We are able to prove that the real parts of the eigenvalues of A are upper bounded, and there exists a constant 1 ) 0 such that sup Re n : n g p Ž A . 4 s y 1
Ž 1 ) 0. .
Ž 5.1.
In fact, it is easy to see from Theorem 3.5 that Re k - 0, Re k - 0. Since lim k ª⬁ k s y 21␦ , lim k ª⬁ k s y⬁, it follows that lim k ª⬁ Re k s y 21␦ and lim k ª⬁ Re k s y⬁. Hence we have the following theorem. THEOREM 5.1. The operator A in Ž2.10. or Ž2.11. is the infinitesimal generator of a C0-semigroup T Ž t . on Hilbert space H ⬘ [ H ⬘, and there are constants M ) 0 and ) 0 such that 5 T Ž t .5 F Mey T Ž t G 0..
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
151
r2. Proof. Ž1. Assume ␦ / yŽ1 for any positive number k. We denote k the eigenvalues and their corresponding eigenvectors of A by n4 and © e n4 , respectively. Let s ysup Re : g Ž A .4 . We see from the discussions above and Theorem 3.5 that for any real number ) y , g Ž A .. For any © u g H ⬘ [ H ⬘, since © e n4 constitutes an unconditional basis of H ⬘ [ H ⬘ Žsee Theorem 4.2., it follows that © u s Ý⬁ns 1 a n© e n , and
Ž I y A.
⬁
y1©
us
1
Ý an y
ns1
Ž I y A.
y1
m
©
u s
s
⬁
©
en ,
n
1
en Ý an Ž y . m © ns1 n m
⬁
an
Ý ns1
1 Ž q . © m m en , Ž y n . Ž q .
Since y s sup n Re n4 - 0 and ) y , we see that <Ž q . m rŽ y n . m < F 1, and
Ž I y A.
m
u F
⬁
1
©
y1
Ž q .
m
en Ý an©
1
s
Ž q .
ns1
m
5© u5.
Thus,
Ž I y A.
y1
m
F
1
Ž q .
m
,
) y ,
m s 1, 2, 3, . . . .
By Theorem 1.5.3 in w11x, it follows that A is the infinitesimal generator of a C0-semigroup T Ž t . on H , and 5 T Ž t .5 F e t , letting M s 1. r2. Ž2. Assume ␦ s yŽ1 for some positive number k. We see from k © Theorem 4.3 that for any u g H ⬘ [ H ⬘
©
us
⬁
nk
Ý Ý ks1 js1 k/k 0
ž
©
©
nk0
Ž2. aŽ1. k j k j q ak j k j q
/
Ý js1
0 bkŽ1.0 k0 j
ž / j
©
q bkŽ2.0 k 0 . j
j
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HOU AND TSUI
Since ©
©
A k 0 s k 0 k 0 , j
0 A k0
ž /
s
j
s
s
ž
I y2 ␦ A
0 yA
k 0
j
y1r2 k 0 k 0j
'
2 k0
j s 1, 2, . . . , jk 0 ,
j
0 k 0
/ž
j
/
s
0 q y1r2 k0 k0
0 ž
k 0
j
y2 ␦ A k 0
0 s
j
k 0
j
y2 1r2 k0 k0j
0
/
j
0 © q k 0 k 0 q Ž y1r2 . . k 0 k0 j
ž / j
©
©
0
0 Hence, the space spanned by k 0 , . . . , k 0 4 j Ž k ., . . . , Ž k 01
nk
1
0
0n
.4 is an k0
invariant subspace of A and of 2 jk 0 dimensions denoted by Mk 0 . Thus we have Ž I y A .y1 Mk 0 ; Mk 0 for g pŽ A . by functional calculus. It follows from w13x that Ž A N M k . ; Ž A .. 0 We now arrange the vectors spanning Mk 0 as follows: 0 0 0 © © k 01 , k 01 , k 02 , k 02 , . . . , k 0 n
ž /
ž / ž /
k0
©
, k0 . nk
0
In this order, let ⺑s
y1r2 k0 0
'
2 k0
q k 0
k 0
0
then A N M k has the form 0
⺑ A N Mk s 0
0
0 ⺑
..
.
⺑
0
Ž there are
jk 0 ⺑’s in the diagonal . .
It is clear that A generates a C0-semigroup T1Ž t .Ž t G 0. in Mk 0 . Since Ž A N M k . ; Ž A . and Ž A N M k . s p Ž A N M k . it follows that there is a 0 0 0 t constant M1 ) 0 satisfying 5 T1Ž t .5 F M1© ey© , where© s© 1 in Ž5.1.. On the other hand, since the family k 1, k 1, . . . , k n , k n 4k / k 0 consists k k of the eigenvectors of A, the subspace Mk spanned by them is an invariant
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
153
subspace of A, and this family is just an unconditional basis of Mk . Thus, we know from the case Ž1. that A generates a C0-semigroup T2 Ž t ., t G 0 in Mk . Since g Ž A . we have
Ž I y A.
y1 © k j
s
1
©
y k
k j ,
Ž I y A.
y1 © k j
s
1
y k
©
k j ,
Ž j s 1, 2, . . . , n k ; k / k 0 . . Hence Ž I y A .y1 Mk ; Mk . Thus, we see from w13x that Ž A N M k . ; Ž A . and therefore there exists a constant M2 ) 0 satisfying 5 T2 Ž t . 5 F M2 ey t .
Ž 5.2.
Note that H s Mk 0 [ Mk , where [ expresses the orthogonal sum in Hilbert space H ⬘ [ H ⬘. Now, let T Ž t . s T1Ž t . [ T2 Ž t . Žobviously, T1Ž t . ⭈ T2 Ž t . s T2 Ž t . ⭈ T1Ž t . s 0.. We shall next prove that T Ž t . is just a semigroup generated by A. In fact, we need to verify the following Ži. ᎐ Živ.: Ži. T Ž0. s T1Ž0. [ T2 Ž0. s Im [ Im s IH ; k0 k Žii. T Ž t q s . s T1 Ž t q s . [ T2 Ž t q s . s T1 Ž t . T1 Ž s . [ T2 Ž t . T2 Ž s . s T1 Ž t . [ T2 Ž t . ⭈ T1 Ž s . [ T2 Ž s . s T Ž t . ⭈ T Ž s. Žiii.
Ž t , s G 0. ;
x g H , x s x k 0 [ x k , x k 0 g Mk 0 , x k g Mk , lim T Ž t . x s limq T1 Ž t . [ T2 Ž t .
tª0 q
tª0
Ž xk
0
[ xk .
s limq T1 Ž t . x k 0 [ T2 Ž t . x k tª0
s limq T1 Ž t . x k 0 [ lim T2 Ž t . x k tª0
s xk0 [ xk s x.
tª0
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HOU AND TSUI
Živ. For any x g DŽ A .. We have that x s x k [ x k , where x k g Mk , 0 0 0 x k g Mk , A x s AŽ xk0 [ xk . s A xk0 [ A xk s limq
T1 Ž t . x k 0 y x k 0 t
tª0
s limq
[ limq tª0
tª0
t
T1 Ž t . x k 0 [ T2 Ž t . x k y Ž x k 0 [ x k . t
tª0
s limq
T2 Ž x . x k y x k
T Ž x. y x t
.
Thus, T Ž t . is the co-semigroup on H generated by A. Taking M s max 1, M2 4 , then we have 5 T Ž t .5 F Mey t. This proves the Theorem 5.1. THEOREM 5.2. The first order homogeneous e¨ olution equation Ž2.11. has a unique solution © uŽ t .. Proof. Since A is the infinitesimal generator of a C0 semigroup T Ž t . Ž t G 0. Žsee Theorem 5.1.. According to the theory of the homogeneous evolution equation we know that the evolution equation Ž2.11. has a unique solution © uŽ t . s T Ž t .© u 0 , where © u 0 s Ž uŽ0., u ˙Ž0..T, uŽ0., u˙Ž0. are the initial values for system Ž2.1. ᎐ Ž2.5.. THEOREM 5.3. The solution uŽ t . of the second order e¨ olution equation Ž2.9. is asymptotically stable. Proof. We have known from Theorem 5.2 and Theorem 5.1 that the solution © uŽ t . of the evolution equation Ž2.11. satisfies the following inequality: 5© u Ž t . 5 E F 5© u 0 5 Mey t
Ž ) 0. .
©
Thus, lim t ª⬁ 5 uŽ t .5 E s 0. It follows from the definitions Ž3.5. and Ž3.4. that 5© u 5 2E G ² A1r2 uŽ1. , A1r2 uŽ1. :H ⬘ s ² AuŽ1. , uŽ1. :H ⬘ G C 5 uŽ1. 5 2H ⬘ . However, since uŽ1.Ž t . s uŽ t . is the solution of Ž2.9., it follows that 5 uŽ t . 5 H ⬘ F
1
'c
5© uŽ t . 5 E
and lim t ª⬁ 5 uŽ t .5 H ⬘ s 0. This implies that the solution uŽ t . of Ž2.9. is asymptotically stable.
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
155
THEOREM 5.4. Suppose for e¨ ery T ) 0, ¨ : w0, T x = L2 Ž0, l . ª L2 Ž0, l . is Lipschitz continuous Ž with constant N . in y on L2 Ž0, l ., then nonlinear e¨ olution equation Ž2.10. has a unique weak solution © u g C Žw0, T x; H .. Proof. In the evolution Ž2.10., it is clear that ©
F Ž t,© u . s Ž 0, ny1 ⍀ ¨Ž t , y . .
T
and ©
5 F Ž t ,© u . 5 E s 5 ny1 ⍀ ¨Ž t , y . 5 H ⬘ F 5 ny1 5 5 ⍀ ¨Ž t , y . 5 H ⬘ .
Ž 5.3.
It follows from the definition of ⍀ and Ž3.2. that 5 ⍀ ¨Ž t , y . 5 2H ⬘ s
l
H0 < x¨Ž t , y . <
2
dx q m m Ž l q c .
q Ž J 0 q mc 2 . J 0 q mc Ž l q c . q Ž JE q me 2 . me Ž l q c . q 2 mŽ l q c .
2
2
2
< ¨Ž t , y . < 2
< ¨Ž t , y . < 2
< ¨Ž t , y . < 2
J 0 q mc Ž l q c . mc < ¨Ž t , y . < 2
q 2 J 0 q mc Ž l q c . me Ž l q c . mce < ¨Ž t , y . < 2 q 2 me Ž l q c . m Ž l q c . me < ¨Ž t , y . < 2 .
Ž 5.4.
Note that every term on the right-hand side of Ž5.4. contains the factor < ¨Ž t, y .< and is independent of x, except for the first term. The first term becomes, for 0 F x F l, l
H0 < x¨Ž t , y . <
2
dx F
l 2 <
H0 l
¨Ž t , y . < 2 dx s l 3 < ¨Ž t , y . < 2 .
Ž 5.5.
Combining Ž5.4. and Ž5.5., we have 5 ⍀ ¨Ž t , y . 5 2H F a20 < ¨Ž t , y . < 2 F a20 N 2 5 y 5 2L2 Ž0 , l . ,
Ž 5.6.
where a02 is the sum of the coefficients from all terms in Ž5.4. and a0 ) 0. Hence it is easy to see that ©
5 F Ž t,© u . 5 E F 5 ny1 5 a0 5 ¨Ž t , y . 5 .
Ž 5.7.
By the assumption that ¨Ž t, y . is uniformly Žin t . Lipschitz continuous in y on L2 Ž0, l ., we have 5 ¨Ž t, y .5 F N 5 y 5 for some N. Note that y s u1 in u s Ž u1 , u 2 , u 3 , u 4 , u 5 . of systems Ž2.1. ᎐ Ž2.5., and 5 y 5 2 s 5 u1 5 2 F 1 5 u 5 2H ⬘.
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HOU AND TSUI
From Ž3.4. and Ž3.5. we get 5 u 5 2H ⬘ F 1c 5© u 5 E . Hence we have ©
5 F Ž t ,© u . 5 E F Ž a0 N 5 ny1 5r c . 5© u5 E .
'
Ž 5.8.
Since ¨Ž t, y . in t is © infinitely differentiable on w0, t x, it must be continuous on w0, t x, and then F Ž t, © u. in t is continuous on w0, t x. Thus, according to Theorem 1.2 we know that the nonlinear evolution equation Ž2.10. has a unique weak solution © u g C Žw0, t x, H ., and this completes the proof. In a similar manner, using Theorem 1.6 of w12x we get THEOREM 5.5. Let T ) 0, ¨: w0, T x = L2 Ž0, l . ª L2 Ž0, l . be continuously differentiable, then © u 0 s Ž uŽ0., u ˙Ž0..T g DŽ A ., and nonlinear e¨ olution equation Ž2.10. has a unique strong solution. In order to investigate the properties of the solution to Ž2.10., we denote C Žw0, q⬁.. s f : f is continuous on w0, ⬁. and 5 f 5 ⬁ s sup t G 0 < f Ž t .< - q⬁4 . It is clear that the space C Žw0, q⬁.. with norm 5 ⭈ 5 ⬁ is a Banach space. We define an operator on C Žw0, q⬁.. by Kg Ž t . s
t y Ž tys.
H0 e
g Ž s . ds,
g g C Ž 0, q⬁ . . ,
where can be found in Theorem 5.1. LEMMA 5.6. The operator K is a linear bounded operator on C Žw0, q⬁.. and 5 K 5 ⬁ F 1r . Proof. It is obvious that K is linear. Next we prove that K is bounded. In fact, for any y g C Žw0, q⬁.., t G 0 we have 5 Kg Ž t . 5 F
t y Ž tys. <
H0 e
F 5 g 5⬁
g Ž s . < ds
t y Ž tys.
H0 e
ds s 5 g 5 ⬁ Ž 1 y ey t . r F
1
5 g 5⬁ .
Therefore 5 Kg 5 ⬁ s max t G 0 < Kg Ž t .< F 1 5 g 5 ⬁ , that is, 5 K 5 ⬁ F 1r . THEOREM 5.7. Suppose ¨: w0, T x = L2 Ž0, l . ª L2 Ž0, l . is uniformly Lipshitz continuous in y on L2 Ž0, l . for any T ) 0 with a Lipschitz constant N - c ra0 M 5 ny1 5. Then the solution © uŽ t . to the nonlinear e¨ olution Ž2.10. decays exponentially so that the solution uŽ t . to the original system Ž2.1. ᎐ Ž2.5. is asymptotically stable in exponential form.
'
Proof. We have seen from Theorem 5.4 that the nonlinear evolution equation has a unique solution © uŽ t ., and the system Ž2.1. ᎐ Ž2.5. has a unique solution uŽ t .. Now, we decompose the solution uŽ t . as follows uŽ t . s Ž t . q Ž t . ,
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
157
where Ž t ., Ž t . satisfy the following equations, respectively,
½
¨Ž t . q 2 ␦ A ˙Ž t . q A Ž t . s 0 Ž 0. s u Ž 0.
and
½
¨ Ž t . q 2 ␦ A˙ Ž t . q A Ž t . s ny1 ⍀ ¨Ž t , y . Ž 0 . s 0.
In space H , the above equations become
¡d Ž t . ©
~
dt
©
s A Ž t .
¢ Ž 0. s u ©
© 0
Ž 5.9. T
s Ž u Ž 0. , u ˙Ž 0 . . ,
©
where s Ž 1 , 2 .T , 1 s , 2 s d rdt, and ©
½
d© Ž t . s A© Ž t . q F Ž t,© u. © Ž 0 . s 0,
Ž 5.10.
where © s Ž1 , 2 .T , 1 s , 2 s drdt, ©
T
F Ž t,© u . s Ž 0, ny1 ⍀ ¨Ž t , y . . . Thus, ©
©
uŽ t . s Ž t . q © Ž t. .
By the semigroup theory of linear operators, we know that ©
Ž t. s
©
t
H0 T Ž t y s . F Ž s, ©u Ž s . . ds.
It follows from Theorem 5.1 and Ž5.8. that t y Ž tys. © 5 5
5© Ž t . 5 E F Ž Ma0 N 5 ny1 5r c .
H0 e
F Ž Ma0 N 5 ny1 5r c .
H0 e
' '
t y Ž tys.
u
E
ds
5 E . ds. Ž 5.11. Ž 5 5 E q 5©
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HOU AND TSUI
The solution of Ž5.9. should satisfy Žsee Theorem 5.1 and Theorem 5.2. ©
©
5 Ž t . 5 E F Mey t 5 0 5 E
Ž t G 0. .
Ž 5.12.
Substitute Ž5.12. into Ž5.11., we have 5© Ž t . 5 E F Ž M 2 a0 N 5 ny1 5r c .
'
t ©
H0 5
q Ž Ma0 N 5 ny1 5r c .
'
5 y t 0 Ee
ds
t y Ž tys. © 5 5
H0 e
E
ds
' © q Ž Ma0 N 5 ny1 5r'c . K Ž 5© Ž t. 5 E.
s Ž M 2 a0 N 5 ny1 5r c . 5 0 5 t ⭈ ey t
and
ž I y Ž Ma
0
N 5 ny1 5r c . K 5© 5 E
'
©
F Ž M 2 a0 N 5 ny1 5r c . 5 0 5 E t ⭈ ey t .
'
Ž 5.13.
We observe Ž5.13. in space C Žw0, q⬁.., using Lemma 5.6 and obtain that 5 Ž Ma0 N 5 ny1 5r c . K 5
'
s Ž Ma0 N 5 ny1 5r c . 5 K 5 F Ma0 N 5 ny1 5r Ž c . .
'
'
But, N - c ra0 M 5 ny1 5, that is, Ma0 N 5 ny1 5r c - 1, and hence the inverse of the operator Ž I y Ž Ma0 N 5 ny1 5r c . K .y1 exists and it is bounded. Thus
'
'
'
I y Ž Ma0 N 5 ny1 5r c . K
'
y1
s
⬁
Ý Ž Ma0 N 5 ny1 5r'c .
n
K n.
ns0
Since K preserves the ordering on C Žw0, q⬁.x, so does w I y Ž Ma0 N 5 ny1 5r c . K xy1. Applying w I y Ž Ma0 N 5 ny1 5r c . K xy1 on both sides of Ž5.13., we get
'
'
5© 5 E F
⬁
Ý Ž Ma0 N 5 ny1 5r'c .
n
Kn
ns0
©
Ž M 2 a0 N 5 ny1 5r'c . 5 0 5 E t ⭈ ey t © s Ž M 2 a0 N 5 ny1 5r'c . 5 0 5 E =
=
⬁
Ý Ž Ma0 N 5 ny1 5r'c . ns0
n
K n Ž t ⭈ ey t . .
Ž 5.14.
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
159
From the known result below y t
K Ž te
t y Ž tys.
. sH e
y s
Ž se
t2
. ds s
2!
0
ey t ,
we obtain that K n Ž t ⭈ ey t . s
t nq 1
Ž n q 1. !
ey t .
Ž 5.15.
Substituting Ž5.15. into Ž5.14., we see that ©
5© 5 E F Ž M 2 a0 N 5 ny1 5r c . 5 0 5 E
'
=
⬁
n
t nq 1
Ý Ž Ma0 N 5 ny1 5r'c . Ž n q 1. ! ey t ns0 ©
s M 5 05 E
⬁
Ý Ž Ma0 N 5 ny1 5r'c .
nq 1
⭈
ns0
©
y1
s M 5 0 5 E eyŽ yM a 0 N 5 n
5 r c .t
'
t nq 1
Ž n q 1. !
ey t
.
Since N - c ra0 M 5 ny1 5, it follows that y Ma0 N 5 ny1 5r c ) 0. Let r s y Ma0 N 5 ny1 5r c , then r ) 0, and
'
'
'
©
5© 5 E F M 5 0 5 E eyr t
Ž r ) 0. .
Ž 5.16.
©
Combining Ž5.2. and Ž5.16., we obtain from © u s q© that ©
©
©
5© u 5 E F 5 5 E q 5© 5 E F M 5 0 5 E ey t q M 5 0 5 Z eyr t and ©
5© u 5 E F 2 M 5 0 5 E eyr t
Ž r ) 0, t G 0 . .
©
Note that in Ž5.9. 0 s © u 0 is the initial value of system Ž2.10., and it follows that 5© u Ž t . 5 E F 2 M 5© u 0 5 E eyr t
Ž r ) 0, t G 0 . .
Ž 5.17.
This implies that the solution © uŽ t . of the Ž2.10. is exponentially decay by the energy norm 5 ⭈ 5 E .
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HOU AND TSUI
Recall the definition of norm, 5 ⭈ 5 E and Ž3.4., we have 5© u 5 2E s ² A1r2 uŽ1. , A1r2 uŽ1. :H ⬘ q ² uŽ2. , uŽ2. :H ⬘ s ² AuŽ1. , uŽ1. :H ⬘ q 5 uŽ2. 5 2H ⬘ G ² AuŽ1. , uŽ1. :H ⬘ G C 5 uŽ1. 5 2H ⬘ . However, uŽ1. s u is just the solution of the system Ž2.1. ᎐ Ž2.5.. Hence we have 5 uŽ t . 5 H ⬘ F 5 uŽ t . 5 H ⬘ F
2 M 5© u0 5 E
'c
1
'c
5© u5 E ,
eyr t
Ž r ) 0, t G 0 . ,
Ž 5.18.
which indicates that the solution uŽ t . of the original robot system Ž2.1. ᎐ Ž2.5. is exponentially decay. THEOREM 5.8.
If we design the controller
e Ž t . s yEIy⬙ Ž 0, t . y Ž Ž t . y 0 . ,
Ž 5.19.
where 0 g w0, 2 x, 0 F F 2r4 Jm . and Jm can be found in Ž2.6., for the system Ž2.1. ᎐ Ž2.6., then the bending ¨ ibration y Ž t, x . and torsional ¨ ibration Ž t, x . of the robot arm can be suppressed to be exponentially stable, and the elastic arm of the robot can be arri¨ ed at any designated position, that is, lim Ž t . s 0 .
tª⬁
Proof. Let e Ž t . s Ž t . y 0 . Substitute Ž5.19. into Eq. Ž2.6. to find Jm ¨e Ž t . q ˙e Ž t . q e Ž t . s 0. Solve the above ordinary differential equation with the initial condition that e Ž0. s Ž1. y 0 , ˙e Ž0. s Ž2. to obtain
e Ž t . s C1 eyp 1 t q C2 eyp 2 t ,
Ž 5.20.
where C1 s
p1 s
p 2 0 y p 2 Ž1. y Ž2.
q
p1 y p 2
'
2
y 4 Jm
2 Jm
,
C2 s
G 0,
p2 s
p1 Ž1. y p1 0 q Ž2. p1 y p 2
y
'
2
y 4 Jm
2 Jm
,
G 0.
CONTROL AND STABILITY OF A TORSIONAL ELASTIC ROBOT ARM
161
We shall see from Ž5.20. that there exist two constants, C0 , p, such that < ¨e Ž t . < F C0 eyp t ,
< ˙e Ž t . < F C0 eyp t ,
< e Ž t . < F C0 eyp t . Ž 5.21.
Considering the first order evolution equation Ž2.10.. We have seen in Theorem 5.5 that for every © u 0 g DŽ A ., there is a unique strong solution given by ©
uŽ t . s T Ž t . u0 q
©
t
H0 T Ž t y s . F Ž s . ds,
Ž 5.22.
where A is the generator of a semigroup T Ž t .. Since 5© T Ž t .5 F Mey t Ž t G 0. © yp t Žsee Theorem 5.1., 5 F Ž t .5 F C0 K 0 e wsee Ž2.10.x, F Ž t . s Ž0, ny1 Ž ⍀ .¨.T , we can evaluate Ž5.22. in norm as 5© u Ž t . 5 E F M 5© u 0 5 ey t q
MC0 K 0
yp
Ž eyp t y ey t . .
Ž 5.23.
It follows from Ž5.23. that lim t ª⬁ 5© uŽ t .5 s 0, and uŽ t . s uŽ1.Ž t ., u ˙Ž t . s uŽ2.Ž t . decay exponentially. This implies that the robot system Ž2.1. ᎐ Ž2.6. is exponentially stable. In other words, the bending vibration y Ž t, x . and torsional vibration Ž t, x . are suppressed to be exponentially stable. Furthermore, in terms of relationship Ž t . s e Ž t . q 0 and Ž5.21., we see that lim Ž t . s 0 .
tª⬁
This means that the robot arm will eventually arrive at any designated position. The proof is complete.
REFERENCES 1. R. A. Adams, ‘‘Sobolev Space,’’ Academic Press, New York, 1975. 2. A. V. Balakrishnan, ‘‘Applied Functional Analysis,’’ 2nd ed., Springer-Verlag, New York, 1981. 3. G. Chen and J. Zhou, ‘‘Vibration and Damping in Distributed Systems,’’ Vol. 1, Chemical Rubber, Boca Raton, FL, 1993. 4. G. Chen and J. Zhou, ‘‘Vibration and Damping in Distributed Systems,’’ Vol. 2, Chemical Rubber, Boca Raton, FL, 1993. 5. R. W. Clough and J. Penzien, ‘‘Dynamics of Structures,’’ Wiley, New York, 1975. 6. C. A. Desoer, ‘‘Notes for a Second Course on Linear Systems,’’ Von Nostrand Reinhold, New York, 1970. 7. I. Erdelyi and R. Lange, ‘‘Spectral Decompositions on Banach Spaces,’’ Springer-Verlag, New York, 1977. 8. H. Goodstein, ‘‘Classical Mechanics,’’ 2nd. ed., Addison-Wesley, Reading, MA, 1980.
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9. T. Kato, ‘‘Perturbation Theory for Linear Operators,’’ Springer-Verlag, New York, 1966. 10. J. L. Lions and E. Magenes, ‘‘Non-homogeneous Boundary Value Problems and Applications 1,’’ Springer-Verlag, New York, 1972. 11. A. E. Taylor and D. C. Lay, ‘‘Introduction to Functional Analysis,’’ Wiley, New York, 1980. 12. A. Pazy, ‘‘Semigroups of Linear Operators and Applications to Partial Differential Equations,’’ Springer-Verlag, Berlin, 1983. 13. Y. Sakawa and Z. H. Lou, Modeling and control of coupled bending and torsional vibrations of flexible beams, IEEE Trans. Auto. Contr. 34 Ž1989., 970᎐977. 14. S. Timoshenko, D. H. Yong, and W. Weaver, ‘‘Vibration Problem in Engineering,’’ 4th ed., Wiley, New York, 1974. 15. S. K. Tsui and X. Z. Hou, A control theory for Cartesian flexible robot arms, J. Math. Anal. Appl. 225 Ž1998., 265᎐288.