Robust Tracking Control

Design for a 6 DOF

Parallel Manipulator
䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇 䢇

Dong Hwan KimU

Seoul National University of Technology
Department of Mechanical Design
172 Kongneung-dong, Nowon-gu
Seoul, 139-743, Korea
e-mail: dhkim@duck.snut.ac.kr

Ji-Yoon Kang
Electro Mechanics Laboratory
Samsung Advanced Institute of Technology
P.O. Box 111
Suwon, 440-600, Korea

Kyo-Il Lee
Department of Mechanical Design and
Production Engineering
Seoul National University
San 56-1, Shinlim-dong, Kwanak-gu
Seoul, 151-742, Korea

Received 25 June 1999; accepted 10 July 2000

The focus of this work is on a robust tracking control design for a 6 DOF parallel
manipulator in the presence of nonlinearity and fast Žor slowly. time-varying uncer-
tainty. Two types of controllers are presented. The controls are based on the Lyapunov
approach and guarantee a practical stability. The controls utilize the information of
link displacements and its velocities. The first control scheme uses the quadratic
Lyapunov function and other uses the geometry dependent Lyapunov function, which
excludes the inverse matrix computation on the inertia matrix. Also, the hydraulic
dynamics is considered in the control design and control performance. The control
performances of the proposed algorithms are verified by simulations and experiments.
䊚 2000 John Wiley & Sons, Inc.

* To whom all correspondence should be addressed.

Journal of Robotic Systems 17(10), 527᎐547 (2000)

䊚 2000 by John Wiley & Sons, Inc.
528 䢇
Journal of Robotic Systems—2000

I. INTRODUCTION tainty. An adaptive control scheme whose controller

gain is regulated by the adaptation law 4,5 is one of
A parallel manipulator like a Stewart platform ŽFig.
the approaches to solving this problem. The control
1. has a high force-to-weight ratio Žhigh rigidity.
mainly dedicates to the time-invariant system or
and is suitable to a highly accurate positioning
slow time-varying system. A time delay control is
system. Since a serial manipulator generally has a
proposed,6 where the control requires the accelera-
long reach and large workspace, it has a lower
tion feedback; hence it may not be appropriate in
stiffness and undesired characteristics, especially at
a high speed and heavy payload due to the flexible case the measurement noise is involved. A variable
structure, compared to a parallel manipulator. Since structure control is applied to the shape memory
the appearance of the Stewart platform, many re- alloy actuator, which is a type of parallel manipula-
searchers have paid tremendous attention to it. The tor.7
system has been applied to flight simulators, ma- As another alternative, a robust control plays a
chine tools, robot manipulators, robot end-effectors, potential role in tackling the time-varying uncertain
and torque sensors, which are mainly dedicated to system using a linear model.8 As for a serial robot,
high-precision robot tasks. Considering a control several robust controls have been reported.9,10 In
aspect, a classical PID control has been employed in this article, we propose a class of robust control
industry but it does not always guarantee a high schemes for a parallel manipulator.
performance for a parallel manipulator. Thus, a so- A robot control stems from two frameworks.
phisticated control scheme is required for an en- One is to design a control on the workspace coordi-
hanced control performance. However, the complex nates Žworkspace-control. and the other is on the
geometry, high nonlinearity, and uncertainty pre- linkspace coordinates Žlinkspace-control.. The link-
vent a genuine control algorithm from being devel- space-control is a conventional control shown in
oped compared to a serial manipulator. As for a Figure 2, and is a kind of tracking control to follow
geometric based control, the control scheme for a 2 the desired link length computed from the position
DOF manipulator,1 the tracking control for the command of the platform by an inverse kinematics.
Stewart platform system,2 and the feedback lineari- Normally, most controllers in applications are based
zation3 are applied to a parallel manipulator. These
on the linkspace coordinates,4,11 which they con-
controls may not be realistic since these rely on the
sider only an approximated manipulator model.
exact knowledge of parameters. In a real situation,
However, the controller design aiming for a high
the payload and parameters may be unknown Žun-
performance requires an almost accurate dynamic
certain. or time-varying; thus it is difficult to design
an appropriate controller counting for the uncer- model of the parallel manipulator, even if some
parameters are not fully known. On the contrary,
the workspace-control12 represents that the control
is designed based on the dynamics expressed by the
workspace coordinates Žtop platform based. as
shown in Figure 3. However, the control based on
the workspace coordinates needs information of a 6
DOF sensor to measure the displacement or velocity
of the top platform. Otherwise, it needs a forward
kinematics solution which relies on the numerical
method or observer design to estimate the 6 DOF
information.
In this article, two types of robust control
schemes based on linkspace coordinates are pro-
posed and the controls guarantee the practical sta-
bility.13 The control schemes handle the dynamics
represented both in the linkspace coordinates and in
the workspace coordinates. This can be done by
transforming the dynamics on the workspace coor-
dinates into the linkspace coordinates and by com-
Figure 1. Coordinates of Stewart platform. puting the corresponding bounding function; thus it
 Kim, Kang, and Lee: Robust Tracking Control Design 䢇
529

Figure 2. Block diagram of control based on link-space.

can manage the dynamics based on both on the at the base platform and a body-fixed frame Ž oxyz .
linkspace and the workspace. at the top platform. The 12 joints’ coordinates are
denoted as follows:
II. DYNAMIC MODEL OF A 6 DOF
PARALLEL MANIPULATOR
Pi P , i s 1, 2, . . . , 6: platform joint vector in
We represent the coordinates of a 6 DOF motion body-fixed frame
with the inertial frame and the body-fixed frame
attached to the moving platform ŽFig. 1.. The 6 DOF Bi , i s 1, 2, . . . , 6: base joint vector in inertial frame
motions are linear and angular motions. Linear mo-
tions consist of the longitudinal Žsurge., lateral If the rotational transformation matrix and the
Žsway., and vertical Žheave. motion. Angular mo- translation vector are represented by R ␣ ␤␥ and D,
tions are Euler angles whose rotational sequences respectively, the relative vector of ith joint is writ-
are x-axis, y-axis, and z-axis. Here, we denote q as ten as
the 6 DOF displacement vector of which elements
are surge Ž u., sway Ž v ., heave Ž w ., roll Ž ␣ ., pitch
Ž ␤ ., and yaw Ž␥ .. l i s R ␣ ␤␥ Pi P q D y Bi Ž2.

qs u v w ␣ ␤ ␥
T
. Ž1. Thus, we can compute the link lengths, i.e., norms
of l i , from the given positions and orientations of
the platform. This problem is called an inverse kine-
In this article, we use the following notations in matics problem of a parallel manipulator. The forward
the model of the parallel manipulator. Referring kinematics problem is the opposite of the inverse
again to Figure 1, we fixed an inertial frame Ž OXYZ . kinematics, i.e., to obtain the positions and orienta-

Figure 3. Block diagram of control based on Cartesian-space.

530 䢇
Journal of Robotic Systems—2000

tions from the given actuator lengths. The solution 2. Uniform boundedness: Given any constant r␰ )
of forward kinematic problem can be analytically 0 and any solution ␰ Ž⭈.: w t 0 , ⬁. ª R n , ␰ Ž t 0 . s
represented as that of a 16th or 40th order polyno- ␰ 0 of Ž4. with 5 ␰ 0 5 F r␰ , there exists d␰ Ž r␰ . ) 0
mial.14 However, it does not mean that it can be such that 5 ␰ Ž t .5 F d␰ Ž r␰ . for all t g w t 0 , ⬁..
solved analytically. Thus, we usually rely on nu- 3. Uniform ultimate boundedness: Given any con-
merical solutions such as the Newton᎐Rhapson stant d␰ ) d␰ and any r␰ g w 0, ⬁., there exists
method 4 in order to solve the forward kinematics
a finite time T␰ Ž d␰ , r␰ . such that 5 ␰ 0 5 F r␰
problem and to obtain the 6 DOF information of the
upper platform. Or an estimator design for forward implies 5 ␰ Ž t .5 F d␰ for all t G t 0 q T␰ Ž d␰ , r␰ ..
kinematics can be utilized.15 4. Uniform stability: Given any d␰ ) d␰ , there ex-
There are several dynamics models of a 6 DOF ists a ␦␰ Ž d␰ . ) 0 such that 5 ␰ 0 5 F ␦␰ Ž d␰ . im-
parallel robot.19 ᎐ 21 We modified the dynamic model plies 5 ␰ Ž t .5 F d␰ for all t G t 0 .
of a 6 DOF parallel manipulator to account for the
inherent uncertainty in the system as follows:
In this article, the norm is Euclidean and the
matrix norm is an induced norm. Thus, 5 ⌸ 5 2 s
M Ž q, ␴ . q¨q C Ž q, q,
˙ ␴ . q˙q G Ž q, ␴ . s J T Ž q . u Ž3. ␭ma x Ž ⌸ T ⌸ ., where ⌸ is a real matrix. ␭minŽmax.Ž ⌸ .
Here, q represents the displacement vector as shown stands for the minŽmax. eigenvalue of the desig-
in Ž1.. M Ž⭈. is the inertia matrix, C Ž⭈. is the Coriolis nated matrix ⌸.
and centrifugal force, GŽ⭈. is the gravitational force,
J Ž⭈. is the Jacobian matrix, and u g R 6 is the general-
ized actuator force at each actuator. ␴ Ž⭈. Žconstant
III. ROBUST CONTROL BASED ON LINKSPACE
or time-varying. denotes the uncertain parameter
COORDINATES (QUADRATIC LYAPUNOV
vector. The detail elements of M Ž⭈., C Ž⭈., GŽ⭈., and
FUNCTION ANALYSIS)
J Ž⭈. are given in the Appendix.
The main focus of this article is to design a A parallel root consists of a top plate and six links
controller to guarantee a high control performance Žcylinders.. The control task for the position and
in the presence of uncertainty. Here, we put an orientation of the top plate to track the desired
assumption regarding to the uncertainty. values may require the information of the top plate’s
position and orientation. This is from the fact that
Assumption 1: The uncertain parameter vector is such the control scheme usually utilizes the dynamics
that ␴ g ⌺ ; R o , where ⌺ is prescribed and compact. represented by the displacement vector q. That kind
of control scheme was defined as a workspace-con-
Next, for the stability analysis we introduce the trol as stated early. To obtain this information we
practical stability. We consider the following class need to compute the forward kinematics or to in-
of uncertain dynamical systems stall an expensive 6 DOF sensor mounting on the
platform. In case of a parallel manipulator, we may
␰˙Ž t . s f Ž ␰ Ž t . , ␴ Ž t . , t . , ␰ 0 s ␰ Ž t0 . Ž4. not be able to adopt a feasible analytic solution of
forward kinematics, which is dedicated to obtaining
where t g R is the time, ␰ Ž t . g R n is the state, the position and velocity information of the upper
␴ Ž t . g R o is the uncertainty, and f Ž ␰ Ž t ., ␴ Ž t ., t . is platform. Therefore, we should rely on a numerical
the system vector. method even if it does not guarantee an exact value
of the platform information, in which it sometimes
Definition 1: The uncertain dynamical system Ž4. is requires much computation effort. Thus, the control
practically stable iff there exists constant d␰ ) 0 such design using the upper platform information has a
that for any initial time t 0 g R and any initial state limitation to the real time application. Hence, it is
␰ 0 g R n , the following properties hold. necessary to design a control scheme based on the
information on link lengths and velocities. The con-
1. Existence and continuation of solutions: Given trol was early defined as a linkspace Žjointspace.
Ž ␰ 0 , t 0 . g R n = R, system Ž4. possesses a so- coordinates control. The control designs directly
lution ␰ Ž⭈.: w t 0 , t 1 . ª R n , ␰ Ž t 0 . s ␰ 0 , t 1 ) t 0 . stem from the dynamics constructed by the link-
Furthermore, every solution ␰ Ž⭈.: w t 0 , t 1 . ª space coordinates.4,5 However, the dynamics em-
R n can be continued over w t 0 , ⬁.. ployed in the control design uses the approximated
 Kim, Kang, and Lee: Robust Tracking Control Design 䢇
531

model. Here, considering the full dynamics, we pro- puted from the neutral position Ž q s 0. and the
pose a control scheme based on linkspace coordi- unknown term as
nates by transforming the dynamics written by the
workspace coordinates into dynamics by the link- My1
1
Ž q, ␴ . s My1
1
Ž 0, ␴ . q ⌬ My1
1
Ž q, ␴ . Ž 12 .
space coordinates. Furthermore, the proposed con-
trol is designed to be robust to the uncertainty. where ␴ g R p ; ⌺ represents the nominal value of
The new dynamic equation on the linkspace parameter vector and ⌬ My1 1
Ž⭈. represents the uncer-
starts from the following property by using the tain portion. Then, Ž11. is written as
Jacobian matrix J Ž⭈..
¨e s My1 y1 Ž
1 u q ⌬ M1 q, ␴ . u q ␾ Ž q, q,
˙ e, ˙e, ˙y d , ¨y d , ␴ .
y1 Ž
q˙s J q. ˙
y Ž5. Ž 13 .
where ˙y g R 6 is the six link velocity vector. Then,
we construct a new dynamic equation on the link- where
space coordinates:
␾ Ž q, q,
˙ e, ˙e, ˙y d , ¨y d , ␴ .
M1Ž q, ␴ . ¨ ˙ ␴ . ˙y q G1Ž q, ␴ . s u
y q C1Ž q, q, Ž6.
¨ d y My1
[ yy 1
Ž q, ␴ . C Ž q, q,
˙ ␴ .Ž ˙e q ˙y d .
where y My1
1
Ž q, ␴ . G1Ž q, ␴ . Ž 14.

M1Ž q, ␴ . s JyT Ž q . M Ž q, ␴ . Jy1 Ž q . Ž7. Also, we express Ž13. in the state space form as

˙ ␴ . s JyT Ž q . M Ž q, ␴ . J˙y1 Ž q .
C1Ž q, q,
˙ ˙y d , ¨y d , ␴ . Ž15.
˙z s Az q Bu q ⌬ Bu q ⌽ Ž e, ˙e, q, q,
q JyT Ž q . C Ž q, q,
˙ ␴ . Jy1 Ž q . Ž8.
Here,
yT
G 1Ž q, ␴ . s J Ž q . G Ž q, ␴ . Ž9.
e 0 I 0
Here, y g R 6 represents the six link displacement zs As Bs
ė 0 0 My1
1
Ž 0, ␴ .
vector. y and ˙ y can be measured directly through a
linear displacement sensor. These signals are dedi- 0 0
cated to the controller design. ⌬ B Ž q, ␴ . s ⌽ Ž⭈. s
⌬ My1
1
Ž q, ␴ . ␾ Ž⭈.
Define the tracking error e g R 6 and its deriva-
e g R 6 in the sense of actuator,
tive ˙ Ž 16 .

esyyyd We construct the uncertain functions hŽ⭈. g R 6

˙e s ˙y y ˙y d Ž 10 .
and matrix EŽ⭈.: R 6 = R p ª R 6=6 as follows:
where y d and ˙ y d represent the desired actuator
Žlink. displacement vector and its velocity vector of h Ž e, ˙ ˙ ˙y d , ¨y d . s M1Ž0, ␴ . ␾ Ž e, ˙e, q, q,
e, q, q, ˙ ˙y d , ¨y d .
each link, respectively. Then, the error dynamic Ž 17 .
equation is given as
E Ž q, ␴ . s M1Ž 0, ␴ . ⌬ My1 Ž q, ␴ . Ž 18.
M1Ž q, ␴ . ¨ ˙ ␴ . ˙e
e q C1Ž q, q,
From Ž16. it can be seen that the matching condition
s yM1Ž q, ␴ . ¨ ˙ ␴ . ˙y d y G1Ž q, ␴ . q u
y d y C1Ž q, q, is satisfied; i.e., the following conditions hold:
Ž 11 .
⌽ Ž e, ˙ ˙ ˙y d , ¨y d . s Bh Ž e, ˙e, q, q,
e, q, q, ˙ ˙y d , ¨y d . Ž19.
Here, M1Ž q, ␴ ., C1Ž q, q,
˙ ␴ ., G1Ž q, ␴ . are not to be
necessarily expressed by the actuator variable y and ⌬ B Ž q, ␴ . s BE Ž q, ␴ . Ž 20.
˙y. This is because the proposed control scheme is
Under the Assumption 1, we can choose a bounding
able to handle the platform information q and q, ˙
which will be explained later. We express the inertia function ␳ Ž⭈.: R 6 = R 6 ª Rq such that for all ␴ g ⌺,
matrix M1Ž⭈. as the sum of the nominal value de-
pendent on the known parameters, which is com- h Ž e, ˙ ˙ ˙y d , ¨y d , ␴ . F ␳ Ž e, ˙e . .
e, q, q, Ž 21 .
532 䢇
Journal of Robotic Systems—2000

Here, we can choose the function ␳ Ž⭈. such that it Now, we show the stability and boundedness of
has a dependency on e and ˙ e only. This is possible that control law through Theorem 1.
since q˙ can be transformed into ˙e by the Jacobian
J Ž q . whose norm can be bounded by a constant Theorem 1: Subject to Assumptions 1 and 2, the sys-
value, and the platform displacement q can be tem Ž15. is practically stable under the control Ž24..
bounded by a constant within the specified work-
space. Proof: See Appendix section II.
For the next process for designing a robust
control, we consider another assumption which em- Remark 1: The condition on the input uncertainty
ploys the condition imposed on the input uncer- shown in Ž22. is limited in applying to a practical
tainty. use. However, this can be relaxed by the following
condition by showing the possible range of the
Assumption 2: There exists ␳ E Ž q . for all q g R 6 such input uncertainty extends to certain range as fol-
that lows:

max E Ž q, ␴ . \ ␳ E Ž q . - 1 Ž 22 . ␭ E Ž q . [ 12 min ␭min w E Ž q, ␴ . q E T Ž q, ␴ .x ) y1

␴g⌺ ␴g⌺
Ž 28 .
This assumption which deals with how much the uncer-
tainty varies over its nominal value, and implies that the
By the ␭ E Ž q ., the new bounding function ␳ 2 Ž⭈. can
nominal value of My1 1
Ž0, ␴ . must not be far from
be modified as
My11
Ž q, ␴ .. If the ratio My11
Ž0, ␴ . to ⌬ My1
1
Ž q, ␴ . is
less than 1 then the condition holds; hence we can 1
compute the value of ␳ E Ž q . which satisfies the Assump- ␳ 2 Ž e, ˙
e. G ␳ Ž e, ˙
e. Ž 29.
tion 2. 1 q ␭E Ž q .

Next, let another bounding function ␳ Ž⭈.: R 6 = First, when 5 ␮ 5 ) ⑀ , the last term of Ž58. becomes
R ª Rq be chosen such that
6
␮T Ž p q Ep q h .
1 y␮ ␮
␳ Ž e, ˙
e. G ␳ Ž e, ˙
e. Ž 23. s ␮T ␳ q ␮T yE ␳ q5 ␮5 ␳
1 y ␳E Ž q . ž 5 ␮5 / ž 5 ␮5 /
Now, we construct a robust control u g R 6 as F y5 ␮ 5 ␳ y ␭ E 5 ␮ 5 ␳ q 5 ␮ 5Ž 1 q ␭ E . ␳
s0 Ž 30 .
e
u s yKz q p Ž e, ˙
e. s y K p Kv q p Ž e, ˙
e . Ž 24 .
ė Second, when 5 ␮ 5 F ⑀ , the last term of Ž58.
becomes from the inequality property used in Ž60.,
Here, K = R 6= 12 , whose elements are K p g R 6= 6 and
K v g R 6= 6 , is chosen such that AŽs A y BK . is Hur- ␮T Ž p q Ep q h .
witz. Thus, there exists a symmetric and positive ␮ ␮
definite matrix P satisfying F ␮T y ␳ 2 y E ␳ 2 q 5 ␮ 5 ␳
ž /
⑀ ⑀
PA q A T P s yQ, Q)0 Ž 25 . Ž1 q ␭ E .
Fy 5 ␮ 5 2␳ 2 q 5 ␮ 5 ␳
⑀
Also, as hinted in ref. 13, pŽ⭈. g R 6 can be taken as
Ž1 q ␭ E .
¡y ␮ Ž e, ė .
␳ Ž e, ˙
e. if ␮ Ž e, ˙
e. ) ⑀
Fy
⑀
5 ␮ 5 2 ␳ 2 q ␳ Ž1 q ␭ E . 5 ␮ 5

e. s
p Ž e, ˙ ~ ␮ Ž e, ė .
F
Ž1 q ␭ E . ⑀
Ž 31 .
␮ Ž e, ė .
¢y
4
␳ 2Ž
e, ˙
e. if ␮ Ž e, ˙
e. F ⑀
⑀ Thus, the system with different input uncer-
Ž 26 . tainty constraint Ž ␭ E ) y1., which is more practical
than the constraint in Ž22., also guarantees the prac-
␮ Ž e, ˙
e . s B T Pz Ž 27 . tical stability.
 Kim, Kang, and Lee: Robust Tracking Control Design 䢇
533

IV. A MODIFIED ROBUST CONTROL SCHEME The bounding function ␳ 1Ž⭈. can also be a function
(GEOMETRIC BASED LYAPUNOV FUNCTION) of e and ˙ e by the geometric boundedness as men-
tioned in section III.
Since the control proposed in section III utilizes the For the proof of the skew-symmetric property of
inverse matrix My1 Ž⭈. to compute the bounding Ṁ1 y 2C1 , we introduce the following lemmas. Here,
function ␳ Ž⭈., a problem in the real time application Lemma 1 is adopted from ref. 2, and Lemma 2 is
due to the computational burden occurs. Here, an partially adopted from ref. 16.
alternative control type is proposed instead. The
control stems from the different Lyapunov function Lemma 1: 2 Ž1. The matrix M is positive definite and
compared to that adopted in section III. The another ˙ y 2C satisfies a skew-symmetric
symmetric. Ž2. The M
assumption is imposed to design the different type property.
of robust control.
Lemma 2: Ž1. The matrix M1 is positive definite and
Assumption 3: There exist positive constants ␴ and ␴ ˙1 y 2C1 satisfies a skew-symmetric
symmetric. Ž2. The M
such that property.

␴ I F M1Ž q, ␴ . F ␴ I ᭙q g R 6 ᭙␴ g ⌺ Ž 32. Proof: Lemma 1 is proven in ref. 2; thus, we prove

Lemma 2 at this stage. Note first that M1 s JyT MJy1
Now, given S1 s diagw S1i x 6=6 , i s 1, 2, . . . , 6, S1 i is also positive definite by Lemma 1, which is obvi-
) 0, and a scalar ⑀ 1 ) 0, we propose a robust control ous. Next, for the skew-symmetric property on M ˙1
u1 g R 6 similar to Ž24.: y 2C1
d d
u1 s Ž yK p 1 e y K v 1 ˙
e q p1 . Ž 33 . ˙1 y 2C1 s
M Ž JyT MJy1 . y 2 JyT M Ž Jy1 .
dt dt
where y 2 JyT CJy1

¡y ␮ 1Ž e, ė .
␳ Ž e, ė .
s JyT Ž M
˙ y 2C . Jy1 q
d
dt
Ž JyT . MJy1
␮ 1Ž e, ė . 1
d
y JyT M Ž Jy1 .
e . g R6 s
p 1Ž e, ˙ ~
if ␮ 1Ž e, ˙ e. ) ⑀1
Ž 34 . dt
␮ 1Ž e, ė .
y ␳ 1Ž e, ė . d
⑀1 ˙ y 2C . w q
s wT Ž M Ž w T . Mw y w T Mẇ
¢ if ␮ 1Ž e, ˙
e. F ⑀1
dt
Ž 40 .

␮ 1Ž e, ˙ e . ␳ 1Ž e, ˙
e . s Ž e q S1 ˙ e . g R6 Ž 35. where w s Jy1 , For any vector x g R 6 , we can ex-
press
K p 1 [ diag w K p 1 i x 6= 6 K p1 i ) 0 i s 1, 2, . . . , 6
Ž 36 . ˙1 y 2C1 . x s x T w T Ž M˙ y 2C . wx q x T w
xT Ž M ˙ T Mwx
K v 1 [ diag w K v 1 i x 6= 6 K v1 i ) 0 i s 1, 2, . . . , 6 y x T w T Mwx
˙
Ž 37 . s xT w
˙ T Mwx y x T w T Mwx
˙ Ž 41 .
Using Lemma 1, and seeing that x T w
˙ T Mwx is a
Here, ␳ 1Ž⭈.: R 6 = R 6 ª Rq is a bounding function ˙ Mwx s x w Mwx,
T T
scalar and thus x w T T
˙ we have
computed from
˙1 y 2C1 . x s 0
xT Ž M Ž 42 .
␾ 1 Ž q, q,
˙ e, ˙e, q˙ , q¨ , ␴ . F ␳ 1Ž e, ˙e .
d d Ž 38. Hence, M1 s JyT MJy1 satisfies a skew-symmetric
property.
where
Theorem 2: Subject to Assumptions 1, 2, and 3, the
␾ 1Ž ⭈ . s yM1Ž q, ␴ . Ž ¨
y d y S1 ˙
e. system Ž15. is practically stable under the control Ž33..

˙ ␴ .Ž ˙y d y S1 e . y G1Ž q, ␴ . Ž39.
y C1Ž q, q, Proof: See Appendix section III.
534 䢇
Journal of Robotic Systems—2000

Remark 2: The proposed control needs an assump- V. MODELING OF HYDRAULIC ACTUATOR

tion which requires that M1Ž⭈. is lower and upper
bounded by constants. This condition is sometimes The hydraulic system for each DOF consists of the
restrictive to applying to a general robot as men- second stage of a two-stage servovalve and hy-
tioned early. For instance, a Scara type robot does draulic cylinder Žshown later in Fig. 5.. In Figure 5,
not satisfy this condition. On the contrary, the con- Ps and Pr represent a supply pressure and return
trol does not impose a constraint on the input pressure, respectively. More specifically, there are
uncertainty EŽ⭈.. That constraint is somehow restric- symmetric electro-hydraulic servovalves and the
tive to the case the uncertainty is not easily esti- dynamics between the input current i v and the
mated. displacement of the spool valve x v is written as

Remark 3: The control scheme does not impose the dx v i

bounding condition on the inertia matrix M1Ž⭈. Tv i q x vi s K vi i vi i s 1, 2, . . . , 6 Ž 44 .
which is usually adopted in robot control. This can dt
be done since the quadratic Lyapunov function is
chosen for the stability analysis from the partially where Tv i is a time constant of the ith servovalve
linear system representation. However, it could and K v i is a gain. The flow rate at each port in the
cause a problem for a system with a prismatic joint ith cylinder is given as18
where the bounding condition on the inertia matrix
does not hold anymore. Fortunately, for the system
taken into consideration in this article, all joints are 1
Q1 i s C d w Ž 1 q sign Ž x v i . .
revolute. 2

2
Remark 4: For time-varying uncertainty, the skew-
symmetry property on M ˙1 y 2C1 does not hold any-
more. However, the control can be manipulated by
= x v i sign Ž Ps y P1 i . ( ␳
< Ps y P1 <
i

computing a modified bounding function ␳ 2 Ž⭈., 1

which is shown later, excluding the skew-symmetry Q2 i s y C d w Ž 1 y sign Ž x v i . .
2
property. The procedure for finding the bounding
function ␳ 2 Ž⭈. is presented here. The derivative of 2
the Lyapunov function candidate V˙1 , which is ex-
pressed in Appendix section III, is modified as
= x v i sign Ž P1 i y Pr . ( ␳
< P1 y Pr <
i

Ž 45 .
1
T Q3 i s y C d w Ž 1 y sign Ž x v i . .
V˙1 s Ž ˙
e q S1 e . ž M S ˙e y M
1 1 1 ¨y d y C1 ˙y d q C1 S1 e 2

˙1 ˙e q M˙1 S1 e
yG1 q u q M 2

qe TŽ
K p 1 q S1 K v 1 . ė
/ = x v i sign Ž Ps y P2 i . ( ␳
< Ps y P2 <
i

T
1
s Ž˙
e q S1 e . ž M S ˙e y M
1 1 1 ¨y d y C1 ˙y d q C1 S1 e Q4 i s C d w Ž 1 q sign Ž x v i . .
2
˙1 ˙e q M˙1 S1 e q p 2
yG1 q M / 2

y e T S1 K p 1 e y ˙
eT K v 1 ˙
e
= x v i sign Ž P2 i y Pr . ( ␳
< P2 y Pr <
i

T
e q S1 e . Ž ␾ 2 q p 2 . y e T S1 K p 1 e y ˙
s Ž˙ eT K v 1 ˙
e Ž 43.
where Q1 i , . . . , Q4 i are flow rate in the servovalve
connected to the ith cylinder as shown in Figure 5.
where ␾ 2 [ M1 S1 ˙ e y M1 ¨
y d y C1 ˙
y d q C1 S1 e y G 1 q Then, we can get the net flow rate between the
˙
M1 ˙ ˙
e q M1 S1 e. The bounding function of ␾ 2 , which servovalve and the chamber of cylinder:
is expressed by ␳ 2 Ž⭈., can also be computed. The
␳ 2 Ž⭈. can be taken in the controller p 1Ž⭈. with the
new bounding function ␳ 2 Ž⭈.. Q A i s Q1 i q Q 2 i Q B i s Q3 i q Q4 i . Ž 46 .
 Kim, Kang, and Lee: Robust Tracking Control Design 䢇
535

The integration of the following continuity equation For a positive constant ␭, choose x v i as follows:
will give the pressure rate at each chamber:
V1 i dP1 i dx p i 1 P1 i P2 i
␤ dt
s Q A i y A1 i
dt
Ž 47 . x vi s
␤ K qi
ž 1
V1 i
y
1
V2 i
ž/ ž
2␤
V1 i
q
V2 i /
V2 i dP2 i dx p i
s yQ B i q A 2 i Ž 48 .
␤ dt dt ␤ ␤
where ␤ is the bulk modulus, A i is the ith cylinder
area, and V1 i , V2 i are each chamber volume. Then,
q
ž V1 i
A21 i q
V2 i / x p i q ␭ e2 i q ˙
A22 i ˙ ␶d i
/ Ž 52 .

the net force of each actuator ␶ i is given as

Then, it becomes ˙ e 2 i q ␭ e2 i s 0; hence e 2 i ª 0 as time
␶ i s P1 i A1 i y P2 i A 2 i s 1, 2, . . . , 6 Ž 49 . goes. Since ˙␶d i is a derivative of u and u1 , hence
u and u1 as differentiable, the former pŽ⭈. or p 1Ž⭈.,
The true control input is the servovalve current i v i
which is expressed by p 2 Ž⭈., changes to 22
rather than ␶ i , which acts as a control input in the
proceeding sections. In simulation, we include the
hydraulic dynamics to find the coherence between ¡y ␮ Ž e, ė .
␳ Ž e, ė .
simulation and experiment. Due to the dynamics ␮ Ž e, ė .
between ␶ i and i v i the proposed controls utilizing
the forces as control inputs need to be modified by e. s
p 2 Ž e, ˙ ~ if ␮ Ž e, ˙
e. ) ⑀
Ž 53 .
including the hydraulic dynamics. The control de- ␲␮ e, ė .
Ž
ysin
ž ␳ 2 Ž e, ė .
/
2⑀
¢
sign including the hydraulic dynamics is written as
follows.19 From Ž49., and Ž47᎐48. it can be seen that if ␮ Ž e, ˙
e. F ⑀
˙␶ i s P˙1 i A1 i y P˙2 i A 2 i
␤ ␤
s Ž Q A y A1 ˙x p . A1 y V Ž yQB q A 2 ˙x p . A 2 VI. SIMULATIONS AND EXPERIMENTS
V1 i i i i i
2i
i i i i

We verify the performance of the proposed con-

␤ trollers through simulations and experiments. The
s Ž K q x v y wK c P1 y A1 ˙x p . A1
V1 i k i i i i i i
motion bed consists of six hydraulic cylinders at-
tached to the top and bottom platform and six
␤ servovalves with each servo amplifier, which con-
y Ž K q x v q 2 K c P2 q A 2 q A 2 ˙x p . A 2
V2 i i i i i i i i i
trol the cylinders. The photocopy of the motion bed
is shown in Figure 4. Each cylinder is a single rod
Ž 50 .
type with a diameter of 75 mm in the high chamber
In general, an electric system is highly faster than and 61.2 mm in the low chamber, and the full stroke
mechanical system; thus the valve displacement can of each cylinder is 0.4 m. The servovalve is of a
be considered as directly proportional to the input Moog 30 series type with a 30 lpm rated flow rate
current of the servovalve. Thus, from Ž44. it can be and a time constant of 0.013 sy1 . Parameters for the
seen that x v i s K v i i v i . Let e2 i s ␶ i y ␶d i , where ␶d i simulation are given in Table I. However, these
represents an ith desired force which is presented in values are not the same as the real values but the
by uŽ⭈. and u1Ž⭈. in Ž24. and Ž33.. Differentiating e 2 i , possible bound of each parameter is known instead.
it can be seen that
A. Simulations
˙e2 i s ˙␶ i y ␶˙d i
The command signals ␣ cm d Ž ␤cmd , ␥cmd . are imposed
␤ ␤ P1 i P2 i with sinusoidal angular motions in Ž54. while the
s y K qi x vi y 2 ␤ q
ž V1 i V2 i / ž V1 i V2 i / translation motions keep zero:

␤ ␤ ␣ cm d Ž ␤cmd , ␥cmd . s ␣ max Ž ␤max , ␥max .w sin Ž 0.2)2␲ t .

y A2l i q ␶d i
x pi y ˙
ž V1 i V2 i /
A22 i ˙ Ž 51 .
q0.3 sin Ž 0.6)2␲ t .x Ž 54 .
536 䢇
Journal of Robotic Systems—2000

Figure 4. 6 DOF motion simulator for experiment.

Figure 5. Hydraulic servo system with hydrostatic sin-
gle-rod cylinder.

where ␣ ma x s 0.01, ␤max s 0.02, and ␥max s 0.03 rad.

Since the motion bed utilizes the hydraulic cylinder scheme. The robust controllers add the pŽ⭈. term in
as a linear actuator shown in Figure 5, the simula- the PD control; thus it compensates the uncertain
tion program includes the dynamic of hydraulic terms. In simulations, the controller parameters ⑀ s
system even if the proposed controllers use the 1 = 10y3 in type I and ⑀ 1 s 30 in type II are as-
hydraulic force as a control input. The fact that the signed. We see that the robust controller type I and
input voltages on the servovalves are proportional type II diminish the tracking error by 50% and 80%,
to the hydraulic forces in each cylinder is generally respectively, compared to the PD scheme. Robust
known. In the modeling of the hydraulic system as control type II has a better performance compared
shown in section IV, the flow equations and the to type I. This is because the control type I has a
continuity equations are used. We compare the con- dependency on the input uncertainty criterion which
trol performances under the PD controller, the ro- is expressed by ␳ E or ␭ E . Moreover, for the type I,
bust control scheme I of section III and another we cannot decrease ⑀ as small as possible to get less
robust control scheme II of section IV. boundedness R z due to a chattering in control. This
Figures 7᎐9 show the histories of the displace- is partly due to the sensitive bounding function
ments of 6 DOF motion tracking errors and corre- ␳ 1Ž⭈. which includes the My1 1
Ž⭈.. This issue is re-
sponding input forces when PD controller, robust laxed in the control type II by excluding the compu-
controller type I, and robust controller type II are tation of My11
Ž⭈.. Thus, between the tracking error
implemented, respectively. The controller gains K p and the chattering a compromise is needed in a real
s 1 = 10 6 and K v s 0.1 are assigned for each control implementation.

Table I. Feature parameters of 6 DOF parallel manipulator.

Variable Description Value Unit

u m , vm , wm maximum excursion of platform 0.320 m

␣ m , ␤m , ␥m maximum angle of platform 25 deg
X max maximum stroke of hydraulic cylinder "0.200 m
m mass of platform 500 kg
I x , I yŽ Iz . mass moment of inertia about x, yŽz. axis 500 Ž1000. kg ⭈ m 2
R pŽ R b . radius of platform Žbase. 0.480 Ž0.600. m
d p, db displacement of adjacent joints of platform and base 0.100 m
 Kim, Kang, and Lee: Robust Tracking Control Design 䢇
537

Figure 6. 6 DOF motion simulator for experiment.

B. Experiments simulations. We see that the robust control guaran-

tees the three angular motions track the command
Figure 6 shows an experimental apparatus of the
signals more accurately than the PD controller.
parallel robot control system. The controller is an
Tracking error is diminished by about 50%; hence
IBM compatible PC Ž486 DX 50 MHz. and its operat-
the tracking performance is enhanced. Between the
ing software implements a real-time system.17 The
simulation result ŽFig. 8. and the experimental re-
real-time software is driven by an interrupt function
sult ŽFig. 9. there are good coherence measures
and has software timer with a 1 ms resolution for a
except the noise involved in the experiment. On the
task switching. The control task is processed in
other hand, even if we propose robust control
every 2 ms and the other three tasks for calculating
schemes based on the assumption that the input
reference signal, emergency handling, and data han-
voltage of servovalve is just proportional to the
dling also run. The AD converter has 12 bits and 20
hydraulic force Žneglecting the hydraulic dynamics.
channels Ž6 for cylinder lengths, 14 for pressure
the control performance has a good shape. The fact
monitoring. and the DA converter has 12 bits and 6
that the time response has some time delay is our
channels to drive the servo amplifier for a servo-
concern. This can be overcome by including the
valve operation. The servovalve is Moog 32 series
hydraulic dynamics as stated in section IV. How-
type whose bandwidth is about 100 Hz. The six
ever, the control scheme considering the hydraulic
cylinder lengths are gathered by an LVDT signal
dynamics includes an acceleration signal. Therefore,
amplifier and the controller uses the error signals
the control may not guarantee better control perfor-
between the cylinder lengths and the reference sig-
mance than the previous one; hence further investi-
nals. Then, the PC sends the control signals via DA
gation is needed at the current stage.
converter to the servo amplifier, hence the hydraulic
actuators move.
Figures 10 and 11 illustrate the experimental
VII. CONCLUSION
results of tracking over sinusoidal command signals
with a PD control and robust control with type I, A class of robust controls in the presence of uncer-
respectively. Here, we put the same control parame- tainty for a 6 DOF parallel manipulator are pro-
ters such as ⑀ , ⑀ 1 , K p , and K v as those adopted in posed. The parallel manipulator system is similar
538 䢇
Journal of Robotic Systems—2000

Figure 7. Simulation results of the tracking histories of 6 DOF motions with PD control.
 Kim, Kang, and Lee: Robust Tracking Control Design 䢇
539

Figure 8. Simulation results of the tracking histories of 6 DOF motions with robust
control Žtype I..
540 䢇
Journal of Robotic Systems—2000

Figure 9. Simulation results of the tracking histories of 6 DOF motions with robust
control Žtype II..
 Kim, Kang, and Lee: Robust Tracking Control Design 䢇
541

Figure 10. Experimental results of the tracking histories of 6 DOF motions with PD
control.
542 䢇
Journal of Robotic Systems—2000

Figure 11. Experimental results of the tracking histories of 6 DOF motions with robust
control Žtype I..
 Kim, Kang, and Lee: Robust Tracking Control Design 䢇
543

to a serial robot but the control needs to be of the geometry of the manipulator is introduced. The
a different form. The control schemes based on control provides an efficient control scheme by ex-
linkspace coordinates are introduced and the perfor- cluding the necessity of computing My1 Ž⭈. for the
mances are verified via simulations and experi- computation of the bounding function that may
ments. The linkspace-control design starts from the cause a computational burden. Both proposed con-
dynamics on the workspace coordinates and utilizes trols rely on the possible bound of the uncertainty.
the dynamics on the linkspace coordinates after The control design considering the hydraulic dy-
transforming the workspace dynamics to the link- namics is proposed even though the control perfor-
space dynamics. A robust control based on the mance via simulation or experiment is not verified
quadratic Lyapunov function is introduced. The yet at this stage. This leaves further investigation.
control handles the time-varying uncertainty and
does not limit a robot type whether it is revolute or
prismatic. However, it uses My1 Ž⭈. computation
and limits the input uncertainty to be within some APPENDIX
constant. On the contrary, a robust control based on
the different Lyapunov function which depends on I. Matrices in 6 DOF Parallel Robot Modeling 2

m 0 0 0 0 0
0 m 0 0 0 0
0 0 m 0 0 0
MŽ q. s 0 0 0 I x C␤ C␥ q I y C␤2 S␥2 q Iz S␤2
2 2 Ž I x y I y . C␤ C␥ S␥ Iz S␤
0 0 0 Ž I x y I y . C␤ C␥ S␥ I x S␥2 q I y C␥2 0
0 0 0 Iz S␤ 0 Iz

0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
C Ž q, q˙. s 0 0 0 K 1 ␤˙q K 2 ␥
˙ ˙ q K 5 ␤˙q K 3 ␥˙
K1 ␣ ˙ q K 3 ␤˙
K2 ␣
0 0 0 ˙ q K 3 ␥˙
yK 1 ␣ K 4␥
˙ ˙ q K 4 ␤˙
K3 ␣
0 0 0 ˙ y K 3 ␤˙
yK 2 ␣ ˙ y K 4 ␤˙
yK 3 ␣ 0

Jacobian J also

uT1 0 1=3 0 1=3 0 1=3 0 1=3 0 1=3

K 1 s yC␤ S␤ Ž C␥2 I x q S␥2 I y y Iz .
0 1= 3 uT2 0 1=3 0 1=3 0 1=3 0 1=3
J1 s 0 1= 3 0 1=3 uT3 0 1=3 0 1=3 0 1=3 K 2 s yC␤2 C␥ S␥ Ž I x y I y .
0 1= 3 0 1=3 0 1=3 uT4 0 1=3 0 1=3 K 3 s 12 C␤ Ž C␥ y S␥ .Ž C␥ q S␥ .Ž I x y I y .
0 1= 3 0 1=3 0 1=3 0 1=3 uT5 0 1=3
K 4 s C␥ S␥ Ž I x y I y .
J2X ’s row s I3= 3 S Ž i . R ␣ R␤ R␥ R ip
K 5 s yC␥ S␥ S␤ Ž I x y I y .
R ␣ S Ž j . R␤ R␥ R ip R ␣ R␤ S Ž k . R␥ R ip
R ␣ ␤␥ Pi q q1 y Bi
ui s R ␣ ␤␥ s R ␣ R␤ R␥
Gs 0 0 ymg 0 0 0
T
5 R ␣ ␤␥ Pi q q1 y Bi 5
544 䢇
Journal of Robotic Systems—2000

0 yv 3 v2 v1 When a ) 0, b are constants, the following in-

S Ž v . s v3 0 yv1 if v s v 2 equality is satisfied for all y:
yv 2 v1 0 v3 2
b b2
yay q by s ya y y
2
ž / q
1 0 0 C␤ 0 S␤ 2a 4a
R␣ s 0 C␣ yS ␣ R␤ s 0 1 0 b2
0 S␣ C␣ yS␤ 0 C␤ F Ž 60 .
4a
C␥ yS␥ 0
When 5 ␮ 5 - ⑀ , it follows from Ž23., Ž26., and the
R␥ s S␥ C␥ 0 inequality Ž60. that
0 0 1
z T PB Ž p q Ep q h .
II. Proof of Theorem 1 s ␮T Ž p q Ep q h .
Define Lyapunov candidate V as ␮ ␮
F ␮T y ␳ 2 q ␮T yE ␳ 2 q 5 ␮ 5 ␳
ž / ž /
⑀ ⑀
V s z Pz 1 T Ž 55 .
2
␳2 5 ␮52
Fy 5 ␮ 5 q ␳E
2
␳ 2 q5 ␮5 ␳
The time derivative of V along the trajectory of the ⑀ ⑀
system Ž15. is given as ␳2
F y Ž1 y ␳E . 5 ␮ 5 2 q 5 ␮ 5Ž 1 y ␳ E . ␳
⑀
V˙s z T Pż
Ž1 y ␳E . ⑀
s z T P Ž Az q Bu q ⌬ Bu q ⌽ . Ž 56. F Ž 61 .
4
From Ž19., Ž20., and Ž24. it can be seen that
Here, the last inequality is satisfied by substituting
a s Ž1 y ␳ E . 1⑀ , b s Ž1 y ␳ E ., and y s ␳ 5 ␮ 5 for the in-
V˙F z T P Az y BEKz q Bp q BEp q Bh
ž / Ž 57 . equality property of Ž60.. Therefore, from Ž58.,
Ž59., and Ž61., V˙ is bounded by
Then, from Ž25. it becomes
V˙F y Ž 12 ␭min Ž Q . y ␳ E 5 PBK 5 . 5 z 5 2
V˙F y z Qz q 5 PBEK 5 5 z 5 q z PB Ž p q Ep q h .
1
2
T 2 T
q z T PB Ž p q Ep q h .
F y ␭min Ž Q . 5 z 5 q ␳ E 5 PBK 5 5 z 5
1 2 2
F y␩ 5 z 5 2 q ⑀
2
Ž 62 .
q z PB Ž p q Ep q h .
T
where
F y Ž 12 ␭min Ž Q . y ␳ E 5 PBK 5 . 5 z 5 2
1
q z T PB Ž p q Ep q h . . Ž 58 . ␩ [ ␭min Ž Q . y ␳ E 5 PBK 5 Ž 63 .
2
When 5 ␮ 5 ) ⑀ , the last term of Ž58. follows from Ž1 y ␳E . ⑀
Ž21., Ž26., and Ž27. ⑀[ Ž 64.
4

z T PB Ž p q Ep q h . Therefore, V˙- 0 for all 5 z 5 ) R z , where

s ␮T Ž p q Ep q h . ⑀

F␮ T
ž y
␮
5 ␮5
␳ q 5 ␮ 5 ␳E ␳ q 5 ␮ 5 ␳
/
Rzs ( ␩
Ž 65 .

Following Ž62. for r z G 0, if 5 z 0 5 r z , we can sat-

F y5 ␮ 5 ␳ q 5 ␮ 5 ␳ E ␳ q 5 ␮ 5Ž 1 y ␳ E . ␳ isfy the requirements of the practical stability 13 with
s0 Ž 59 . the uniform boundedness Žball size of d z Ž r z .., the
 Kim, Kang, and Lee: Robust Tracking Control Design 䢇
545

uniform ultimate boundedness Žball size of d z G d z is identical to z defined in Ž16.. Since ⍀ 1 i ) 0, ᭙ i, V1

and a finite escaping time Tz Ž d z , r z .., and the uni- is positive definite:
form stability Žball size of ␦ x Ž d z .. with the argu- 6
ments V1 G 12 Ý ␭min Ž ⍀ 1 i . Ž e i2 q ė i2 .
is1
Ž ␥y1
1 (␥ 2
.Ž R z . if r z F R z
d z Ž rx . s
½ Ž ␥y1
1 (␥ 2
.Ž r z . if r z ) R z
Ž 66 . G ␶ 1 5 z1 5 2 Ž 72 .

¡
where
␭min Ž P .

~␥
0 if r z F d z ( ␭ma x Ž P .
␶ 1 [ 12 min min ␭min Ž ⍀ 1 i . , i s 1, 2, . . . , 6
i
½ ␴ g⌺
5 Ž 73 .

Tz Ž d z , r z . s Ž r z . y Ž ␥ 1 (␥y1 Ž 67 .
2 (␥ 1
2
.Ž d . Next, with respect to the bound from the above
condition in Assumption 2,
¢ Ž ␥ 3 (␥y1
2 (␥ 1
otherwise
.Ž d .
V1 F 12 ␴ 5 ˙
e q S1 e 5 2 q 12 e T Ž K p 1 q S1 K v 1 . e
␦z Ž d z . s R z Ž 68 . 6
s 12 ␴ Ý Ž ˙e i2 q 2 S1 i ˙e i e i q S12i e i2 .
where ␥ 1Ž z . s ␭minŽ P .5 z 5 , ␥ 2 Ž z . s ␭max Ž P .5 z 5 ,
1
2
2 1
2
2
is1

␥ 3 Ž5 z 5. s ␩ 5 z 5 2 , d z s ␥y1 Ž . y1
1 (␥ 2 R z , R z s ␥ 2 (␥ 1 d z .
Ž . 6
q 12 Ý Ž Kp 1i
q S1 i K v 1 i . e i2
is1
III. Proof of Theorem 2
6 ei
Define Lyapunov function candidate V1 as s 12 Ý ei ė i ⍀ 1 i Ž 74 .
is1
ė i
T
V1 s 12 Ž ˙
e q S1 e . M1Ž ˙
e q S1 e . q 12 e T Ž K p 1 q S1 K v 1 . e where
Ž 69 .
␴ S1i2 q K p 1 i q S1i K v 1 i ␴ S1 i
⍀1i [ Ž 75 .
To see that V1 is a legitimate Lyapunov function ␴ S1 i ␴
candidate, we shall prove that V1 is positive defi-
nite and decrescent. By Assumption 2, it can be seen Therefore, we have
that
6

V1 G ␴ 5 ˙
1
2e q S1 e 5 q e Ž K p 1 q S1 K v 1 . e
2 1 T
2
V1 F 12 Ý ␭max Ž ⍀ 1 i . Ž e i2 q ė i2 .
is1
6
s 12 ␴ F ␶ 2 5 z1 5 2 Ž 76 .
Ý Ž ˙e12 q 2 S1 i ˙e i e i q S12i e i2 .
is1
where
6
q 12 Ý Ž Kp q S1 i K v 1 i . e i2
1i ␶ 2 [ 12 max max ␭max Ž ⍀ 1 i . , i s 1, 2, . . . , 6 . Ž 77.
½ 5
is1 i ␴ g⌺
6 ei
s 12 ei ė i ⍀ 1 i Ž 70 . The derivative of V1 along the trajectory of the
Ý ė i system Ž11. is given by
is1

T
where V˙1 s Ž ˙
e q S1 e . M1Ž ¨
e q S1 ˙
e.
T
␴ S12i q K p 1 i q S1 i K v 1 i ␴ S1 i q 12 Ž ˙ ˙1Ž ˙e q S1 e .
e q S1 e . M
⍀1i [ Ž 71.
␴ S1 i ␴ q e T Ž K p 1 q S1 K v 1 . ė
T
where e i and ˙e i are the ith components of e and ˙ e, s Ž˙
e q S1 e . Ž M1 S1 ˙e y M1 ¨y d y C1 ˙y d y C1 ˙e
respectively. For the sake of not arising confusion,
we introduce a state variable z 1 s w e ė x T even if it yG1 q u . q e T Ž K p 1 q S1 K v 1 . e
546 䢇
Journal of Robotic Systems—2000

q 12 Ž ˙
e q S1 e . M
T
˙1Ž ˙e q S1 e . Ž 78 . Following Ž82. for r z 1 ) 0, if 5 z 1 0 5 F r z 1, we can sat-
isfy the requirements of the practical stability 13 with
According to Ž33., Ž39., and the skew-symmetric the uniform boundedness Žball size of d z 1Ž r z 1 .., the
˙1 y 2C1 shown in Lemma 2, it can be
property on M uniform ultimate boundedness Žball size of d z 1 ) d z 1
seen that
and a finite escaping time Tz 1Ž d z 1, r z 1 .., and the uni-
V˙1 s Ž ˙
e q S1 e .
T
Ž M1 S1 ˙e y M1 ¨y d y C1 ˙y d q C1 S1 e form stability Žball size of ␦ z 1Ž d z 1 ..

~¡Ž ␥
yG1 q p 1 . y e T S1 K p e y ˙
eT K v ˙
e y1
4 (␥ 5 if r z 1 F R z 1
1 1
.Ž R z .
d z 1Ž r z 1 . s
¢
1
T T Ž 86 .
e q S1 e . ␾ 1 q Ž ˙
F Ž˙ e q S1 e . p 1 Ž ␥4y1 (␥ 5 .Ž r z . if r z 1 ) R z 1
1

y ␭min Ž S1 K p 1 , K v 1 .Ž 5 e 5 2 q 5 ˙
e5 2 . Ž 79 .
¡0 ␶1
If 5 ␮ 1 5 ) ⑀ 1 , the first two terms in Ž79. become by
if r z 1 F d z 1 ( ␶2
the p 1 in Ž34.
ž / ~ ␥ Žr
Tz 1 d z 1 , r z 1 s 5 z1
. y Ž ␥4 (␥y1
5 ( ␥4
.Ž d . Ž 87 .

¢ Ž ␥6 (␥y1
5 ( ␥4
T T
e q S1 e . ␾ 1 q Ž ˙
Ž˙ e q S1 e . p 1 .Ž d .
otherwise
˙e q S1 e
F5˙
e q S1 e 5 ␳ 1 q Ž ˙
T
e q S1 e . y ␳
ž 5˙e q S1 e 5 1 / ␦z1 d z1 s R z1
ž / Ž 88 .

s0 Ž 80 .
where ␥4Ž z 1 . s ␶ 1 5 z 1 5 2 , ␥ 5 Ž z 1 . s ␶ 2 5 z 1 5 2 , ␥6 Ž5 z 1 5. s
If 5 ␮ 1 5 F ⑀ 1 , those become from the inequality prop-
erty of Ž60. ␩1 5 z 1 5 2 , d z s ␥4y1 (␥ 5 Ž R z 1 ., R z 1 s ␥y1
5 ( ␥4 d z 1 .
Ž .

T T
e q S1 e . ␾ 1 q Ž ˙
Ž˙ e q S1 e . p 1 This paper is supported by the research fund of Seoul
National University of Technology.
˙e q S1 e
F5˙
e q S1 e 5 ␳ 1 q Ž ˙
T
e q S1 e . y ␳ 12
ž ⑀1 /
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e q S1 e 5 ␳ 1 y 5˙
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