1987 Harashima - Tracking Control of Robot Manipulators Using Sliding Mode.pdf

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. PE-2, NO. 2, APRIL 1987 169
Tracking Control of Robot Manipulators Using
Sliding Mode
FUMIO HARASHIMA, SENIOR MEMBER, IEEE,
JIAN-XIN XU, AND HIDEKI HASHIMOTO, MEMBER, IEEE
Abstract-A new methodology using the theory of variable structure computing inaccuracies will be caused if the system consystems
is developed for accurate tracking control of robotic manipu- tains unknown parameter variations. An adaptive method
lators. The proposed method, based on a sliding mode controller, pro- tajectory contro maniators in taskoreted
duces a discontinuity control input so that the nonlinear interactions
and influence caused by unknown parameter variations can be de- ordinates is also proposed by [6]. This method requires
pressed whereas the resulting control law is simple and easy to apply neither knowledge of the parameters of the system nor
to on-line computer control. The digital simulation results of a three- complicated calculations. The shortcoming of the method
degree-of-freedom manipulator show the validity of accurate tracking is that the whole design procedure is implemented under
capability and robust performance of the system with the developed the assumption that the velocity of the desired trajectory
control scheme.
should be sufficiently small.
To realize on-line manipulator control with accurate
INTRODUCTION tracking capability while in high-speed motion, the de-
IN THE ADVANCED control of robotic manipulators, signers should consider the following two points which
it is important for manipulators to track arbitrary trajec-
are the key problems in choosing a control law:
tories in a wide range of work space. However, if high 1) avoidance of complex computations such as those
speed and accuracy in tracking motion are required, the for the nonlinear compensation;
control of manipulators is difficult to realize using con- 2) insensitivity of the manipulator control to the unventional
methods such as proportional integral derivation known parameter variations or modeling ambigui-
(PID). This is because a multijoint manipulator is a highly
ties.
nonlinear system that includes nonlinear interactions between
each arm. Moreover, the interactions vary over a To overcome these difficulties, the authors developed a
wide range corresponding to the positions and motions of sliding mode controller based on the theory of variable
the manipulator.
structure systems (VSS). The main feature of sliding mode
In general, the dynamics of each joint can be described control [7] is to let a sliding mode occur on a prescribed
by a coupled second-order differential equation with Cor- switching surface. While in sliding mode the system is
iolis and centrifugal forces represented by coupling terms. only govemed by the sliding equation; thus the system
As to such complicated dynamics, quite a number of pa- will be forced to "slide" along the desired phase trajecpers
have been presented on the subject of manipulator tories or near the vicinity of the switching surface. The
control which are based on various control concepts. Lin- system is then robust and insensitive to those interactions
ear feedback control using linearized models as the basis as well as unmodeled dynamics [8], [9].
for decoupling and control has been proposed [2], [3]. The control scheme proposed in this paper is mainly
This method, however, is only valid in case some condi- directed to the tracking problem in high-speed motion. By
tions are satisfied. For example, the rate-linearized con- choosing a lot of switching gains, the system first shows
trol method will not perform well while manipulators are the capability to suppress the nonlinear interactions existin
high-speed motion. A nonlinear control system based ing in the manipulator's joints and next shows the capaon
complete decoupling [4] is effective in high-speed mo- bility to suppress the influences resulting from unknown
tion, but the nonlinear compensations are complex and parameter changes.
costly to compute. The Newton-Euler method [5], using
the relationships of moving coordinate systems, solves the DYNAMIC MODEL OF ROBOTIC MANIPULATORS
on-line computational problem of mechanical manipula- The dynamic model of industrial robots or manipulators
tors, although such a scheme still suffers from the fact that is derived from Lagrange's equations [10], which have
the form
Manuscript received October 3, 1985; revised July 15, 1986. This work
was originally presented at the Power Electronics Specialists Conference, d (t3L AL N
Gaithersberg, MD, June 18-21, 1984, under the title, "'Arbitrary Trajec- dt -AJ J - 0 =iF 1, * , (1I)
n
tory Tracking Characteristics of Sliding Mode Controlled Servo System.''"
The authors are with the Institute of Industrial Science, University of wee0= ~...6 [ an* . T 6] Tar
Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106, Japan. theregeneralze coordinates andveoiy respectively,on and
IEEE Log Number 8612106.tegnrlzdcodntsan eoiy epciey n
0885-8993/87/0400-0169$01.00 ©C 1987 IEEE

170 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. PE-2, NO. 2, APRIL 1987
T = [Tl, , Tn I" is the generalized external force. If the control inputs i1H and u; are selected such that si
The Lagrangian L is expressed as and si always have the opposite signs in the neighborhood
L = K(, O)-P(H) (2) of si = 0, that is,
>
where P(6) denotes the potential energy of the system , 0 lim i < , (9)
and K(6, 0) is the kinetic energy which is written as
then sliding mode will occur on the switching surface si
K = IOTR(6)b (3) = 0. Therefore, the state trajectories of the system will
slide and remain on the surface, which implies that the
where the inertia matrix R (6) is symmetric and positive motion of the system (6) is dominated by the sliding equadefinite
for all 0.
tion (7). Consequently, the original system, though it may
Substituting (2) and (3) into (1), the equation of motion be higher order or nonlinear, behaves like a lower order
for robotic manipulators can be written as and linear system defined by the sliding equation si = 0.
(fn M( \ To
I
guarantee the existence condition (7) of the sliding
R(H)H + L 0.
0 -
mode, one need only know the bounds of system param-
e _R(6)&
Ki=1 36j / 2 a6 eters for determining the control inputs (8). This clearly
a means that the system is insensitive with respect to pa-
+ - P(O) = T. (4) rameter uncertainties and other disturbances.
Note that the second and third terms of the left side of the TRACKING CONTROL USING SLIDING MODE
equation represent the Coriolis and centrifugal forces, and
the last term denotes the gravitational forces. Equation (4) Two kinds of motions, the point-to-point (PTP) motion
can take a more brief form and the tracking motion, should be considered. Usually
the PTP motion may be regarded as a special case of the
R(O)H + M(0) t + g(6) = T (5) tracking motion. Therefore, only the tracking problem is
where
considered here. The tracking capability of advanced ma-
=}[06 0 0 022
T nipulators is indispensable for the smooth motion required
g = [,3P/0l, * * *aP/aOn]
in various industrial tasks, such as in an assembly line
T process. In tracking motion the values of desired position
Od(t), desired velocity Ad(t), and the desired acceleration
M = {mi,(6)}nx
of the manipulator on the entire path are provided by a
supervisory computer.
1 = ln(n + 1). In this paper a new approach is developed to tracking
motion control of manipulators.- Each arm is directly con-
SLIDING MODE CONTROL trolled in the manipulator's coordinates (joint space).
Moreover, the proposed method can be easily imple-
Sliding mode control based on the theory of variable mented in a task-oriented space
structure systems has a number of attractive features [7]. When manipulator motion is in low speed the super-
Therefore, it is widely applied to solve various control visory computer is able to neglect those quadratic terms
problems [11]-[ 15]. Consider the following system: 6,6A of
i
. Then (5) would become
H =f(O, t) + B(H, t)u (6) R(60)0 + g(0) = T. (10)
where 6 e Rn x I, fe Rn x', B E Rn xm, and u E Rmxl. To The desired input torque is computed from
obtain the desired dynamic performance of the system, a
set of sliding mode equations is previously selected as T = R(6) {I -Kd -Kpe} + g(6) (11)
si(0, t) = 0, i = 1, * , m (7) where Kd and Kp are constant gain matrices. Equations
which defines the switching surfaces. Then the control in- (10) and ( 1) give
puts with discontinuities have been decided according to R(6) { e + Kde + Kp e } = 0 (12)
the switching logic as where e = 6 -
(u+(6, t), if s460, t) < 0 Since the inertia matrix R(6) is nonsingular except in
u1(6, t) = _.(8) some special cases, we may choose Kd and K,p such that
yu, (6, t), If si(6, t) > 0 the characteristic roots of (12) have negative real parts,
where u1 (6, t) is the ith component of u. The preceding and the error e approaches zero asymptotically.
system with discontinuity control is termed a variable Notice that the effectiveness of the preceding control
structure system since the feedback structure of the sys- algorithm is grounded on the premise that the manipulator
tem is switching alternatively according to the state of the is in low-speed motion, whereas in most cases the manipsystem.
ulator is required to track preplanned trajectories in high

HARASHIMA et al.: TRACKING CONTROL OF ROBOT MANIPULATORS 171
speed, so the influences of Coriolis and centrifugal terms
Gravity
M(6) t should also be taken into account. G(8) Compensation
Control Scheme In High-Speed Motion
To derive the control laws in high-speed motion, the ld
authors proposed a considerably simpler method based on X
the theory of VSS instead of a complete nonlinear com-
Feedback
pensation which would need much more computing time.
Compensation
Note that the system state variable 6 and the correspond- ri /
ing differential H can be directly measured. The control Mo,
r
inputs are selected as follows:
T = g(O) + R(){G6d -Ce + rF} (13) S=o
Switching
where C = diag [c, , ca], and r = h,'j}l1. In Surfaces
this case (5) could become
+d'd
R(O){e + Ce} = r(6){F - R1(0) M(H)}H (14) Fig. 1. Block diagramof sliding mode control system.
or
posed to contain some unknown parameter variations,
e + Ce = {r - M'(o)}t (15) e.g., the payload change. Taking into account the influ-
X I = R-1(0() M (0). ences caused by the unknown variations of system param-
where M'(0) = { m!,(0) }
Define the sliding surfaces as follows:
eters, the equation of motion for the robotic manipulators
can be written as
si = ei + ci i = 1, ** n. (16) {R(6, 4) + AR(, A4))} + {M(6, 4)
The existence condition of sliding mode
51S, < 0, 0, 0d (17) + AM(0, A))} + {g(0, 4)) + Ag(0, A4)} T
guarantees that the system states continue on si = 0, so (22)
that ei satisfies the equation
where A4) indicates the unknown variations of system paei
= -ciei. (18) rameters 4), and AR, AM, and Ag represent the changes
Now that the switching surfaces have been given, one caused by A4) with respect to the R, M, and g, respecneed
only decide the switching logics and the correspond- tively.
ing control gains to guarantee the asymptotic stability of To determine the control law, rewrite (22) in the form
the system (15). Since the matrix M'(0) only depends on RO + g = T - ARO - (M + AM) - Ag. (23)
the state variable 0, the upper and lower bounds of each
component of m!-(6) can be calculated in advance. As- According to the philosophy of VSS theory described
sumingXX
previously, the following control law is selected:
m,j < m1-(0) _ me v (19) T= g + R{0d-C + AO +rF + d}. (24)
then by considering (15) and (17) the next inequality could Substitution of (24) into (23) results in
be obtained
R(e + Ce) = R{(A - AR*)0 + (r -m*)
E(,yjj - m!) jsi < O, i = 1, * ,n (20) + (d - Ag*)} =0 (25)
and the control gains of Ir be determined as
QY:t > fii, if (jsi < 0 or
= ~~~~~~~~(21)
Qry < iM if (sSi > 0. + Ce = (A-AR*)6 + (r-M*)t + (d -g*)
This inequality guarantees that the states of system (5)(
move towards and then slide along the given trajectories. (6
Fig. 1 shows the block diagram of the sliding mode con- where
trol system.
Control Scheme in the System with Unknown Parameter
AR* = {ri(6, A4)}
= R1AR
Variations M* {m1 (6, A))}nx = R'l(Az\M + M )
Following the preceding design procedure, a case willT
be considered in which manipulator dynamics are sup- Ag* = [A gj (O, A4)), ***, Ag* ( , A4 ))] = R-1Ai\g .
nXm
I

172 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. PE-2, NO. 2, APRIL 1987
Under the assumption that a sliding mode starts at and gardless of the parameter changes; that is, the robust percontinues
on the intersection of switching surfaces si = formance of the system will be achieved.
. .
0, (i = 1, *, n), the value of si in this case can be
regarded as approximating zero, therefore the following
SIMULATION AND RESULTS
relation is satisfied:
In this section the design guideline described in former
..=- 6d- Ce (27)
sections will be used to design a sliding mode controller
for a three-degree-of-freedom manipulator with direct-
Substituting (27) into (26) yields drive arms as shown in Fig. 2.
s=(A - AR* ) (Od - Ce )
inThe parameters of the manipulator dynamics are listed
in Table I, where m, s, and L represent the total mass, the
+ (F -M**) + (d - Ag*). (28) position of the center of mass, and the length of each arm,
respectively. Ix, I., and I, are the inertia of each arm about
...
Let sji < 0, (i = 1, , n ), then the center of mass in its own coordinates. The inertia man
I trix R(0) and matrix M(0) of the manipulator dynamics
Z (XAij - AriJ ) (Hdj - §jej)si + >Z (rj - miJ ){jsi (5) have the forms
+(d Ag)S < 0 i = 1, , n. (29) F= r22(0) r23(0)
1
~~~~~~R
(0) = 220) 230
The goal here is then to find a set of control gains A, 10 r23(0) r33(0)
r, and d such that the inequality (29) is always satisfied.
One can see that these control gains are easy to determine
if
Arz
M(0)
_ Avr,*' A(01\))_ < Aig 0 m12(0) m13(0) 0 0 0 1
My < m, (0, A4t) < mii m21(0) 0 0 0 m25(0) m26(0)I
Ag_* < lAg*(0, A4) < Akg (30) Lm31(0) 0 0 m34(0) 0 0]
where all the upper and lower bounds are constants.
where
Note that the maximum and minimum values of Ar (0, = 0. 6 + 1.35 sin2 02 + 0. 1 sin2 (02 + 03)
A4)), mJ(O, A4)), and Ag*(O, A4)) can be obtained by + 0.4 sin 02 sin (02 + 03))
previous calculation of every component of
R-1(0, 4)) AR(0, A4) r22 = 20.45 + 0.4 cos 03
R1(0, 4){M(0, 4)) + AM(0, A4)} r23 = 0.2 + 0.2 cos 03
R'l(0, 4)) Ag(0, A4))r = 0.2
for all 0. Thus to determine control gains in (29) it is only M12 = 1.35 sin 202 + 0.1 sin 2 (02 + 03)
necessary to know the bounds of AA+ on the system param- + 0.4 sin 02 sin (02 + 03)
eters.
Finally, from (29) and (30) the switching gains have M13 = 0.4 sin 02 COS (02 + 03) + 0.1 sin 2(02 + 03)
the forms =
{k± > AyiJ, if si(Od;- Cj) < 0 2 = -0.4 sin 03
l j > if Si(6dj - cjej) > 0 M226 = -0.2 sin 03
in31
= -l/
ZJ > mii, if Sit < 0 in34 = -2/
&y1( < Ain, if > and the gravity terms are
dj> Ag, if s 0. (31
= -39.2 sin 02 -4.9 sin (02 + 03)
As is indicated by the above design procedure, all switch- 5= -4.9 sin (02 + 03)
ing gains are determined by the variation bounds of the To examine the validity of the proposed method, the
system parameters. So it is clear that the precise tracking whole system is simulated on a digital computer. The decapability
of such manipulators will be guaranteed re- sired trajectory in the three-dimensional work space is il-

HARASHIMA et al.: TRACKING CONTROL OF ROBOT MANIPULATORS 173
z
y
-Q
2
O 02
0.2
7~~~~~~~~~~~~~~~P
0.5
Fig. 2. Configuration of manipulator with three degrees of freedom. Fig. 3. Desired trajectory in work space (meter).
x
TABLE I
MANIPULATOR PARAMETERS
elements of (34) are decided. In this case the r is
Link m I IY L S = { L
Number (kg) (kg M2) (kg iM2) (kg i2) (m) (m)
1 13 - - 0.35 0.5 - 20 17.5 6.456 60 0 0
2 10 0.3 0.3 0.15 0.4 0.2 213 0 6 1.168 1.309 0.888
3 5 0.15 0.15 0.10 0.3 0.1 2.174 2.384 1.542
lustrated in Fig. 3. The end of the manipulator starts from
{ )ij}
position P, goes along the straight line, and stops at po- Similarly, the control gains A and d can be determined
sition Q.
in the same way as that of F
The following experiment contains the unknown parameter
variation A4, which mainly results from a pay- 4.386 0 0
load change. The payload located at the end of the ma- 0 0.965 -0.677
nipulator is supposed to vary from 0 to 10 kg. The
switching surfaces are chosen as 0 1.079 0.280j
s = + Ce (32) -4.619 0 0
where C = diag (20, 20, 20) and e = 0 - A- = 0 -0.835 -0.677
The control gains r, A, and d are selected according to L 0 -1.481 0.9471
(30) and (31). First, note that all elements of matrix
M*(O, A4) can be obtained from equation =-[0 34.85 79.79]
M*(O, A4) = R'(0))[M(O) + AM(0, A4)] d= -d+.
O Mi2 Mi3 0 0 0 Next, the system's performance will be evaluated in the
= o0 0 m ~m following two cases.
|m 1 O m24 m25 * m26
Case 1: The first motion pattern of the manipulator's
Lm* 0 0 Mi34 M3 M end is shown in Fig. 4, where the maximum acceleration
(33 is about 2.7 m/s2 and the maximum velocity is about 1.3
min/s. Fig. 5 shows the tracking error of the manipulator's
Then the control gain matrix F can be determined in terms end in work space corresponding to motion pattern one.
of (33), i.e.,
Fig. 5(a) is the case without payload variation, and in Fig.
0 'Y12 OY13 0 0 0 5(b) a 4-kg unpredicted payload variation exists. The il-
~~261. ~
lustrated results confirm the fact that the proposed control
r= Y2i 0 0 OY24 'Y25 Y2* (3) algorithm is insensitive to uncertain parameter variations.
_'Y31 ° O34 735 PY36_ Case 2: To illustrate the robustness of the developed
sliding mode controller in high-speed motion, the second
Since the upper and lower bounds his, rne of inJ (O, motion pattern about the manipulator's end is selected,
z4) can be calculated previously, by ensuring (31) all the which is shown in Fig. 6. Fig. 6(a) is the acceleration

174 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. PE-2, NO. 2, APRIL 1987
(a) . (a)
r2: (n/s) t~~~~.
ir (m/s)
1
(sec) ~~~~~~0. 3 06 .9(sec)
WS)~~~~~~~~~~~~~~~~~~~~~~~~~~~b
0.5 1.0 1. (sec) 0
(b)
Fig. 4. Motion pattern I of manipulator's end. (a) Acceleration. (b) Ve- Fig. 6. Motion pattern two of manipulator's end (higher speed motion).
locity. (a) Acceleration. (b) Velocity.
0~ ~ ~~~~~~O
(me ter )
(N *m)
0.01 30
(a)
(a)
-01(X 0.5 1.0 1.5 (sec)
0~ ~~ ~~~~~~~~~~~3
(meter)
0.01 30
|\ ' _ @~~~~~~~~~~
Torque due to gravity
0 .2
Q Total torque
0 0.45 0.9 (sec)
® Torque due to coupling inertia
(i Torque due to effective inertia
(d3 Torque due to centrifugal and Coriolis forces
-0.01 '.. .
0 0.5 1.0 1.5 (sec) Fig. 7. Torque components of link'3. Payload: 4 kg. (Motion pattern two.)
(b)
Fig. 5. Tracking error of manipulator's end in work space (motion pattern byteculniera)ndhefth(usdycntfone).
(a) Payload variation: 0 kg. (b) Payload variation: 4 kg. gal and Coriolis forces) cannot be neglected compared
with other terms. In this case the tracking responses of
pattern with a maximum value of 7.5 m/s2, and Fig. the three joints to the desired trajectories and the corre-
6(b) is the velocity pattern with a maximum value of 2.1 sponding input torques resulting from sliding mode conrn/s.
The torque components of link 3 with a 4-kg pay- trol are shown in Figs. 8, 9, and 10, respectively.
load variation in motion pattern two are shown in Fig. 7. As shown in these figures, the manipulator shows per-
Due to the direct-drive manipulator, the influences of fect tracking capability as there are almost no tracking
nonlinear coupling terms such as the third term (caused errors visible between the desired trajectories and the sys-

HARASHIMA et al.: TRACKING CONTROL OF ROBOT MANIPULATORS 175
(rad)
3 t
(rad)
3 03d
2 2 \
0D3
1 / 1 1
t (sec)
0 3 0.6 0.9 0.3 0.6
0.3 0.6
0.9
0 .9
t (sec)
(a)
(N m) (N m)
(a)
600 300
300- ttsec) 150- t(sec)
0 .3
- 30
-600 -300
0.6 0.9 -150 0.3 0.6 0 .9
(b)
(b)
Fig. 8. Tracking response of first joint to desired trajectory. Payload vari- Fig. 10. Tracking response of third joint to desired trajectory. Payload
ation: 4 kg. (a) Tracking motion. (b) Input torque.
variations: 4 kg. (a) Tracking motion. (b) Input torque.
(rad)
servo sampling period is equal to 5 ms. The sensor reso-
3 lution is assumed to be infinite.
2 2 902d
CONCLUSION
The algorithm presented in this paper is for the accurate
02 t (sec) tracking control of robotic manipulators. By using the im-
0 0.3 proved sliding mode control law, the nonlinear dynamic
0 . 3
0.6 0 . 9 interactions of the manipulator joints are effectively sup-
-1 pressed, and the system is insensitive to the physical parameter
variations. The simulation results demonstrate the
(N.m)
(a) perfect tracking property of a robotic manipulator with
three degrees of freedom. The nonlinear coupled manipulator
dynamics show well-behaved linear uncoupled
300 characteristics in high-speed motion. In addition, the de-
150 t (sec) sign of the sliding mode controller needs less a priori in-
-150 0.43 0.609PI formation about the parameter changes owing to the fact
-300 that control gains need only to satisfy some inequality
300
f conditions associated with parameter bounds. Since the
(b)
variations of the control gains are determined according
Fig. 9. Tracking response of second joint to desired trajectory. Payload to simple switching logic, computation for completing
variation: 4 kg. (a) Tracking motion. (b) Input torque.
control inputs can be reduced to the degree that real-time
tracking control is realizable. As for measuring practical
signals, the sensor resolution is always finite and gives
tem's, i.e., the developed control system isin every way rise to the problem of affecting the actual control chatterrobust
in high-speed motion even with the existence of ing frequency. Therefore, in future research we will take
unknown parametric variations,
Due to the actuator dynamics a simulation has been performed
by introducing first-order filters between each
controller output and the corresponding manipulator input
into account the limitations of sensor resolution.
REFERENCES
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176 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. PE-2, NO. 2, APRIL 1987
prostheses," IEEE . rans. Man-Mach. Syst., vol. MMS-10, pp. 47- Fumio Harashima (M'75-SM'81) was born in
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69-76, June 1980. in 1978 from the Society of Instrument and Con-
[6] M. Takegaki and S. Arimoto, "An adaptive trajectory control of ma- trol Engineers (SICE) of Japan. In 1983 he received the Best Paper Award
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[10] P. Paul, Robot Manipulators: Mathematics, Programming, And Con- in 1982 and the M.S. degree from the University
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o oy,Tko aa,i 96 ei urnl
[11] K. D. Young and H. G. Kwatny, "Variable structure servomecha- a graduate student in the doctora1 p rogram at the
nism design and applications to overspeed protection control," Au- a University of Tokyo. His main interests art in
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31, pp. 313-316, 1984. ment and Control Engineers of Japan, the Robot-
[13] A. Sabanovic and D. B. Izozimov, "Application of sliding modes to ics Society of Japan and the Society of Artificial Intelligence of Japan.
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41-49, 1981.
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system theory applied to sub-time optimal position control with an
Hideki Hashimoto (S'84-M'87) was born in Toinvariant
trajectory," Trans. Inst. Elec. Eng. Japan, vol. 104, no. kyo, Japan, on August 15, 1957. He received the
3/4, Mar./Apr. 1984. B.E., M.E., and Dr.Eng. degrees from the Uni-
[15] F. Harashima, H. Hashimoto, and S. Kondo, "MOSFET converter- versity of Tokyo, Tokyo, Japan, in 1981, 1984,
fed position servo system with sliding mode control," IEEE Trans.
and 1987, respectively.
Ind. Electron., vol. IE-32, pp. 238-244, 1985. He is currently a Lecturer at the University of
[16] F. Harashima, H. Hashimoto, and K. Maruyama, "Sliding mode con- Tokyo, working on control engineering, robotics,
trol of manipulator with time-varying switching surfaces, " Trans. Soc.
and artificial intelligence.
Instrum. Contr. Eng., vol. 22, no. 3, pp. 335-342, Mar. 1986. Dr. Hashimoto is a member of IEE of Japan,
[17] -, "Practical robust control of robot arm using variable structure the Society of Instrument and Control Engineers
system," Proc. 1986 IEEE Int. Conf. Robotics and Automation, vol. of Japan, the Robotics Society of Japan, and the
1, pp. 532-539, 1986. Society of Artificial Intelligence of Japan.