IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 41, NO. 1, FEBRUARY 1994

25

Path Tracking Control of a Manipulator
Considering Torque Saturation
Hirohiko Arai, Member, IEEE, Kazuo Tanie, Member, IEEE, and Susumu Tachi, Member, IEEE

Abstract-When the minimum-timetrajectory of a manipulator
along a geometrically prescribed path is planned taking into
consideration the manipulator’s dynamics and actuator’s torquelimits, at least one of the joints should be at the torque limit.
The execution of such a trajectory by a conventional feedback
control scheme results in torque saturation. Consequently, thetracking error cannot be suppressed and the manipulator may
deviate from the desired path. In this paper, we propose a
feedback control method for path tracking which takes the torquesaturation into account. Based on the desired path, a coordinate
system calledpath coordinates is defined. The path coordinatesare
composed of the component along the path and the components
normal to the path. The equation of motion is described in termsof the path coordinates. Control of the components normal tothe path is given priority in order to keep the motion of themanipulator on the path. Simulationsof a two-degree-of-freedom
manipulator show the effectiveness of this method.

I. INTRODUCTION

H

IGH-SPEED manipulation is one of the most important
performance requirements of a robot manipulator. It isespecially necessary in industrial applications. However, joint
actuators should be as small and as light as possible. In the
future, robot applications will spread into the medical field,
domestic life, and various other areas in which robots operate
in a common space with humans. In those applications fromthe viewpoint of safety, it is not desirable that robots haveactuators as powerful as the ones of current industrial robots.
In space manipulators in order to save energy, it is required thatlong arms be controlled by small actuators efficiently. Thus,
a control scheme which provides faster operation of a weaker
manipulator by making the best of dynamic characteristics ofthe manipulator will be more important
There are two types of minimum-time control problems
in manipulator dynamics. One is where only the starting
point and final point of the path are given, and the path andcontrol input are determined [1]-[3]. The other is where thepath geometry as well as the starting and final points are
specified. The latter is more practical when it is combinedwith path planning algorithms for obstacle avoidance, etc.
In that case, the solution of the problem is represented as
an acceleratiotddeceleration profile along the desired path.
Several off-line planning algorithms for the minimum-time
Manuscnpt received June 6, 1992; revised August 18, 1993. This paper waspresented in part at the 1992 IEEERSJ International Conference on Intelligent
Robots and Systems (IROS’92), Raleigh, NC, July 1992.
H. Arai and K. Tanie are with the Robotics Department, MechanicalEngineering Laboratory, MITI, 1-2 Namiki, Tsukuba, Ibaraki 305, Japan.
S. Tachi is with RCAST, University of Tokyo, 4-6-1 Komaba, Meguro-ku,
Tokyo 153, Japan.
IEEE Log Number 9214344.

trajectory with a specified path and with torque limits of theactuators have previously been proposed [4]-[6].
When the minimum-time trajectory of a manipulator along
a geometrically prescribed path is planned taking into consideration the manipulator’s dynamics and its actuators’ torquelimits, at least one of the joints should be at the torque limit.
Trajectory execution is usually based on the position feedback
following of a target point on the desired path. The executionof minimum-time trajectory by such a control scheme results in
torque saturation. Consequently, the control has no margin to
suppress the tracking error and the manipulator may deviate
from the desired path.
In this paper, we propose a feedback control method forpath tracking which takes the torque saturation into account.
Dah1 and Nielsen [7] proposed an on-line path following
algorithm, in which the time scale of the desired trajectoryis modified in real time according to the torque limit. Tam[SI used a perturbation scheme of modifying switching timeand nonsaturated torque for the same purpose. We define acoordinate system, called the path coordinates [9], based on
the desired path. The path coordinates are composed of thecomponent along the path and the components normal to thepath. The equation of motion is described in terms of the pathcoordinates. Control of the components normal to the path isgiven priority in order to keep the motion of the manipulator
on the path. In this method, dynamics of the manipulator
are described not only on the path, but also out of the path.
Thus, the control is guaranteed to converge to the desired pathunder torque saturation and a transient response can be chosen.
Furthermore, in this method, it is not required that the nominal
trajectory be the minimum-time trajectory because this method
does not depend on characteristics of the nominal trajectory(e.g., bang-bang characteristics). The simulations of a twodegree-of-freedom manipulator show the effectiveness of thismethod.
OF MOTIONIN TERMS OF PATH COORDINATES
11. EQUATION

A. Path Coordinates

Control of an n DOF manipulator in n-dimensional operational space is considered here. A mathematical description of
the desired path is considered first. The desired path is geometrically specified as a continuous curve in operational space.
It is not associated with a time variable. In the minimumtime trajectory planning problem, this type of path is often
parameterized by a path parameter [4]-[6]. A position of apoint on the path is represented as a vector function of a scalar02784046/94$04.00 0 1994 IEEE-”.-.

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 41, NO. 1, FEBRUARY 1994

26

%
A

Path coordinates: Cylindrical coordinates_--x2

q(P> = a(k1, 2 2 , S I T >= [x1 cos ( u s ) ZO, 5 1 sin ( u s )

+

S

(n=3)

+ yo, 221.

The desired path is represented as

Fig. 1. Path coordinate system.

z = [a,%ITparameter. When the operational space is n-dimensional, apoint q E Rn on the path is represented as
so I s I Sfq = q(s),(1)

where s is the path parameter. q(s0) is the starting point of
the path and q ( s f )is the final point. s can be considered the
distance along the path. Since s is a scalar value, this method
can represent a point only on the path itself.
The path tracking control considered here is one in which
the feedback control makes the manipulator return to the path
if the manipulator deviates from the path due to disturbances.
Therefore, points which are not on the path as well as points
on the path should be represented. Furthermore, the tracking
error should be measured. We propose the concept of pathcoordinates [9], which is an extension of path parameter.
A curvilinear coordinate frame is defined in the operational
space. The coordinates are composed of a component s along
the path and components, 21, . . . ,5,- 1 which are normal tos. These coordinates are called path coordinates (Fig. 1). A
point p E R" represented in terms of the path coordinates is

The desired path is represented as
2 = q(constant),

so 5 s 5 Sf

(3)

= b o , Z0IT

in terms of the path coordinates.
Note that there exist many path coordinate systems for one
desired path. In the case of this example, spherical coordinates
which have an origin at the center of the desired path can also
be path coordinates.
B. Equation of Motion
The equation of motion of the manipulator in joint spacecan be written as follows:q e ) B + b(e, 4) =

(6)

where

q e , 4) = h(o, 4) + re + g(e)8 E W": joint angle vector
E Rn: joint torque vectorg ( 0 ) 5 W": gravity torque vectorh(0, e) E W": Coriolis and centrifugal torque vector
M ( 0 ) E WnX": inertia matrixr E Rnxn: viscosity friction matrix.
T

The equation of motion is rewritten in terms of path coordinates p defined in Section 11-A. From (5), the operational
coordinates q and the path coordinates p are related as q =q ( p ) . The operational coordinates q is calculated by forward
kinematics, q = q(0). Whenin terms of the path coordinates. The desired path is also
represented asq = q.([z$,)'ISso

I s I Sf

the Jacobian matrix, J ( 0 ) E R"'"in terms of the operational coordinates. Equation (4) is extended to all the points in the operational space. A point q in
the operational space is represented as a vector function of thepath coordinates p:(5)= q(P>*of path coordinate p for(4) joint coordinates 0 is represented as
J = -aP
=

J-~J

ae

"

Consequently, p and 8 are related as(7)

p = JB.

When (7) is differentiated with respect to time, we obtain

Equation (5) represents the coordinate transformation from thep = Je+Je.
(8)path coordinate space to the operational space. The motion on
If J is nonsingular,
the path is of the s component only and has one degree of
freedom. It corresponds to motion on the straight line 2 = Zd
4 = J-'(# - j b ) .
(9)in the path coordinate space. It means that the z components ofHere, p , M ,b, and H
J-' are partitioned as follows:
the path coordinates remain constant Zd, irrespective of time if
the manipulator is on the path. z- Zd represents the deviationof the point p = [zT .IT from the desired path z = Zd.
Example: Operational coordinates: Cartesian coordinates(n = 3).
H = [ H , $1".
(10)
Desired path: A circle of radius T O , centered at [ZO, yo, ZO],
n-1
parallel to the xy plane
When (9) and (10) are substituted in (6),we obtain+q ( s ) = [ T COS(WS)~xO,TO sin(ws)+ YO, ZO].
. ..

mi(H,j;+h,i-HjB)+b;

=

~

i

( i = ~ , . . * , n )(11).

21

ARAl et al.: PATH TRACKING CONTROL OF A MANIPULATOR

The equation of motion (11) relates accelerations of pathcoordinates to the torque of each joint actuator.

III. MINIMUM-TIMETRAJECTORY P L A N " G
An off-line algorithm for minimum-time trajectory planning[4]-[6]is reviewed in this section. A minimum-time trajectoryis planned for the case when the torque of the joint actuators islimited. The planned trajectory is used as a nominal trajectory,
and the path tracking control in the following sections isapplied to it. The torque limit is represented as a domain T
in torque vector space Rn:
T

ET,T = { T 17Yn5 T; 5 T?

(2 = 1 , .

= (mihS)-'(T;- m;hsi- b;)

Fig. 2. 'bo-degree-of-freedom manipulator.

. . ,n)}. (12)From (3), zcomponents of the path coordinates are constantwhen the manipulator moves along $e desifed path. Thus,
i: = 0 and k = 0 on the path. Since H J = -HJ and p = JO,
acceleration along the path isS

L = 0.3m(i = 1,- * * ,n)Fig. 3. Desired path.

(13)

from (1 1). The maximum value sqs and the minimum valuesqinof the path acceleration s for the available torque of eachjoint are obtained when the torque in (13) is limited according
imax]
for the wholeto (12). The admissible region of s, [smin,manipulator is obtained as a product of admissible regions foreach joint torque limit:

-

0
0.0

0.5

,,o Time (sec)(a) Path vclalty

nrqin,
n[gmin, imax] =

qnx]

i=l
I
Time (sec)

0.5

0.0

(bJTaqus I

z2
Note that the range of the region [smin,smax]depends on thepath velocity i.If the velocity is too large, the admissible
region of s cannot exist and the motion along the path
27
0.0
0.5
,:o Time (sec)is impossible. It determines the admissible velocity. The
Toquc 2minimum-time trajectory is composed by connecting maximudminimum acceleration trajectory segments so as to obtain Fig. 4. Minimum-time trajectory. (a) Path velocity. (b) Torque 1. (c Torquemaximum velocity without exceeding the admissible velocity. 2.
An example of the trajectory planning for a two DOF
horizontally articulated manipulator (Fig. 2) is shown. The
Iv. PATH TRACKING CONTROL
joint torque limit is -2 5 7 1 5 2, -1 5 7 2 5 1 ( N . m).
In the trajectory planned by the algorithm in Section 11, theThe desired path is a circle (Fig. 3; center: [0.4, 01, radius:
0.1 m). In this case, the path coordinate system is a polar torque of at least one joint is saturated. Under conventional
coordinate system whose origin is at the center of the path. feedback control, it may be possible that the manipulator
The s component along the path is angle a, and the 2 can deviate from the desired path. Path tracking control
component normal to the path is radius T . In the planned which takes into account torque saturation is proposed in thistrajectory, the velocity d! along the path (Fig. 4(a), bold line) section. In this method, the tracking capability is achieved by
is obtained by switching maximum acceleratioddeceleration independently controlling the zcomponents normal to the path
without exceeding the admissible velocity (Fig. 4(a), thin line). and the s component along the path. Since the characteristics
One of the joint torques is always saturated [Fig. 4(b), (c)]. of a minimum-time trajectory (e.g., bang-bang joint torques)
Therefore, a conventional feedback control cannot be used.
are not used, this method can also be applied to the nominal
In this section, the equation of motion is represented in terms trajectories which are not minimum time.
of the path coordinates. However, (13) can also be rewritten as
The equation of motion in terms of path coordinates is
S = (miq')-'(T. - mifli2 - b i )(i = 1,.. . ,n>
M ( H z Z hss- H j b ) b = T .
(15)since h, = &/ds = q' on the desired path. Thus, thisThis equation is rewritten as follows:
expression is equivalent to the expressions in [4]-[6] using
the path parameters.
Mzk mss = 7
(16)
N

(c)

+

+

+

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 41, NO. 1, FEBRUARY 199428

domain U by the inertia matrix M . For example, T is arectangular prism and U is a parallel hexahedron for n = 3.
2 = zc is a straight line parallel to the 3 axis in the acceleration
M=MH, Mz=
space. zc can be realized when z = 2c intersects with domain
U.Intersection between U and z = Zc means an admissible
region of S.
i, = b - M j d ,On the other hand, if U and 2 = zc do not intersect,
the admissible region of S does not exist and 3c cannot be
obtained. It is necessary to limit He and the actual acceleration
30 should be determined. It is desirable that Za is as close to
The joint torque limit iszCas possible; thus, $0 is determined so that 120 - &I2 is
T = { f I ?Fin 5 ?a 5 ?Fax ( i = l , . . . , n ) } minimum. Projection 0' of domain U on the plane S = 0
f E(18) is considered. 20 should belong to 0' on the plane S = 0.
where
Therefore, 20 is chosen as the closest point to SC in the region
U'.20 is the closest acceleration to zc which can be achieved
?-,mm
=
,y
&.
f i p n = , p n - &,(19)by the available torque.
The domain T is moved parallel to T by b in the torque vector
Equation (20) is not valid since 2 # zc. However, it
space.
does not directly mean that the manipulator deviates from thedesired path. When one of xi, xi, and xi - X d i is positive(negative) and the others are negative (positive), the tracking
A. Control of z Components
error of the ith component is inclined to decrease. If the
The control of the z components normal to the path is given
priority in order to keep the motion of the manipulator on the domain U' includes the origin z = 0, each component of 2
path. The tracking error is suppressed by the following PID can have either a positive or negative value. Then there exists
2 which satisfies the condition describe! above. Since k = 0
feedback:on the nominal trajectory, the domain U includes 2 = 0 near%c = -Kyk + K p ( z d - 2) K i ( ~ -dZ) dt.
(20) the nominal trajectory, and the condition of convergence tothe path can be satisfied.
Zd is the desired position and z is the measured position of
the z coordinates.
is the acceleration to suppress the error.
If acceleration of the z coordinates is 3c,
B. Control of s Component

where

rz.,

I

+

+Ki

I

( ~ -d 2)dt = 0 . (21)Next, the s component is controlled and the acceleratioddeceleration along the path is determined. The path acceleration S, is calculated from the planned nominal velocity,B, nominal acceleration i n ,and measured velocity 9. We used
proposed in [7]:the feedback of i2

Dynamic characteristics of z are determined by the gain
matrices K v ,K p , and K i . z converges to zd and thes, = Sn(s) K,(S,(s)2- 92).
(23)
manipulator tracks the path if K y , K p , and K; are selected
appropriately.
The differentiation of s2 with regard to s is
In fact, it may be possible that the ac_celeration 3c can!ot
ds2 2 dsdB
d2s = 2;.
be generated due to torque limit (Vs, M z z c fisS 9 T).
- - . - = 2ds
dt ds
dt2
The possibility to obtain zc is judged by the geometrical
relation in the acceleration space. If zc cannot be obtained, the The feedback of (23) provides a linear system with regard to
and B2 converges to :9 if 5 = S,.
acceleration is limited to the best value that can be realized i2,
The actual acceleration sa is determined by the torque limit
and the actual acceleration 20 is determined.
The acceleration Bc is substituted in (16), If f i s i # 0,
in the same way as in the case of za. If ji.c can be generated,
the admissible region [Smin, Sma] of S is already determinedS =sa- mzijic).
(22) (see Section IV-A). If jlc is limited, the admissible region
If the torque of (22) is restricted according to (18), the of S is determined by substituting 20 to 2, of (22). In the
maximum acceleration Syxand the minimum acceleration latter case, the admissible 3 is generally a unique value. Since
SFin along the path are determined for the torque limit of each 3 is a scalar value, the closest value to S, in the admissible
joint. If there exists-a common admissible region [Smln, S"]region of S is chosen as 8,. The limitation described above
for all the joints, M z x c + m s S E T for this region. In this means that the acceleration of s is limited in order to realize
case, z c can be obtained and 20 = xc. (If %,; = 0, there is the acceleration $0. In particular, when the admissible region
Pa]for the acceleration zc exists, Zc itself can be
no limit of s for joint i. In this case, zc can be realized when [Pin,f p n 5 Mzzc 5 y a x . )realized even if S, is limited. It corresponds to the case
The inertia matrix k can be considered a linear transforma- when one joint torque is saturated. The tracking capabilitytion from the torque space Rn to the acceleration space R". is guaranteed by (21). Note that sn and 9, are functions ofTorque limit T is transformed to an admissible acceleration s and are not functions of time. The time trajectory of s can

+

+

~

29

ARAI ef al.: PATH TRACKING CONTROL OF A MANIPULATORTorque Space

Acceleration Space(a) Proposed Method

Fig. 5. Limit of accelerations.

(b) Conventional Method

I-

I KinematicsI------'

Fig. 7. Simulation results with joint friction. (a) Proposed method. (b)Conventional method.

Fig. 6 . Path tracking control system.

be different from that of the nominal trajectory when S, islimited. (The planning of s, and B, is based on the shape ofthe desired path and the dynamics of the manipulator, which
are functions of s. Consequently, S, and 4, can be represented
as functions of s.)
Fig. 5 illustrates the limit of the accelerations when n = 2.
can be regarded as a linear transformation
The matrix
from the torque vector space to_the acceleration vect2r space.
The admissible torque domain T is transformed by M to theadmissible acceleration domain U . Line segment U', which
is a projection of U on the ? axis, represents the admissible
region of 2 . If xc obtained by feedback (20) is outside U',
the actual acceleration is limited to 5,. In this case, S, isdetermined uniquely. If 5c is inside U', S, is limited within
the admissible region of S.
The joint torque can be calculated by substituting the accelerations Za and S, in the equation of motion (16). (Generally,
the torque is already calculated by previous procedures.) Fig.
6 illustrates the control system. All the inputs of the system
do not depend on time. zd and the torque limits are constantvalues. S, and i, are functions of s. Therefore, this control
system can stretch or shrink the time axis in order to achieve
path tracking.
As the time to complete the path tracking may be different
from that of the nominal trajectory, the end of the tracking
should be watched:

I

0.0

0.5

1.0 Time(bJTorqw I

1

24

0.5

0.0

I:o Time (sec)I(cJ Torque 2

Fig. 8. Simulation results with joint friction. (a) Path velocity. (b) Torque1. (c) Torque 2.

++[Joint 1: 0.2 sgn(4) 0.48 ( N . m), Joint 2: 0.1 sgn(8) 0.24(N.m)] were applied to the joints as disturbances. Friction was
not considered in the dynamic model used for the trajectory
planning algorithm and the controllers. Fig. 7 shows thesimulation results. Although the manipulator deviates fromthe desired path in the case of the conventional method [(b),s - sothe computed torque method in operational space], tracking
X = -.
(25)
Sf - s oerror in the case of the proposed method (a) is small. Fig.
The value of X is monitored during the path tracking. If the 8(a) shows the velocity along the path for tracking by the
manipulator is on the path, 0 5 X 5 1. The path tracking proposed method. In the result, the velocity is slowed downin comparison with the nominal trajectory (Fig. 4) and thecontrol is terminated when X = 1.
tracking is achieved. However, the joint torques (b), (c) are
V. SIMULATIONS
almost saturated and the capacities of the actuators are fully
Path tracking simulations were carried out with a two DOF utilized.
In the next simulations, the initial position is not on the pathhorizontally articulated manipulator (Fig. 2). A minimum-time
trajectory (Fig. 4) was planned as the nominal trajectory for a (Fig. 9). The initial error ( T - ~ d ) is 0.015 m. The conventionaldesired path of Fig. 3. Viscosity friction and coulomb friction method (b) expands the error due to the torque limit. In theIEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 41, NO. 1, FEBRUARY 199430

ponent along the path are independently controlled considering
the torque limit. Since the inputs to the control system are notfunctions of time, path tracking can be achieved by modifying
the time axis of the nominal trajectory. In simulations witha two-degree-of-freedom manipulator, tracking error is well
suppressed even if disturbances or initial errors exist.
Presently, we plan to apply this method to a real manipulator and to confirm the tracking ability experimentally. Thismethod depends on the dynamic model of a manipulator, andit requires real-time calculation of dynamics. We are also
investigating methods to deal with model variations and to
reduce calculations.

(a) Proposed Method

ACKNOWLEDGMENT(b) Conventional Method

Fig. 9. Simulation results with initial error. (a) Proposed method. (b)Conventional method.

The authors would like to express their appreciation to
Dr. N. Oyama, Former Director of the Robotics Department,
and Dr. T. Nozaki, Director of the Robotics Department,
Mechanical Engineering Laboratory, for their constant support;
and the members of the Biorobotics Division and the Cybernetics Division, Mechanical Engineering Laboratory, for theirassistance. The authors would also like to thank Prof. B. Roth,Stanford University, for polishing the English in this paper.
REFERENCES

-3 ’

0.0

0.5

Time (sec)(b)Torque I

Z * l ......................................
5

,.

I

8 o-~”~+/#-.

N

E.,..........................................

71

0.0

0.5

I

I:o Time (sec)1‘) Torque 2

Fig. 10. Simulation results with initial error. (a) Path velocity. (b) Torque1. (c) Torque 2.

proposed method (a), the error is suppressed quickly, and themanipulator can subsequently track the path. By shifting thetime axis, the velocities and torques (Fig. 10) after the error
suppression nearly coincide with the minimum-time trajectoryof Fig. 4.
When the feedback gain is increased, the tracking error
increases by the conventional method since the difference
between the desired torque and the actual torque grows inboth simulations. On the other hand, the tracking error by
the proposed method decreases when the feedback gain is
increased.
VI. CONCLUSIONS
A feedback control method for path tracking which takes
the torque saturation into account is proposed. The equationof motion of the manipulator is described in terms of the pathcoordinates. The components normal to the path and the com-[ l ] H. P. Geering, L. Guzzella, S. Hepner, and C. H. Onder. “Time-optimal
motions of robots in assemby tasks,” IEEE Trans. Automat. Contr., vol.
31, no. 6, pp. 512-518, 1989.
[2] V. T. Rajan, “Minimum time trajectory planning,” in Proc. IEEE Inc.
Cont Robotics Automation, 1985, pp. 759-164.
[3] G. Sahar and J. M. Hollerbach, “Planning of nunimum-time trajectories
for robot arms,” Inc. J. Robotics Res., vol. 5 , no. 3, pp. 90-100, 1986.
[4] K. G. Shin and N. D. McKay, “Minimum-time control of robotic
manipulators with geometric path constraints,” IEEE Trans. Automat.
Contr., vol. AC-30. no. 6, pp. 531-541, 1985.
[5] J. E. Bobrow, S. Dubowsky, and J. S. Gibson, ‘Time-optimal controlof robotic manipulators along specified paths,” Inc. J. Robotic Res., vol.
4, no. 3, pp. 3-17, 1985.
[6] F. Pfeiffer and R. Johanni, “A concept for manipulator trajectory
planning,” IEEE Truns. Robotics Automation, vol. RA-3, no. 2, pp.
115-123, 1987.
[7] 0. Dahl and L. Nielsen, “Torque limited path following by on-line
trajectory time scaling,’’ IEEE Trans. Robotics Automation, vol. 6, no.
5, pp. 554-561, 1990.
[8] H. Y. Tam, “Minimum time closed-loop tracking of a specified path by
robot,” in Proc. 29th IEEE Con5 Decision Contr., 1990, pp. 3132-3137.
[9] H. Arai, K. Tanie, and S. Tachi, “Dynamic control of a manipulator with
passive joints in operational space,” IEEE Trans. Robotics Automation,
vol. 9, no. 1, pp. 85-93, 1993.

Hirohiko Ami (M’91) was born in Tokyo, Japan,
on July 9, 1959. He received the B.E. and Ph.D.
degrees in measurement and control engineeringfrom the University of Tokyo in 1982 and 1993,
respectively.
From 1982 to 1984 he was with Honda Engineering Company, Saitama, Japan. He joined the
Mechanical Engineering Laboratory, Ministry of
Intemational Trade and Industry, Tsukuba, Ibaraki,
Japan, in 1984. He is currently a Senior Researcher
at the Biorobotics Division of the Robotics Denartr ment. He was a Visiting Scholar at the Mechanical Engineering Department,
Stanford University, CA, from March 1993 to February 1994. His researchinterests include dynamic control of manipulators, and man-machine interface
for teleoperation.
-

31

AFUl et al.: PATH TRACKING CONTROL OF A MANIPULATOR

Kazuo Tanie (M’85) was bom in Yokohama, Japan,
on November 6, 1946. He received the B.S., M.S.,
and Dr.Eng. degrees in mechanicalengineeringfrom
Waseda University, Tokyo, Japan, in 1969, 1971,
and 1980, respectively.
He joined the Mechanical Engineering Laboratory, Ministry of Intemational Trade and Industry,
Tsukuba, Ibaraki, Japan, in 1971, where he is currently the Director of the BioroboticsDivision of the
Robotics Department. He has also been a Professor
in the Graduate School of the Universitv of Tsukuba
since 1992, and was a Visiting Scholar at the Biotechnolog; Laboratory,
University of Califomia, Los Angeles, from August 1981 to August 1982. His
research interests currently relate to tactile sensors, control for multifingered
hands, force control for robotic arms, virtual reality and its application totelerobotics, and the practical application of robotics.
Dr. Tanie is a member of the Board of Directors of the Robotics Societyof Japan and the Society of Instrumentation and Control Engineers. He is a
mmber of the Advisory Board of the Society of Biomechanisms.

. .

Susumu Tachi (M’82) was born in Tokyo, Japan,
on January 1, 1946. He received the B.E., M.S.,
and Ph.D. degrees in mathematical engineering andinformation physics from the University of Tokyoin 1968, 1970, and 1973, respectively.
He joined the Faculty of Engineering of the
University of Tokyo in 1973, and in 1975 joined
the Mechanical Engineering Laboratory, Ministry ofInternational Trade and Industry, Tsukuba Science
City, Japan, where he served as the Director of
the Biorobotics Division. In 1989 he reioined the
University of Tokyo as an Associate Professor. Since 1992 he-has beena Professor in the Research Center for Advanced Science and Technology(RCAST) of the University of Tokyo. From 1979 to 1980 he was a Japanese
Government Award Senior Visiting Scientist at the Massachusetts Institute ofTechnology, Cambridge. His scientific achievements include intelligent mobile
robot systems for the blind, called Guide Dog Robot (197&1983), and anational large-scale project on advanced robotics, especially advanced humanrobot systems with a real-time sensation of presence, known as Tele-Existence(1983-1990). His present research covers robotics information systems, teleexistence, and virtual reality.
Dr. Tachi is a Founding Director of the Robotics Society of Japan, and is
a Fellow of the Society of Instrument and Control Engineers. Since 1988 hehas been Chairman of the IMEKO (IntemationalMeasurement Confederation)
Technical Committee on Robotics.

. ,.