Voluntary Movements for Robotic Control
Chi-haur Wu, Kuu-Young Young, Kao-Shing Hwang, and Steven Lehman
A neuromuscular-like nonlinear
mathematical model for controlling robotic
limbs has been developed. The model consists
of two muscle-reflex models representing a
pair of muscles acting as an agonist and an
antagonist in moving a load. The responses of
the m o d e l have been compared with
experimental data from human wrist movements in order to identify command signals in
the form of a series of rectangular pulses for
effecting the measured movements with two
different loads. These results can be used to
obtain several empirical rules for muscle
modulation control. The proposed control
strategy can also compensate for unexpected
disturbances, which is an essential capability
for compliance control.
interface between limb and environment.
Modeling the spring-like behavior of the
neuromuscular system, while the limb is in
contact with the environment, allows successful impedance control to provide compliance
for a robot. The concept of flexion and extension in muscular system is also adopted by
Jacobsen etal. [3] to control manipulator links
with two tendon-driven actuators. Their control algorithms in position and force control
showed good results experimentally. The suc-
cess of these efforts indicates that a better
design of robotic compliance control may
benefit from modeling the mechanisms of
biological limbs.
Results from neurophysiological studies
[4],[5] strongly suggest that nonlinear dynamics in the neuromuscular system have a substantial contribution to superior adaptability
and facile performance in limb control. The
nonlinear features of the neuromuscular system have been simulated and studied through
Industrial Robot Compliance
After two decades of designing active
compliance controllers, the performance of
industrial robots is still crude in providing
compliance [I]. This deficiency remains one
of the major problems limiting the scope of
robotic applications. It is widely recognized
that the primate limb has much superior performance for delicate, skillful maneuvers,
especially in adaptability to different loads
and capability to execute compliant tasks. The
design of a good compliance control for
robotic applications could emulate the compliant capability of the neuromuscular system.
However, few developed schemes for robotic
compliance control take advantage of findings
in biological limb research. Hogan's impedance control [ 2 ] relates to the spring-like
Presented at the I991 IEEE International
Conference on Robotics and Automation,
Sacramento, CA, April 7-12,1991.This work
M'US supported by ONR Contract N00014-88K-0339. Chi-haur Wu and Kao-Slzing Hwang
are with the Department of Electrical Engineering and Computer Science at
Northwestern Universiry,Evanston, IL 60208.
Kuu-Young Young is currently with the
Department of Control Engineering, National
Chiao Tung Uni\,ersio, Hsinchu, Taiwan.
Steven Lehnian is currently with the Department of Physical Education and the Bioengineering Graduate Group, University of
California,Berkeley, CA.
0272-1708/92/$03.0001992IEEE
8
/€€E Control Systems
~~
w
w
Spindle-like
I
,
,
+
&
producing opposing torques on a hand. The
hand rotates about the wrist with a single
degree of freedom, and the moment arms of
the muscles are fixed, so "force" will be substituted for "torque" in this description. The
muscle forces f m and extemal forces f w act on
the hand and any loads attached to it, represented by "Load" in the figure, to determine a
wrist position xw. For example if the hand is
assumed to be an inertial loadM with a damping
constant B, the system can be represented as:
xw I
I
rant
T
I
Spindle-like
Fig. I . Neuromuscular-like model for a single joint.
phase-plane analysis [6]. Results showed that
a system with nonlinear dynamics has the
capability of adapting to changes of loads and
to motion constraints, and can bring the movement to graceful termination. Based on these
results, a muscle-reflex model was developed
[7] through fitting a set of experimental data
recorded from two types of involuntary movements. Although the study of involuntary
movements demonstrated that a muscle-reflex
model has appealing features for robotic compliance control, the application of this model
to robotics is not feasible until an understanding of the pattems of motor commands
to the neuromuscular system is developed.
Human beings change control strategies
under different conditions, and consequently
pattems of motor commands may vary. To
understand the relationship between motor
command pattems and control strategies, research has been devoted to finding proper
controlled variables, such as movement time,
speed, and duration of electromyographic
(EMG) activation of muscles, and their
relationship with task variables, such as distance, speed, load, and instruction [8]-[l l].
Many different control strategies and controlled variables have been proposed to account
for the observed phenomena under different
experimental conditions. However, all the
various effects of the different proposed control strategies can be accounted for through
modulating height and duration of motor command pattems.
Here we focus on the characterization and
control of voluntary movements by analyzing
motor command pattems. One approach to
finding motor command pattems corresponding to voluntary movements is to assume some
February 1992
optimality criterion, such as minimum-jerk
[ 121, minimum-time [13], minimum-torque
[15], then to find the command pattern that
minimizes that criterion. Our approach is to
infer the motor commands by minimizing the
difference between force measured during actual movements and force produced by the
neuromuscular models, given the position
trajectories as inputs. In our study of limb
involuntary movements [6], the nonlinear viscosity plays a dominant role in regulating the
behavior of the neuromuscular system. Therefore, this study is based on a nonlinear
neuromuscular-like model. To summarize our
findings, certain empirical rules for muscle
modulation control will be proposed for possible applications in robotic control. Simulations will then be used to demonstrate
compliant control using the control strategies.
Neuromuscular-Like
Two-Muscle Model
In our previous study of involuntary limb
movements [7], a muscle-reflex model was
developed to emulate the response of a single
muscle. Voluntary movements require torques
in two directions (flexion and extension), and
therefore two muscles. Involuntary movements are driven by extemal forces, but voluntary movements are dictated by the central
nervous system (CNS) through motor commands as well as by external forces. Based on
these basic distinctions between voluntary and
involuntary movements, we developed a
neuromuscular-like model with a pair of
agonist and antagonist muscles.
The proposed neuromuscular-like model,
shown in Fig. 1, represents two muscles
where nw and xwrepresent the wrist velocity
and acceleration, respectively. The muscle
force is produced by two muscle-reflex
models, representing an agonist and an antagonist.
Each muscle-reflex model [6],[7] consists
of a simple muscle stiffness mechanism,
driven by the sum of a voluntary command
and a reflex signal produced by a spindle-like
model. Muscles are assumed to be spring-like,
with spring constant K m multiplying three
components to produce force f m as follows:
The first component represents the length-tension relationship of the muscle, as well as
spring-like behavior due to reflex components
not represented in the spindle model. haasoand
ha,", represent lumped neural inputs to the
muscles, where two subscripts "ago" and "ant"
represent agonist muscle and antagonist
muscle, respectively.
A low-pass filter (LPF) is included in each
muscle-reflex model to induce the bell-shaped
velocity profile for the voluntary movement.
It might represent smoothing of motor commands by motor neuron pools. This effect is
represented as follows:
where hb,,, and hb,,, are the neural signals
before filtering, and the function LPF represents a low-pass filter with a time constant of
RC. The time constant RC is about 0.03 to 0.05
s based on the experimental data in [4].
Each k is the sum of a voluntary motor
command C and a reflex signal r generated by
a receptor in the muscle called the spindle:
where ragoand rant are the discharge rates of
the spindle-like models for the agonist and
9
antagonist, respectively, scaled through a
reflex gain coefficient H .
The spindle senses both muscle length and
rate of change of muscle length, and produces
a firing rate reflecting both measurements. The
discharge rate for each spindle can be represented by a nonlinear equation as follows [6]:
demonstrate a sequence of burst pattems: a
first agonist burst, followed by an antagonist
burst, then a second agonist burst. Correspondingly, there are three altemating phases
of force: fiist in the direction of movement,
followed by an opposing force against the
movement, then in the movement direction
again. On the basis of these observations, we
will assume that basic pattems of input motor
were used as templates for identifying pattems
of motor commands through our neuromuscular-like model. Position-time plots are the
dashed curves in Fig. 4. These data were used
as inputs to the muscle-like models in the
command identification.
The experimental data of forces in Fig. 2
and the recorded electromyograms (EMGs)
show that rapid voluntary movements
4
are the internal
wherex,,;/,,,, .&,080, and xpooRc,
position, velocity, and bias position of the
spindle-like model for the agonist. (The antagonist counterparts are subscripted “ant.”)
xR,represents the wrist position; and Kr is the
reflex stiffness and B,, is a scaled damping
coefficient for each spindle-like model.
The template data of involuntary movements used in our previous study [6] were used
to identify all parameters of the model, except
the command inputs. After fitting the transient
responses of four step forces recorded from
the experiments in [6], a set of parameters
were obtained as follows: H = 0.00166, Km =
240 N/m, L = 0.5, Kr = 720 N/m, B, = 950
N/m.(s/m)’’’, xpougo
= xpoa,,,= -0.01 m, and RC
= 0.03 s. After fitting the transient responses
of two ramped stretches, the parameters were
obtained as follows: H = 0.00166, Km = 231
N/m, L = 0.5, K,. = 1150 N/m, Bp = 2200
N/m.(s/m)’”, x,,~,,, = xpo,,., = -0.008 m, and
RC = 0.03 s.
The mean square error obtained from this
neuromuscular-like model is similar to that
from the muscle-reflex model [6]. The inclusion of the low-pass filter does not have
much effect on the response of the involuntary
movement. However, it does smooth the input
from the motor command.
‘\ U
\
\
\
I-
-401
0
.
/
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I
0.2
I
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I
0.6
0.4
time (sec)
Fig. 2. Force-time plots of experimental data: light load (solid line) and heavy load (dashed line).
0.16
0.12
0.08
0.04
0
0
2
4
position (cm)
6
8
Fig. 3. Functional plot between the motor command and the corresponding equilibriumposition.
Motor Command Pattern
Identification
To identify pattems of motor commands,
the experimental data recorded from fast
voluntary movements at the wrist were
studied. The movement distance of the hand
in all cases was 3.0 cm. The instruction was to
move as quickly as possible. Hand position,
force, and electromyograms (EMG) of both
wrist flexors and extensors were recorded
during well-practiced wrist flexions against
two different loads. In each case, the load was
primarily an inertia, of 2.15 or 21.5 kg. Loads
also included a small nonlinear viscous component.
The force-time plots for both the light and
heavy loads are shown as in Fig. 2. These data
10
2.5
1.5
0.5
-0.5
-1.5
0
0.2
0.4
time (sec)
0.6
Fig. 4. Position-time plots of experimental data (dashed lines) and emulated limbpositions (solid
lines)for both light and heavy loads.
0.3
I
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-
-
-
0.1 -
U
2E
-
8
-0.1
E
-0.3 -
5
-0.5
1
1
-
I
- .- - - - - -
I
displayed in Fig. 6 illustrate how motor commands regulate the limb motion. The vertical
dashed lines in Fig. 6 correspond to different
time intervals that the amplitudes of motor
commands change.
By comparing the pulse durations of two
motor commands in Fig. 5, the first two pulse
durations of heavy load, 0.12 s and 0.136 s,
are two times that of the light load with 0.06
and 0.068 s, correspondingly. Though there is
not enough evidence, the ratio of these command durations implicates that human beings
adjust the command duration to adapt to different loads. Depending on the strength of the
limb, this load-duration ratio is determined by
a load-duration function learned by human. In
our experiment data, this load-duration ratio
learned by the subject seems to be two for the
tested two loads. We do not have enough data
to verify this load sensitivity function. Some
researchers [2],[8] used square root of load
ratio as a load-duration function to estimate
the load-duration ratio. However, we believe
that a higher order nonlinear function, such as
cubic root or fifth root of load ratio, may be
more close to estimate the load-duration function of the neuromuscular system and more
efficient for adapting to different loads.
No second set of motor command seems to
be needed for the heavy load. Observe in the
experimental data that at the time the first set
of motor commands is finished, the limb position for the heavy load was over the target
position and converging to the target; on the
other hand, the limb position for the light load
was behind the target position and going away
from the target. This behavior indicates an
empirical rule for artificially generating the
second set of motor command as follows: At
the time the first set motor command finishes,
if the current limb position is behind the target
position and going away from the target, a
second set motor command should be
generated to correct the movement. The es-
I
I
I
I
I
Fig. 5. Motor commandsfor heavy load movement (dashed line} and light load movement (solid
line). Both agonist commands startfrom time 0.
commands will be in the form of combinations
of rectangular waves with different heights or
durations, plus a step command at the end of
the motor command to maintain a final equilibrium position for a voluntary movement.
Mapping of Motor Command and
Equilibrium Position
The mapping between different stable
wrist forces and corresponding equilibrium
positions was obtained from a set of loading
and unloading curves [6] generated from our
neuromuscular-like model. These curves represent the nonlinear neuromuscular stiffness.
Fig. 3 shows different equilibrium positions
after applying various step motor commands
for a duration of 0.7 s. The period of 0.7 s is
chosen to match the settling period of experimental data shown in Fig. 4. With this
curve, the corresponding step size of the motor
command pattem can be deduced for maintaining a desired equilibrium position after 0.7 s.
Motor Command Patterns
To identify motor command pattems, experimental position-time data was used as
input to the model. Then, model force-time
trajectories were fit to experimental forcetime data by varying pulse heights and widths
of a train of square pulses in altemate directions. Although the experimental loads included a small unknown nonlinear viscous
component, two constant loads of 2.15 kg and
2 1.S kg were used in our model to simulate
light load and heavy load, respectively. This
nonlinear dynamics will not affect the emulating capability of our model. Actually, this
unknown nonlinear dynamics can be compensated by our model and motor commands.
For the light load, the estimated motor command consists of atrain of five pulses, in altemate
directions, with consecutive durations of 0.06,
0.068,0.06,0.098, and 0.03 s. As for the heavy
February 1992
load, there are two main pulses, in altemate
directions, with durations of of 0.12 and 0.136
s, followed by a small pulse with a duration of
0.03 s. The final step command for either the
light load or heavy load is found to be very
close to 0.0585 This value matches the estimated value from the curve in Fig. 3 for a 3
cm movement after 0.7 s.
Comparing these two types of motor commands, it seems that there are two consecutive
sets of motor commands for the light load,
while only one set for the heavy load. We
assert that to compensate for the overshoot of
the rapid response in the case of the light load,
there is a need of two consecutive motor commands.
The optimal responses by minimizing the
mean-square errors are displayed as solid curves in Fig. 4.The estimated motor command
pattems are demonstrated in Fig. 5. The same
amplitudes of 0.27 for the agonist and -0.41
for the antagonist are generated for both heavy
load and the first set motor command of light
load. In case of light load, amplitudes of the
second set command are also found to be
0.116 and 4.094 for the agonist and the antagonist, respectively. The phase-plane curves
-
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0.6
0.4
L .'
0
-0.2
/
\
\
t
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-0.4
0
I
0.01
I
I
0.02
position (m)
I
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0.03
I
I
0.04
Fig. 6 . Phase-plane plots between emulated position and velocityfor both heavy load (solid line)
and light load (dashed line) movements.
JI
I
I
I
I
I
Fig. 7. Emulated movements of 3 , 4 , and 6 cm for both heavy load and light load: dotted line3 cm movement for light load; middle dashed-dotted line - 4 cm for light load; upper
dashed-dotted line - 6 cm movement for light load; solid line -3 cm for heavy load; larger
dashed line -4 cm for heavy load; and small dashed line -6 cm for heavy load.
timated second set motor command for the
light load shows that the total duration of this
second is the same as the first one. However,
the counter command for the antagonist starts
30-ms later and finished 30-ms earlier than the
agonist command. Interestingly enough, this
lag-time and lead-time are both half of the
duration of the agonist pulse leading the antagonist pulse in the first motor command set,
60 ms. Therefore, a simple rule to determine
the duration of this second motor command
set is also proposed as follows: The total duration of the second set of command is the same
as the first one; however, the agonist command will start T/2 ahead and end T/2later
than the antagonist command, where T represents the duration of the agonist pulse leading
the antagonist pulse in the first motor command set.
On the basis of the above estimated pattems for motor commands, a muscle modulation control (MMC) will be proposed for
controlling our neuromuscular-like model in
the following section.
modulation, the load sensitivity compensation, the second motor command set for compensating overshoot, and the compensation
for disturbances. Based on the identified
motor commands, we propose the following
empirical rules for the generation of motor
commands.
Control Strategies of MMC
Rule A - Final step command adjustment:
For a given equilibrium position, the final step
command will be defined through the mapping of motor commands and equilibrium
points shown as in Fig. 3 . This mapping actually represents the nonlinear stiffness of the
emulated limb.
Rule B - Height modulation: For the same
load, every pulse duration of the motor com-
mand is the same for all distances. However,
command amplitudes of both agonist and antagonist muscles are adjusted proportional to
that of the amplitude adjustment for the final
step command in Rule A, in comparison with
the studied basis movement.
Rule C - Second motor command set: This
second motor command set will be generated,
if at the time the first motor command set
finishes, the current limb position is behind the
target position and going away from the target.
The total duration of this set of command is
the same as the first one. However, the agonist
command will start T/2ahead and end T/2later
than the antagonist command, where T represents the duration of the agonist pulse leading
the antagonist pulse in the first motor command set. (In our studied 3-cm movement, T
is 60 ms in light load.)
Rule D - Load sensitivity adjustment: For
the same movement amplitude but different
loads, pulse durations of motor commands
will be adjusted, without changing heights.
Pulse durations will be adjusted according to
a load-duration function of the emulated limb,
which is estimated from the studied basis
movement with different loads.
Rule E - Disturbance compensation: Disturbance forces such as gravity will cause the
system to miss the target position. To reach
and maintain at the desired equilibrium position, a larger or smaller height for the motor
command is needed depending on the direction of the disturbance. This effect can be
treated as an extra motor command which is
needed for countering the disturbance. The
size of this compensated command can be
obtained via the mapping of disturbance forces and motor commands. Therefore, the correct final step command for the equilibrium
point will be the original final step command
plus the compensated motor command. Since
Muscle Modulation Control (MMC)
To generalize the identified motor commands to any arbitrary movement amplitude,
we will adopt the concept of equilibrium point
control. Under this concept, the height
modulation of motor commands can be applied to generate different movements. The
equilibrium assumption is that the height
modulation is proportional to the movement
amplitude. This modulation needs a studied
movement as a basis. Therefore, to move to
any given final position, the proposed control
strategies include the adjustment of the final
step command size, the command height
12
Fig. 8. A 3-cm movement without (solid line) and with (dashed line) 3.6 N step disturbanceforce,
and after height modulation compensation (dotted line).
/€€E Control Systems
1.2
ments have different sizes of commands (different speeds), both results show that after
hitting the simulated wall, the limb oscillates
first and then stabilizes at the contact point
with the wall. This simulation demonstrates
that a neui-omuscular-like system can adapt to
the transition from position control to force
control without a serious stability problem.
This property is extremely useful for robotic
compliance control.
1JhV
Discussion and Conclusion
0
0.2
0.4
0.6
time (sec)
Fig. 9. Emulated movements hitting a wall at position 1.76 cm with a small motor command of
3 cm (solid line) and a large motor command of 6 cm (dashed line).
t h i s final step command is c h a n g e d ,
amplitudes of the agonist and antagonist will
also be adjusted accordingly, following the
height modulation in Rule B.
Simulation of Movements
According to the proposed control strategies, simulations of moving 3,4, and 6 cm in
0.7 s are executed for an emulated wrist with
the studied light and heavy loads. The 3-cm
movement is the basis movement for the
others. For the same movement, the command
for heavy load will have the same amplitude
as light load, except pulse duration adjustment
according to MMC. According to Rule A of
MMC, the final step commands for movements of 4 cm and 6 cm are determined from
Fig. 3 as 0.076 and 0.1 12, respectively. Based
on Rule B, in comparison with the basis 3-cm
movement, pulse amplitudes of 4-cm and 6cm movements can be determined. For both
movements, amplitudes of the agonist for the
first motor command set are adjusted to 0.351
and 0.513, respectively; as for the antagonist
commands, they are 4 . 5 3 4 and -0.779,
respectively. In case of light load, the second
motor command set is also needed. For the
same 4-cm and 6-cm movements, amplitudes
are 0.15 1 and 0.22 for the agonist, respectively, and -0.122 and -0.179 for the antagonist,
respectively. With motor commands assigned
as the above, position-time plots of 3-cm,
4-cm, and 6-cm movements for both loads are
displayed in Fig. 7.
Simulations of Constrained Motions
For a neuromuscular-like model to be used
for compliant applications, it must adapt to
different constraints, such as gravity disturbances and compliant motions. To simulate
Februaw 1992
the effect of disturbance forces, such as
gravity effect, a simulated joint with the light
load as defined earlier was required to perform
a 3-cm movement with a step disturbance
force 3.6 N imposed from the beginning of the
movement. Fig. 8 shows the position-time
plots of the movements with and without this
disturbance shown as the dashed and solid
curves, respectively. Because of this disturbance, the final position deviates from the
desired target. To compensate for the effect of
this disturbance force, height modulation in
Rule E of MMC is applied. The effect of this
disturbance will change the equilibrium position by 0.76 cm. Therefore, to compensate this
effect, the adjusted motor command is found
to be 0.07 137 from Fig. 3. In comparison with
the original command of 0.0585, the adjusted
ratio is 1.22. Therefore, the height adjustment
ratio for the other parts of motor commands is
also 1.22. The compensated position-time plot
is plotted as the dotted curve in Fig. 8. The
simulation result shows that the disturbance
force can be compensated through a modified
motor command easily. If the disturbance forces are not constant, a more complicated compensating function is needed for modulating
the motor command.
The final simulation is a case that the limb
will hit a wall in the process of a movement.
This condition means that the motion changes
from unconstrained to constrained, immediately. A linear control system would experience a serious stability problem when the
condition occurs at a fast speed. However, our
neuromuscular-like model can adapt to the
environment through the nonlinear damping
property and reduce the effect of this stability
problem. Fig. 9 shows the position-time plots
of the emulated limb hitting a simulated wall
at 1.76 cm with 3-cm movement and 6-cm
movement. Although these two fast move-
Limitations in current robotic compliance
controllers motivate our analysis of voluntary
and involuntary limb movements. Because of
the capability of the biological limb system,
the findings are beneficial to robotic compliance control. Due to different mechanisms
involved in involuntary and voluntary movements, a neuromuscular-like model with two
muscle-reflex models is proposed, which represents an agonist and antagonist pair that can
perform operations for both extensor and
flexor. By fitting the experimental data
recorded from two fast voluntary movements,
basic motor command patterns are estimated
by exploring the inverse relation from responses of limb movements to input commands.
The estimated motor command pattems
consist of separate motor commands for the
agonist and antagonist muscles and afmal step
command for maintaining the equilibrium
position. Based on this command pattern, a
control approach named Muscle Modulation
Control (MMC) with several empirical rules
is proposed for controlling an emulated limb
with the neuromuscular-like model. To
generate different movements, MMC will
regulate heights of motor commands for the
agonist and antagonist muscles and the final
step command for the desired equilibrium
point. It is also demonstrated that the
neuromuscular-like model can adapt to the
transition from position control to force control without a serious stability problem. This
capability is essential in designing a compliance control.
Although the result demonstrated from the
neuromuscular-like model shows promising
applications in controlling limb movements,
the parameters in our model were set up based
on the limited data. There are many remaining
questions: How variable are the identified
parameters among subjects? How sensitive is
the identified motor command to modeling assumptions [ 16]? How sensitive are the system's
behaviors to changes in parameters? How would
closed-loop control of this neuromuscular-like
system compare to the proposed MMC
strategies? Are these MMC strategies valid for
controlling more general loads?
13
Current industrial robots are very accurate,
but poor in compliant control and adaptability.
The neuromuscular system seems not so accurate but very good in tasks requiring compliance and adaptation. Based on the research
presented in this paper, the expectation is that
the design of robotic control may benefit from
modeling the properties of the biological limb.
Before a neuromuscular robot system can be
realized, further research is needed to identify
applications of a such neuromuscular-like
model.
Acknowledgment
The authors would like t o thank the
reviewers for their invaluable comments.
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tegration, and integrated industrial automation. Currently he is involved in designing a CAD-driven,
surgical robot system. During 1985, he was honored
with the Outstanding Young Manufacturing Engineer Award from the Society of Manufacturing
Engineers.
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M.L. Latash, "Organizing principles for single-joint
movements 111. A speed-insensitive strategy as a
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1990.
1121T. Flash, and N. Hogan, "The coordination of
arm movements: An experimentally confirmed
mathematical model," J . Neuroscience, vol. 5, pp.
1688-1703, 1985.
1131 S.L. Lehman and L.W. Stark, "Simulation of
linear and nonlinear eye movement models: Sensitivity analyses and enumeration studies of time
optimal control," J . Cybern. Info. Sei., vol. 4,pp.
21-43, 1979.
1141 T. Flash, "The control of hand equilibrium
trajectories in multi-joint arm movements," Biol.
Cybern., vol. 57. pp. 257-274, 1987.
1151 M. Kawato, Y. Uno, M. Isobe, and R. Suzuki,
"Hierarchical neural network model for voluntary
movement with application to robotics," IEEE ControlSyst. Mag., vol. 8, pp. 8-16, Apr. 1988.
1161 S.L. Lehman, "Input identification depends on
model complexity," in Multiple Muscle Systems, J.
M. Winters andS. L-Y. Woo, Eds. New York:
Springer-Verlag, 1990, pp. 94-100.
Chi-huar Wu is an As-
sociate Professor of Electrcal Engineering and Computer
Science
at
Northwestern University,
Evanston, IL. He received
the B.S. degree in electrical
engineering from National
Taiwan University, Taiwan,
in 1973, the M.S. degree in
electrical engineering from Viginia Polytechnic Institute and State University in 1977, and the Ph.D.
degree in electrical engineering from Purdue
University in 1980. After graduating from Purdue,
he joined Unimation Inc., Danbury, CT. During that
period, his job involved designing robot motion
control algorithms and digital servo systems for
PUMA robots and hydraulic-servo Unimate robots.
Since September 1983, he has been with
Northwestem University. His areas of interest are
robotics, robot accuracy and calibration, neural networks and learning control, motor control of limb
arm and roboticc, compliance and part assembly
strategy, computer graphics and CAD/CAM in-
and Ph.D. degrees in
electrical engineering from Northwestem University in 1987 and 1990, respectively. Between 1983
and 1985, he served as an electronic officer in the
ChineseNavy. Since 1990,he has been an Associate
Professor in th Department of Control Engineering
at National Chiao-Tung University, Hsinchu,
Taiwan, R.O.C. His current research interests include biological control systems, fuzzy control systems, robot path planning, and accuracy analysis.
Kao S. Hwang received
the B.S. degree from the
Department of Industrial
Design, National Cheng
Kung University, Taiwan,
in 1981, and the M.M.E.
degree from Northwestem
University in 1988.
Presently, he is workring
toward the Ph.D. degree in
the Department of Electrical Engineering and Comptuer Science at Northwestern University,
Evanston, IL. His interests include robotics, neural
networks, and CAD/CAM.
Steven L. Lehman was
born in Portland, OR, in
1948. He receivedthe B.S.
degree in mathematics
from Stanford University
and the Ph.D. degree in
biophysics from the
University of California,
Berkeley.He joined the
faculty of the Department
of Physical Education at Berkeley in 1983, and is
also an active member of the U.C. San Francisco/U.C. Berkeley Bioengineering Graduate Group.
His research interests are in neuromotor control,
biomechanics, and neuromuscular fatique.
/E€Contra/ Systems