Proceedings of the 2003 IEEERSI
Inti. Conference on Intelligent Robots and Systems
Las Vegas. Nevada . October 2003
Estimation and modeling of the harmonic drive transmission in
the Mitsubishi PA-IO robot arm*
Christopher W. Kennedy and Jaydev P. Desai’
Program for Robotics, Intelligent Sensing and Mechatronics (PRISM) Laboratory
3141 Chestnut Street, MEM Department, Room 2-115
Drexel University, Philadelphia, PA 19104, USA
Email: {cwk,desai}@coe.drexel.edu
Abstract
The purpose of this paper is to present our results in
developing a dynamic model of the Mitsubishi PA-IO
robot arm for the purpose of low velocity trajectory
tracking using very low feedback gains. The novelty
of this research is therefore the development of a
systematic algorithm to extract the model
parameters of a harmonic drive transmission in the
robot arm to facilitate model-based control. We
have chosen the elbow pitch joint (joint 4J of the PA-
I O robot arm for estimation and modeling purposes.
We have done several experiments to identify the
various parameters of the harmonic drive system.
We conclude with a sample trajectory tracking task
whereby the feedback torque required to do
trajectory tracking with and without the parameter
identification of the HDT is significantly different.
1 Introduction
Accurate modeling of the inherent dynamics of a
robot manipulator is essential in many manipulation
tasks. Through careful modeling, low feedback
gains can be used along with a feedforward model
to accurately follow a desired trajectory for tasks
requiring low interaction forces while manipulating
and interacting with objects in the environment.
The purpose of this paper is to present our results
in developing a dynamic model while taking into
account the inherent dynamics of the Mitsubishi PA-
10 robot arm for the purpose of low velocity
trajectory tracking using very low feedback gains.
The PA-IO is ideal for precision manipulation tasks
due to the backdrivability, precise positioning
capabilities and zero backlash afforded by its
harmonic drive transmission. However, the
compliance and oscillations inherent in harmonic
’This material is based upon the wolk supported by the National
Science Foundation grants EIA 0079830. and CAREER award
IIS 0133471 and the American Healt Association grant
m60368u.
‘Corresponding author
drive systems make the development of an accurate
dynamic model of the robot extremely challenging.
The novelty of this research is therefore the
development of a systematic algorithm to extract the
model parameters of a harmonic gear driven
transmission in the Mitsubishi PA-IO robot arm. This
robot is significantly used in research laboratories [ l -
31 and we believe that this paper is the first of its kind
to address the transmission modeling and
implementation on the Mitsubishi PA-IO robot arm in
a research environment.
There is a substantial body of previous research in
the area of modeling harmonic drive systems 14-61,
Tuttle [4] presents an excellent overview of modeling
and parameter identification of harmonic drive
systems and Kircanski 151 provides a detailed analysis
of the nonlinear behavior of harmonic gears due to
compliance, friction and hysteresis. Although these
research efforts provide significant insight into the
physical phenomena that characterize harmonic drive
behavior, all of their experimental work has been
periormed on customdesigned. elaborate test-beds
that allow direct measurement of many important
system parameters such as compliance and
kinematic transmission error. In addition, many
control schemes for harmonic drive systems rely on
torque sensors mounted directly to the transmission
components [7-91. To our knowledge, there is no
published work to date describing an efficient means
of modeling, parameter identification, and control of
the HDT in the Mitsubishi PA-IO robot arm.
This paper is organized as follows: in section 2, the
PA-IO system is described, including the control
system architecture, section 3 offers a brief overview
of harmonic drive transmissions along with a model of
harmonic gearing, section 4 outlines our methods for
experimentally determining the necessary parameters
for model-based control of the PA-IO robot arm,
section 5 details our experiments designed to verify
the effectiveness of our model, and finally in section 6
concluding remarks are presented.
0-7803-7860-1/03/$17.00 @ 2003 IEEE 3331
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2 The Mitusubishi PA-IO Robot
The Mitsubishi PA-10 robot arm is a 7 degree+f-
freedom robot arm with an open control architecture
and is manufactured by Mitsubishi Heavy Industries
(see Figure la). The four layer control architecture
is made up of the robot arm, servo controller,
motion control card, and the upper control
computer. The host computer runs the QNX real-
time operating system and achieves communication
rates of up to 700 Hz with the robot servo driver
through the ARCNET motion control card.
ARCNET is a token passing LAN protocol
developed by Datapoint Corporation.
The robot joints are actuated through three-phase
AC servo motors and harmonic drive gear
transmissions. Joint positions are measured
through resolvers at the joint output axis, with a
resolution of 0.000439'' over +/- 3 output
revolutions.
Control of the robot can be achieved in either
'Velocity mode' or 'Torque mode'. In 'Velocity
mode', the desired velocity for each joint is sent to
the servo driver from the host computer. A high-
gain digital PI feedback loop running at 1538 Hz on
the servo driver then controls the joint velocity. In
Torque mode' the desired joint torque (in this case
the motor toque constant times the motor current,
before conversion by three-phase), is sent to the
servo driver. The expression for communicating a
desired torque to the robot arm through the servo
driver in this system is given by:
T m =kT *I=Td'(O.OOINm/dig) (1 )
where T,,, is the motor torque, kr is the motor torque
constant. I is the motor current, and Td is the
desired torque written to the servo driver in the form
of a 2 byte integer.
3 Harmonic Drive Systems
Harmonic drive gears, also called strain-wave
gearing, were developed by C. Walton Musser in
the 1950's primarily for aerospace applications.
They are compact, light-weight, and have torque
ratios between 30:l and 320:1, making them ideal
for robotics applications. Harmonic drives are
composed of three components: the wave
generator, the flexspline and the circular spline (see
Figure 1). The wave generator is an elliptical ball
bearing assembly and is nested inside the
flexspline. The teeth on the nonrigid flexspline and
the rigid circular spline are in continuous
engagement. Since the flexspline has two teeth fewer
than the circular spline, one revolution of the input
causes relative motion between the flexspline and the
circular spline equal to two teeth. With the circular
spline rotationally fixed, the flexspline rotates in the
opposite direction to the input at a reduction ratio
equal to one-half the number of teeth on the
flexspline. The displacement, velocity and torque
relationships between the transmission elements in
the ideal case are therefore given by:
1
( N + l ) - - N
T --Tfs
1
Twg =-
where N is the transmission ratio, e,, is the wave
generator angle, 9- is the circular spline angle, BtS is
the flex spline angle, w,, is the wave generator
angular velocity, wFI is the circular spline angular
velocity, wtS is the flexspline angular velocity, Ty is
the wave generator torque, T, is the circular spline
torque, and Te is the flexspline torque. In general
either the flexspline or the circular spline can be fixed.
However, in our robot the circular spline is fixed while
the flexspline rotates with the joint.
WLe-m Fdlpl" cm!Lsplm
Figure 1. Harmonic Drive Components.
The flexibility inherent in the harmonic drive systems
provides advantages such as zero backlash due to
natural preloading. However, there are also several
disadvantages such as nonlinearity due to friction,
alignment error of the components, and transmission
losses due to compliance in the system. All of these
were found to be critical in the modeling of the
Mitsubishi PA-10 robot arm.
In the following subsections we will describe in detail
our methodology for estimation and modeling of the:
a) friction in a HDT, b) torsional stiffness, and c) the
kinematic transmission error. This will be followed by
a model of the HDT for the Mitsubishi PA-10 robot
arm.
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3.1 Friction In Harmonic Drives
Because we are primarily concerned with
controlling the robot at low velocities, we model
friction in the robot joints using a Stribeck curve at
low velocities and a linear relationship at high
velocities. The expression for this model is given
by:
Td(6) = Fo +be + Fce -(6N,)2 (5)
where Tvl is the viscous friction torque, F., F, and V,
are the Stribeck coefficients, 0 is the rotational
velocity, and b is the viscous friction term.
Friction in harmonic drives is strongly position
dependent due to the kinematic error. Therefore,
we model coulomb friction as the torque required to
maintain a very slow velocity over the entire range
of motion of the robot.
3.2 Torsional Stiffness in Harmonic Drives
Harmonic drives exhibit significant compliance
when externally loaded. Experimental tests in [4, 51
indicate that stiffness in harmonic drives increases
with increasing load. In addition, harmonic drives
display a behavior known as 'soft wind-up' which is
characterized by very low stiffness at low applied
loads. The reader is referred to Kircanski's paper
151 for an excellent overview of harmonic drive
torsional stiffness behavior.
Our experimental tests on the Mitsubishi PA-IO
robot arm reveal that wave generator compliance
has a significant effect on the robot dynamics. The
effect of the non-linear stiffness profile of the wave
generator is prevalent when the load on the wave
generator is below a critical point where a dramatic
decrease in stiffness occurs.
3.3 Klnematlc Transmission Error
Harmonic drives display kinematic error that causes
the torque transmission characteristics of the drive
to deviate from the ideal transmission model. This
error is caused by a number of factors such as
tooth-placement errors on both the circular spline
and flexspline. out-of-roundness in the three
transmission components, and misalignment during
assembly (41. The resulting error signature can
display frequency components at two cycles per
wave-generator revolution and several subsequent
harmonics. Based on the above. the error function
including two harmonics of wave generator rotation is
given by:
eem = Alsin(ew + 'pl )+A2sin(29wg + 'p2 )
where are the amplitudes of the sinusoids, pi are
the phase shifts and e,, is the wave generator
position.
This expression is of limited use in our case, because
it is not possible to measure the wave generator
rotation. Furthermore, although kinematic error has a
significant effect on the torque transmission
characteristics of HDTs, we found that compensating
for coulomb friction using the toque required to
maintain slow velocity eliminated almost all the effects
of kinematic error. Therefore, we neglect the effect of
kinematic error in the feedforward implementation of
our model.
3.4 Harmonic Drive Model
We consider the model of the harmonic drive to be
composed only of friction, gravity, externally applied
torque, and stiffness. Because we are concerned
primarily with low-velocity applications, we are
neglecting acceleration terms in our model (thus the
only torque component from the dynamics will be the
gravitational torque).
We have modeled the non-linear expression for
torque transmission in harmonic drives in our robot
arm as:
(6)
J,N = Td (e) + Td (8) + TgP) + Tc (Td, Td, Tg) (7)
where TI. is the input torque, N is the transmission
ratio, T@ is the coulomb friction, Td is the viscous
friction torque calculated using equation (9, T, is the
gravity toque, and T, is the torque used to deform the
wave generator. A schematic for the proposed
control system is shown in Figure 2.
... . . . . . . . "'"""?.??? .......
0
w
I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I /
\
Figure 2. Proposed control system including model
of friction, gravity and compliance of the wave
generator.
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4 Experimental Methods
We have chosen the elbow pitch joint (joint 4) for
analysis because it not only is strongly influenced
by gravity and friction, but also is the most affected
by the oscillations induced by kinematic error.
During normal operation with the high gain
controllers, the joint emits a chattering noise due to
vibration. This joint will therefore benefit most by
accurate modeling. Although we present results for
the elbow pitch joint only, we have obsenred similar
behavior in the other joints with the exception of the
chattering noise.
4.1 Friction Characterization .
Friction is typically characterized by its relationship
with velocity. To determine the friction-velocity
relationship, the robot was commanded to move at
a constant velocity and the mean toque required to
maintain the velocity was taken to be the friction for
that value of velocity. After collecting data for
velocities from 0.005 radls to 0.5 radls in
increments of 0.05 radls, the data was fit to the
friction model in equation (5). To collect this data,
the robot was mounted on the wall as shown in
Figure 3 to negate the effect of gravity. The results
for the dynamic friction are shown in Figure 4.
friction experiments.
To obtain our model of coulomb friction, torque data
was collected in 'Velocity mode' to maintain a
velocity of 0.005 rad/% This data was then fed-
forward in the form of a look-up table to represent
coulomb friction.
L p.6 : A zo4p--_.,. :"r"; I
020 Roll8nalVelocd&dfs)
Figure 4.
Mitsubishi PA-IO robot arm.
Friction data for elbow pitch joint o
4.2 Estlmatlon and modeling of Stiffness,
Gravlty Compensation, and Kinematic
Error
To determine the effect of stiffness and gravity, we
commanded the elbow pitch joint to move at a
constant velocity of 0.01 radls from -90' to + 90' with
weights from 0 to 10 Ibs attached to apply torque to
the drive. The output torque required to track this
trajectory for weights of 0, 5, and 10 Ibs is shown in
Figure 5.
100
E z
?fa
b
50
a
v
=
n -
-50
Posffion @d) ' ' -1 -0.5
Figure 5. Constant velocity experiments with
varying weights.
The effects of gravity and stiffness are clearly evident
from these plots. Note the sharp decrease in torque
when the output torque approaches 0 Nm. This is
due to the 'soft wind-up' behavior of the harmonic
drive system. As the load on the wave generator
decreases, the stiffness if the wave generator
decreases, hence more torque output is required to
sustain motion.
We determined the torque used to deform the wave
generator during the motion in Figure 5 by equating
the output torque during the motion with the right side
of equation (7). The friction torque for these motions
is nearly identical to the coulomb friction, hence we
can eliminate friction from equation (7) by subtracting
the coulomb friction torque from the total toque (as
noted previously, coulomb friction torque was
computed from the arm motion at low velocity while it
was attached to the wall). The gravity torque is also
known from the estimated masses of the robot arm
links and the known weights attached to the robot.
We can therefore subtract the gravity torque from the
right side of equation (7). This leaves the remaining
torque equal to the torque used to deform the wave
generator. The successive plots of each step of this
process are shown in Figure 6.
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80, I
. .
Figure 6. Steps in model identification process.
To verify that this approach is valid, the stiffness
torque calculated as described above was plotted
as a function of the external torque (gravity plus
mean coulomb friction) for each of the three weight
trials in both the positive and negative directions
(Figure 7).
Figure 7. Stiffness torque as a function of external
torque for three weight trials in positive and
negative directions.
As can be observed from Figure 7, the model of
stiffness torque as a function of external load
appears to be fairly repeatable. The region where
the torque decreases or increases sharply due to a
rapidly changing stiffness coefficient seems to be
modeled effectively. We model this function as
composed of three linear regions for each direction.
For the feedforward implementation, we use the
data collected in the previous experiments to
generate a relationship between the external load
and the input torque required to generate the
necessary output torque to compensate for the
load.
5 Results
To verify our model, we fed forward the computed
torques to track a trapezoidal velocity profile at 0.1
rad/s. The gains were set to 0.5 N and 0.15 Ns for
the proportional and derivative gains respectively.
Velocity measurements were taken by differentiating
the position measurements and filtered using a fifth
order Bunerworth filter with a cutoff frequency of 200
Hz. This filter was chosen because it was empirically
determined to give the best results. The feedback
toques generated while tracking the desired
trajectory are shown in Figure 8 for the low gain
implementation with the feedforward model, and for
the high gain implementation with no feedforward
model. The magnitude of the feedback torques is
approximately 1 Nm in the region of high wave
generator stiffness and approximately 6 Nm in the
region of low wave generator stiffness for the low gain
feedforward case.
- 0 5 0 0.5 1 1.5 ’
PwIion [nd)
Figure 8. Feedback torque results for low gain
controller with model based control and high gain
controller without model based control.
The tracking error for the low gain feedforward case is
shown in Figure 9. The mean tracking error is
approximately 0.02 radians in the region of high wave
generator stiffness, and 0.2 radians in the region of
low wave generator stiffness.
1.51 I
Figure 9. Desired trajectovand actual trajectory of
robot during trajectory tracking experiment.
5 Discussion
Our approach in developing a dynamic model for the
HDT in the Mitsubishi PA-10 robot arm can be
described as follows: 1) we developed a
mathematical model for the HDT accounting for
friction, transmission compliance and gravity,
resulting in equation (6), 2) we characterized friction
in the transmission using constant velocity
experiments, 3) we commanded the to robot move
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along a test trajectory with weights of known
magnitude attached to determine the effect of
transmission compliance, and 4) we determined the
model parameters based on the experimental data
and then fed forward our model to the robot for the
purpose of tr
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