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Systems Analysis Model Simul, 200?, Vol. 00, No. 0, pp. 114
MODELING AND ANALYSIS OF HYSTERESIS INHARMONIC DRIVE GEARS
RACHED DHAOUADIa,* and FATHI GHORBELb
aDivision of Electrical, Electronics and Computer
Engineering,American University of Sharjah, P.O. Box 26666,
Sharjah, UAE;
bDepartment of Mechanical Engineering, Rice University, 6100
Main Street,Houston, TX 77005, USA
(Received 5 February 2002)
In this article, a mathematical model and its parameter
identification scheme are proposed for harmonic drivegears with
compliance and hysteresis. The hysteresis phenomenon in harmonic
drives is described by anonlinear differential equation
representing the torquedisplacement relationship across the
flexpline of theharmonic drive. The representation is equivalent to
having the combination of nonlinear stiffness andnonlinear viscous
damping. Numerical simulations along with experimental data have
been used to validatethe proposed modeling concept.
Keywords: Harmonic drive gear; Hysteresis; Nonlinear stiffness;
Nonlinear ordinary differential equation
1. INTRODUCTION
Harmonic drives have been designed and used in demanding
industrial andinstrumentation servo systems such as industrial
robots and medical equipment,where they provide high velocity
reduction in a relatively small package permittinghigh torque
amplification with only small motors. Numerous contributions have
beenmade to the intuitive understanding and analytical description
of harmonic drives.However, their inherent nonlinear
characteristics have not been clearly analyzed. Thethree main
nonlinear transmission attributes in harmonic drives responsible
formotion transmission performance degradation include nonlinear
stiffness, friction,and kinematic error. The transmission
compliance and the internal dynamic frictionmechanisms, resulting
in hysteresis curves when torque is plotted against
angulardisplacements, are controversial issues regarding the
primary source of energy storageand dissipation in harmonic drives
[10,11,1315]. The accurate modeling of a totalharmonic drive system
(including the actuator, harmonic drive, sensors, and load)presents
therefore a difficult problem. In much of the literature, the
actuators providingthe drive torques are modeled as pure torque
sources, or as first-order lags. Numerousmodels have been proposed
also to represent either the general system dynamics or
*Corresponding author. E-mail: [email protected]
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ISSN 0232-9298 print: ISSN 1029-4902 online 200? Taylor &
Francis LtdDOI: 10.1080/0232929032000115137
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some aspects of nonlinear friction and compliance effects. The
majority of these modelshave been either too complicated, with
parameters that are difficult to determine, or toosimple, assuming
a linearized model and neglecting the nonlinear effects. The
physicalrealities of the system have therefore limited the
acceptance of these models. There istherefore a need to better
understand the kinematic, dynamic, and transmission proper-ties of
harmonic drive gears, and their interaction with actuators and
external loads.The hysteresis phenomenon has been also studied in
many other areas of engineering.
The most familiar example is the ferromagnetic hysteresis. The
magnetic hysteresismodel admits descriptions in terms of hysteresis
operators [12], or in terms of differentialequations. The former
description is used during mathematical analysis in order to
getexistence, uniqueness, and regularity results. The latter
description is very useful fornumerical computations and
construction of the global model in terms of partialmodels given by
dynamic equations. Bouc [1] used differential equations to model
thehysteresis relationship. His model is based on the variation of
the multivalued sign func-tion. The problem of describing a
material with hysteresis can reduce to that of finding anonlinear
or a piecewise linear function of the input signal v and the output
signal w, sothat w forms a classical hysteresis loop when v is a
sinusoid. The work of Hodgdon [8,9]and Coleman and Hodgdon [5,6]
shows that Boucs model is useful in applied electro-magnetics
because the functions and parameters can be fine tuned to match
experimentalresults in a given situation. Choua and Stromsoe [2,3]
and Chua and Bass [4] alsopresented another general theory of
hysteresis, considering constitutive models thattake the form of
first order differential equations. The main advantages of
theirmodels over existing models is its simplicity and the
constructive procedure availablefor determining the nonlinear
functions describing the model.This article deals with the
mathematical modeling of hysteresis in harmonic drives for
the purpose of developing effective controllers for
electro-mechanical actuators with har-monic drives. Our proposed
approach uses differential equations to model the
hysteresisrelationship, which is resulting from the combined effect
of the nonlinear flexibility of theflexpline and friction. In our
case, position and speed satisfy an Euler-like differentialequation
describing the system dynamics. The representation of the
hysterisis phenom-enon by a differential equation is a useful
approach to describe the overall harmonicdrive system with ordinary
differential equations that are smooth and well posed [1,3].The
problem of describing the harmonic drive hysteresis can reduce to
that of finding
two nonlinear functions of the angular displacement and speed,
one is representingthe nonlinear stiffness and the other the
nonlinear viscous damping, so that the combi-nation of both forms a
classical hysteresis loop when the displacement is a sinusoid.This
article is organized as follows. Section 2 presents the harmonic
drive system and
the experimental setup. Section 3 presents the proposed dynamic
model of hysteresis.The dynamics of the setup including the new
harmonic drive model is given inSection 4 and the parameters
identification procedure in Section 5. The simulationand
experimental results with the model validation, discussion, and
conclusions aregiven in Sections 6 and 7.
2. HARMONIC DRIVE SETUP
The harmonic drive system considered for our analysis is
composed of a motor actua-tor, a harmonic drive gear, and an
inertial load. The harmonic drive gear consists of the
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mechanical assembly of three components: a rigid Circular
Spline, an elliptical WaveGenerator, and a nonrigid flexible spline
or Flexspline, which form together a compacthigh-torque,
high-ratio, in-line gear mechanism. A harmonic drive test apparatus
wasdesigned and built at Rice University as a platform to perform
various types of experi-ments on the harmonic drive and to
characterize the different errors inherent in itsoperation while
preventing any external error component from being imposed [7].The
system is shown in Fig. 1. It has its major axis of motion in the
vertical plane toavoid the radial loading problem. A special design
of vertical support plates and circu-lar steel pipe sections with a
highly stable platform was also used to maintain torsionalintegrity
of the system. Effort was also put into making the linkage joining
the motor,harmonic drive, and torque sensor very rigid. This aspect
is important since the objec-tive was to avoid any torsion in the
system produced by elements other than theflexpline and the
harmonic drive as a hole. The harmonic drive system is driven byan
AC servo motor with a dedicated power supply and controller. The
total systemis controlled with an IBM PC to which it is interfaced
through a DSP board madeby dSPACE [16]. Position feedback from the
motor and the load are provided withhigh resolution optical
encoders. The torque sensor used is a DC-operated noncontacttorque
sensor with a large capacity matching that of the harmonic drive.
The signalsand feedback are processed by the dSPACE board and may
be displayed at the terminalin real-time. The system has also the
ability to store acquired data for later processing.This data will
be loaded into Matlab for further analysis and evaluation. The
programsnecessary for the operation and control of the system were
developed so that the usercould communicate with the system through
a Windows interface and dSPACE.
3. Formulation of the Dynamic Model of Hysteresis
Our approach of modeling consists of postulating the following
mathematical represen-tation relating two variables x(t) and y(t)
[3]:
dy
dt hygxt f yt, 1
FIGURE 1 View of the harmonic drive test apparatus.
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where f(), g(), and h() are real-valued continuous and
differentiable functions withcontinuous first-order derivatives and
satisfying
f 0 > 0, g0 > 00 < h < 1
where the prime denotes differentiation with respect to the
functions argument, and and are finite positive constants.With an
appropriate selection of f, g, and h, Eq. (1) can be designed to
exhibit the
desired nonlinear phenomena of hysteresis. The conditions
imposed on f, g, and hwill insure the existence and uniquences of a
solution of (1) when x(t) is a continuousvariable. This property is
very important, since our objective is to obtain a
reliablehysteresis model that can be integrated easily in the
global harmonic drive model toyield a well-posed set of ordinary
differential equations.The describing characteristic of interest
for our present purpose is the graph of
the torque applied to the harmonic drive flexpline as a function
of the angulardisplacement across the flexpline, relative to a
fixed reference position
1N 2, 2
where 1is the wave-generator position, 2 is the load position at
the end side of theflexpline and N is the reduction ratio. If we
replace the variable x by the torsionaltorque and the variable y by
the angular displacement , Eq. (1) becomes:
d t dt
hg t f t: 3
This equation can be rearranged into the form
t g1 tht
f t: 4
Equation (4) can be interpreted as the mechanical dynamic
equation across theflexpline describing the parallel combination of
a nonlinear torsional spring and anonlinear viscous damping. The
function f() determines the stiffness curve while thefunction g1()
represents the nonlinear dynamic friction as shown in Fig. 2.
FIGURE 2 Proposed mechanical analog of the hysteresis model.
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The validity of the nonlinear model (4) will then be
established. This consists of firstshowing that the postulated
model exhibits the same significant properties as the actualsystem
and then verifying that the model gives realistic responses to one
or more testsignals.
4. FORMULATION OF THE EQUATIONS OF MOTIONOF THE HARMONIC
DRIVE
In order to study the dynamic behavior of the complete harmonic
drive system, themodel of hysteresis will be combined with the wave
generator and load dynamicmodels. The following set of equations
represent the complete model of the harmonicdrive:
J1 1 B1 _1 f _, N
m 5
J2 2 B2 _2 L _, 0 6
_, g1_
h
f 7
1N 2 8
where J1 is the total motor and wave generator inertia, J2 is
the total load inertia,1 is the motor position, 2 is the load
position at the end side of the flexpline, N isthe reduction gear
ratio, B1 and B2 are the viscous damping coefficients at the
motorside and the load side, is the transmitted torque across the
flexpline, m is the drivingtorque applied by the electric motor,
and L is the load torque. f represents a dryfrictional torque
component at the bearings of the wave generator which is
acombination of the necessary torque to initiate motion from rest
(static friction) andthe friction present during stabilized motion
(sticktion).We note that the transmitted torque across the harmonic
drive represents a
nonlinear coupling factor between the motor side and the load
side dynamics.
4.1. Locked Rotor Case
When the output of the drive is locked 2 0, the motor side can
still rotate within alimited angular range allowed by the
flexibility of the harmonic drive gear. Themodel of the setup in
this case becomes
2 0 9
1N
10
J B _ _, n 11
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J N2J1 12
B N2B1 13
n Nm f 14
5. NONLINEAR FUNCTIONS IDENTIFICATION
In order to identify the nonlinear functions f, g, and h, a pair
of waveforms {(t), (t)}must be measured. If (t) is selected as the
excitation signal and d/dt is known, thenEq. (4) is reduced to an
algebraic relationship to find the output signal (t). To beable to
perform the proposed experiments, the experimental setup shouldbe
configured in a way to allow the angular displacement to be
manipulated as theexcitation signal. This consists in having the
load side of the harmonic drive lockedand then forcing the desired
angular displacement through a feedback position controlloop. The
position control loop gains are adjusted so as to get the desired
accuracy ofthe following reference waveform ref.The analysis will
proceed by carrying out a sequence of well instrumented and
carefully performed laboratory tests in which the excitation
signal ref with a givenfrequency and amplitude is assumed. These
experiments produce (t) versus t and (t)versus t graphs in pairs
which lead to a complete (t) versus (t) system characteristic.Given
one specific hysteresis loop, the procedure to construct the
nonlinear functions
f, g, and h is as follows [3]: If the displacement signal (t) is
a cosine waveform with aperiod T, then for each value of , there
exist two instants of time t1 and t2 such that:
t1 t2 0 t1, t2 2 0,T 15
_t1 _t2 16
Then with the function g odd, we have:
g1_t1
h t1
g1_t2
ht2
Xd 17
In view of Eq. (4), we note that ((t1), (t1)) and ((t2), (t2))
represent points on thehysteresis loop with the same ordinate:
t1 t2 2Xd 18
t1 t2 f 1 t1 t22
f 1Xm 19
Geometrically, Xm represents the midpoint of the two points on
the hysteresis loopcorresponding to t1 and t2 and Xd is the
horizontal distance from the edge of thehysteresis loop to the
midpoint as shown in Fig. 3. Therefore, the locus of the variableXm
determines the function fwhile the locus of the variableXd
determines the function g.
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Next, the locus of each of the functions f and g is fitted to an
analytical odd poly-nomial function. Assuming that h is a unity
function (h() 1), the f and g parameterscan be estimated through a
nonlinear least-square fit.
Xm X51
a21 "f 20
Xd X51
b _ 21 "g 21
where (a) and (b) are the function parameters and "f, "g are the
model equation errors.For each equation the optimum parameters in
the least-square sense are determined
to minimize the criterion functions
Jf Xni1
"2f i Xni1
Xmi X51
a i 21" #2
22
Jg Xni1
"2g i Xni1
Xdi X51
b _ i 21" #2 23
where n is the number of data points in the hysteresis loop.The
estimated parameters are next used to find the estimated nonlinear
functions f^f , g^g.
f^f X51
a^a21 24
g^g1 _ X51
b^b _21 25
FIGURE 3 Measurement of stiffness and damping curves.
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The estimated transmitted torque ^ at the output of the harmonic
drive is finallyexpressed as:
^, _ g^g1 _ f^f 26
6. EXPERIMENTAL RESULTS
Various experiments have been carried out on the harmonic drive.
Figure 4 showsthe measured motor position which was controlled to
follow a sinusoidal referencesignal with 10 amplitude and 0.005Hz
frequency. Figure 5 shows the resultingtransmitted torque across
the flexpline. It can be seen that the torque is not a puresine
wave, which reflects the nonlinear relationship with the
displacement. Figure 6
FIGURE 4 Measured angular displacement.
FIGURE 5 Measured transmitted torque.
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shows the hysteresis curve obtained when the torque is plotted
as a function of thedisplacement. The results obtained also show
that the hysteresis curve depends onthe amplitude of the
displacement. It follows also that the resulting torque dependson
all the previous angular displacements which have been applied on
the elasticbody of the harmonic drive.In order to identify the
nonlinear functions f and g, the pair of waveforms ((t), (t))
is used. Given the measured hysteresis loop, the procedure is to
construct the locusof the points Xm and Xd representing
respectively the midpoint of the hysteresis loopand the horizontal
distance from the edge of the loop to the midpoint. The locus ofthe
variable Xm determines the function f while the locus of the
variable Xd determinesthe function g.To plot the function g, the
angular velocity d/dt is needed. The measured velocity is
given in Fig. 7(a). The high frequency noise in the speed signal
is a result of the differ-entiation of the angular position
measured from the position sensor. Because of thehigh frequency
noise, the accuracy of the parameters estimates of the function g
willbe affected. On the other hand, filtering the speed signal will
introduce a phase shiftwhich will also affect the accuracy.
Therefore the motor velocity is replaced with an esti-mated signal
obtained from the angular position . Since is assumed to follow a
puresine wave, its derivative will also be a sine signal displaced
by 90 or equivalently acosine function as shown in Fig. 7(b). The
experimental values of the f and g functionsare shown in Figs. 8
and 9. The estimated parameters of the analytical odd
polynomialfunction are listed in Table I.
7. SIMULATION RESULTS AND MODEL VALIDATION
To validate the proposed hysteresis model, a simulation of the
harmonic drive system isperformed with the locked output shaft. The
data of the identified parameters is used torepresent the stiffness
and viscous damping nonlinear functions of the flexspline.The
transmitted torque across the flexspline is computed using the
actual displacementangle and the estimated stiffness and damping
functions as given by Eq. (26). Figure 7(b)
FIGURE 6 Measured steady state hysteresis curve.
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FIGURE 7 Measured and estimated angular velocity.
FIGURE 8 Measured and stiffness function.
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shows the results of simulation in comparison with experiments.
We clearly observe avery good match of experimental results with
those of simulation. This proves thatthe proposed model is very
well suited for the purpose. The proposed model isshown to be
useful because the functions and parameters can be fine tuned to
matchexperimental results in a given situation. The nonlinear
ordinary differential equationhas guaranteed existence and
uniqueness of solution. The nonlinear functions arealso strictly
monotonically increasing and differentiable functions. Thus, the
resulting
FIGURE 9 Measured viscous damping function.
TABLE I Identified parameters
Parameter Value Unit
a^a1 14.133 Nm/(deg)
a^a2 3.71 103 Nm/(deg)3a^a3 1.471 105 Nm/(deg)5a^a4 3.778 106
Nm/(deg)7a^a5 3.370 107 Nm/(deg)9b^b1 65.594 Nm/(deg/s)
b^b2 5.571 107 Nm/(deg/s)3b^b3 2.485 1012 Nm/(deg/s)5b^b4 5.615
1016 Nm/(deg/s)7b^b5 4.673 1020 Nm/(deg/s)9
TABLE II A characteristics of the harmonicdrive system
Characteristic Value
Gear rated output torque 226NmGear maximum input speed 2800
rpmGear reduction ratio 50Wave generator inertia 0.436 103Motor
rated output torque 3.8NmMotor rated speed 4000 rpmMotor inertia
0.29 103 kgm2
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FIGURE 10 Measured and estimated stiffness function ():
measured; (- - -): estimated.
FIGURE 11 Measured and estimated viscous damping function ():
measured; (- - -): estimated.
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differential equation will be easily analyzed by standard
nonlinear systems stabilityanalysis tools and control
methodologies. see also Table II, Figs. 812.
8. CONCLUSION
A mathematical model for the hysteresis phenomenon in harmonic
drives has beenpresented. The proposed model is described by a
nonlinear differential equationrepresenting the torquedisplacement
relationship across the flexpline of the harmonicdrive. A
mechanical analogy obtained through the proposed methodology
amounts tohaving the combination of a nonlinear stiffness and a
nonlinear viscous damping.Numerical simulations and experiments
have been used to test this modeling concept.
References
[1] R. Bouc (1971). Modele mathematique dhysteresis. ACUSTICA,
24(3), 1625.[2] L.O. Chua and K.A. Stromsmoe (1970 Nov). Lymped
circuit models for nonlinear inductors exhibiting
hysteresis loops. IEEE Trans. on Circuit Theory, CT-17(4),
564574, .[3] L.O. Chua and K.A. Stromsmoe (1971). Mathematical
models for dynamic hysteresis loops. Int. Journal of
Eng. Science, 9 435450.[4] L.O. Chua and S.C. Bass (1972 Jan). A
generalized hysteresis model. IEEE Trans. on Circuit Theory,
CT-19(1), 3648.[5] B.D. Coleman and M. Hodgdon (1986). A
constitutive relation for rate-independent hysteresis in ferro-
magnetically soft materials. Int. Journal of Eng. Science,
24(6), 897919.[6] B.D. Coleman and M. Hodgdon (1987). On a class of
constitutive relations for ferromagnetic hysteresis.
Archive for Rational Mechanics and Analysis, 99(4), 375396.[7]
S. Hejny and F. Ghorbel (1997 May). Harmonic Drive Test Apparatus
for Data Acquisition and Control.
Internal Report ATP96-2, Dynamic Systems and Control Laboratory.
Rice University Department ofMechanical Engineering.
FIGURE 12 Measured and estimated hysteresis loop (): measured;
(- - -): estimated.
HYSTERESIS MODEL 13
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[8] M. Hodgdon (1988 Jan). Application of a theory of
ferromagnetic hysteresis. IEEE Trans. on Magnetics,24(1),
218221.
[9] M. Hodgdon (1988 Nov). Mathematical theory and calculations
of magnetic hysteresis curves. IEEETrans. on Magnetics, 24(6),
31203122.
[10] N. Kircanski, A.A. Goldenberg and S. Jia (1993). An
experimental study of nonlinear stiffness, hysteresisand friction
effects in robot joints with harmonic drives and torque sensors.
Third InternationalSymposium on Experimental Robotics, pp. 147154.
Oct. 2830, Kyoto.
[11] G. Legnany and R. Faglia (1992 March). Harmonic drive
transmissions: the effects of their elasticity,clearance and
irregularity on the dynamic behavior of an actual SCARA robot.
Robotica, 10(1),369376,.
[12] J.W. Macki, P. Nistri and P. Zecca (1993 March).
Mathematical models for hysteresis. SIAM Review,35(1), 94123.
[13] T. Marilier and J.A. Richard (1989). Nonlinear mechanic and
electric behaviour of a robotic axis with aharmonic drive gear.
Robotics and Computer Integrated Manufacturing, 5(23), 129136.
[14] W. Seyfferth, A.J. Maghazal and J. Angeles (1995).
Nonlinear modeling and parameter identificationof harmonic drive
robot transmissions. IEEE International Conference on Robotics and
Automation,30273032.
[15] T. Tuttle and W. Seering (1993). Modeling a harmonic drive
gear transmission. Proc. 1993 InternationalConf. on Robotics and
Automation, 624629.
[16] dSPACE (1993). Digital Signal Processing And Control
Engineering GmbH. DSP-CITeco LD31/LD31NET Users Guide.
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