MODELING AND ANALYSIS OF HYSTERESIS IN HARMONIC DRIVE GEARS · MODELING AND ANALYSIS OF HYSTERESIS...

Systems Analysis Model Simul, 200?, Vol. 00, No. 0, pp. 1–14

MODELING AND ANALYSIS OF HYSTERESIS IN

HARMONIC 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 torque–displacement 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 [10,11,13–15]. 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]

[25.6.2003–6:12pm] FIRST PROOFS {GandB}Gsam/GSAM-31049.3d Systems Analysis Model Simul, (GSAM) Paper: GSAM-31049 Keyword

ISSN 0232-9298 print: ISSN 1029-4902 online � 200? Taylor & Francis Ltd

DOI: 10.1080/0232929032000115137

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 Bouc’s 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 forthe 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 findingtwo 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 andthe 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

2 R. DHAOUADI AND F. GHORBEL


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

dt¼ hðyÞg½xðtÞ � f ðyðtÞÞ�, ð1Þ

FIGURE 1 View of the harmonic drive test apparatus.

HYSTERESIS MODEL 3


where f(�), g(�), and h(�) are real-valued continuous and differentiable functions withcontinuous first-order derivatives and satisfying

f 0 > 0, g0 > 0

0 < � � h � � < 1

where the ‘prime’ denotes differentiation with respect to the function’s argument, and� and � are finite positive constants.

With an appropriate selection of f, g, and h, Eq. (1) can be designed to exhibit thedesired 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 ofthe torque � applied to the harmonic drive flexpline as a function of the angulardisplacement � across the flexpline, relative to a fixed reference position

� ¼�1N

� �2, ð2Þ

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� tð Þ

dt¼ hð�Þg½� tð Þ � f ð�ðtÞÞ�: ð3Þ

This equation can be rearranged into the form

�ðtÞ ¼ g�1 �ðtÞ

hð�ðtÞÞ

� �þ f ð�ðtÞÞ: ð4Þ

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 g�1(�) represents the nonlinear dynamic friction as shown in Fig. 2.

FIGURE 2 Proposed mechanical analog of the hysteresis model.



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 MOTION

OF 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 þ B1_��1 þ �f þ

�ð _��, �Þ

N¼ �m ð5Þ

J2 €��2 þ B2_��2 þ �L � �ð _��, �Þ ¼ 0 ð6Þ

�ð _��, �Þ ¼ g�1_��

hð�Þ

� �þ f ð�Þ ð7Þ

� ¼�1N

� �2 ð8Þ

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 anonlinear coupling factor between the motor side and the load side dynamics.

4.1. Locked Rotor Case

When the output of the drive is locked �2¼ 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 ¼ 0 ð9Þ

� ¼�1N

ð10Þ

J €�� þ B _�� þ �ð _��, �Þ ¼ �n ð11Þ

HYSTERESIS MODEL 5


J ¼ N2J1 12ð Þ

B ¼ N2B1 ð13Þ

�n ¼ Nð�m � �f Þ ð14Þ

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 andcarefully 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 functionsf, 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:

�ðt1Þ ¼ �ðt2Þ ¼ �0 t1, t2 2 ½0,T � ð15Þ

_��ðt1Þ ¼ � _��ðt2Þ ð16Þ

Then with the function g odd, we have:

g�1_��ðt1Þ

hð� t1ð ÞÞ

� �¼ �g�1

_��ðt2Þ

hð�ðt2ÞÞ

� �¼ Xd ð17Þ

In view of Eq. (4), we note that (�(t1), �(t1)) and (�(t2), �(t2)) represent points on thehysteresis loop with the same ordinate:

�ðt1Þ � �ðt2Þ ¼ 2Xd ð18Þ

�ðt1Þ ¼ �ðt2Þ ¼ f �1 �ðt1Þ þ �ðt2Þ

2

� �¼ f �1ðXmÞ ð19Þ

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.



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(�)¼ 1), the f and g parameterscan be estimated through a nonlinear least-square fit.

Xm ¼X5�¼1

a��2��1 þ "f ð20Þ

Xd ¼X5�¼1

b� _�� 2��1

þ "g ð21Þ

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 ¼Xni¼1

"2f ðiÞ ¼Xni¼1

XmðiÞ �X5�¼1

a� �ðiÞ½ �2��1

" #2

ð22Þ

Jg ¼Xni¼1

"2g ið Þ ¼Xni¼1

XdðiÞ �X5�¼1

b� _�� ið Þ� �2��1

" #2

ð23Þ

where n is the number of data points in the hysteresis loop.The estimated parameters are next used to find the estimated nonlinear functions ff , gg.

ff �ð Þ ¼X5�¼1

aa��2��1 ð24Þ

gg�1ð _��Þ ¼X5�¼1

bb�ð _��Þ2��1

ð25Þ

FIGURE 3 Measurement of stiffness and damping curves.

HYSTERESIS MODEL 7


The estimated transmitted torque �� at the output of the harmonic drive is finallyexpressed as:

��ð�, _��Þ ¼ gg�1ð _��Þ þ ff ð�Þ ð26Þ

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.



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 isgiven 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.

HYSTERESIS MODEL 9


FIGURE 7 Measured and estimated angular velocity.

FIGURE 8 Measured and stiffness function.



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

aa1 14.133 Nm/(deg)

aa2 3.71� 103 Nm/(deg)3

aa3 � 1.471� 105 Nm/(deg)5

aa4 3.778� 106 Nm/(deg)7

aa5 � 3.370� 107 Nm/(deg)9

bb1 65.594 Nm/(deg/s)

bb2 5.571� 107 Nm/(deg/s)3

bb3 � 2.485� 1012 Nm/(deg/s)5

bb4 5.615� 1016 Nm/(deg/s)7

bb5 � 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� 10�3

Motor rated output torque 3.8NmMotor rated speed 4000 rpmMotor inertia 0.29� 10�3 kgm2

HYSTERESIS MODEL 11


FIGURE 10 Measured and estimated stiffness function (—): measured; (- - -): estimated.

FIGURE 11 Measured and estimated viscous damping function (—): measured; (- - -): estimated.



differential equation will be easily analyzed by standard nonlinear systems stabilityanalysis tools and control methodologies. see also Table II, Figs. 8–12.

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 torque–displacement 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 d’hysteresis. ACUSTICA, 24(3), 16–25.[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), 564–574, .[3] L.O. Chua and K.A. Stromsmoe (1971). Mathematical models for dynamic hysteresis loops. Int. Journal of

Eng. Science, 9 435–450.[4] L.O. Chua and S.C. Bass (1972 Jan). A generalized hysteresis model. IEEE Trans. on Circuit Theory,

CT-19(1), 36–48.[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), 897–919.[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), 375–396.[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


[8] M. Hodgdon (1988 Jan). Application of a theory of ferromagnetic hysteresis. IEEE Trans. on Magnetics,24(1), 218–221.

[9] M. Hodgdon (1988 Nov). Mathematical theory and calculations of magnetic hysteresis curves. IEEETrans. on Magnetics, 24(6), 3120–3122.

[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. 147–154. Oct. 28–30, 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),369–376,.

[12] J.W. Macki, P. Nistri and P. Zecca (1993 March). Mathematical models for hysteresis. SIAM Review,35(1), 94–123.

[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(2–3), 129–136.

[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,3027–3032.

[15] T. Tuttle and W. Seering (1993). Modeling a harmonic drive gear transmission. Proc. 1993 InternationalConf. on Robotics and Automation, 624–629.

[16] dSPACE (1993). Digital Signal Processing And Control Engineering GmbH. DSP-CITeco LD31/LD31NET User’s Guide.



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