MODELING AND ANALYSIS OF HYSTERESIS IN HARMONIC DRIVE GEARS

Transcription

1 Systems Analysis Model Simul, 200?, Vol. 00, No. 0, pp MODELING AND ANALYSIS OF HYSTERESIS IN HARMONIC DRIVE GEARS RACHED DHAOUADI a, * and FATHI GHORBEL b a Division of Electrical, Electronics and Computer Engineering, American University of Sharjah, P.O. Box 26666, Sharjah, UAE; b Department 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 drive gears with compliance and hysteresis. The hysteresis phenomenon in harmonic drives is described by a nonlinear differential equation representing the torque displacement relationship across the flexpline of the harmonic drive. The representation is equivalent to having the combination of nonlinear stiffness and nonlinear viscous damping. Numerical simulations along with experimental data have been used to validate the 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 and instrumentation servo systems such as industrial robots and medical equipment, where they provide high velocity reduction in a relatively small package permitting high torque amplification with only small motors. Numerous contributions have been made to the intuitive understanding and analytical description of harmonic drives. However, their inherent nonlinear characteristics have not been clearly analyzed. The three main nonlinear transmission attributes in harmonic drives responsible for motion transmission performance degradation include nonlinear stiffness, friction, and kinematic error. The transmission compliance and the internal dynamic friction mechanisms, resulting in hysteresis curves when torque is plotted against angular displacements, are controversial issues regarding the primary source of energy storage and dissipation in harmonic drives [10,11,13 15]. The accurate modeling of a total harmonic drive system (including the actuator, harmonic drive, sensors, and load) presents therefore a difficult problem. In much of the literature, the actuators providing the drive torques are modeled as pure torque sources, or as first-order lags. Numerous models have been proposed also to represent either the general system dynamics or *Corresponding author. rdhaouadi@aus.ac.ae ISSN print: ISSN online ß 200? Taylor & Francis Ltd DOI: /

2 2 R. DHAOUADI AND F. GHORBEL some aspects of nonlinear friction and compliance effects. The majority of these models have been either too complicated, with parameters that are difficult to determine, or too simple, assuming a linearized model and neglecting the nonlinear effects. The physical realities of the system have therefore limited the acceptance of these models. There is therefore a need to better understand the kinematic, dynamic, and transmission properties 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 hysteresis model admits descriptions in terms of hysteresis operators [12], or in terms of differential equations. The former description is used during mathematical analysis in order to get existence, uniqueness, and regularity results. The latter description is very useful for numerical computations and construction of the global model in terms of partial models given by dynamic equations. Bouc [1] used differential equations to model the hysteresis relationship. His model is based on the variation of the multivalued sign function. The problem of describing a material with hysteresis can reduce to that of finding a nonlinear or a piecewise linear function of the input signal v and the output signal w, so that 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 electromagnetics because the functions and parameters can be fine tuned to match experimental results in a given situation. Choua and Stromsoe [2,3] and Chua and Bass [4] also presented another general theory of hysteresis, considering constitutive models that take the form of first order differential equations. The main advantages of their models over existing models is its simplicity and the constructive procedure available for 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 harmonic drives. Our proposed approach uses differential equations to model the hysteresis relationship, which is resulting from the combined effect of the nonlinear flexibility of the flexpline and friction. In our case, position and speed satisfy an Euler-like differential equation describing the system dynamics. The representation of the hysterisis phenomenon by a differential equation is a useful approach to describe the overall harmonic drive 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 representing the nonlinear stiffness and the other the nonlinear viscous damping, so that the combination 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 in Section 4 and the parameters identification procedure in Section 5. The simulation and experimental results with the model validation, discussion, and conclusions are given in Sections 6 and HARMONIC DRIVE SETUP The harmonic drive system considered for our analysis is composed of a motor actuator, a harmonic drive gear, and an inertial load. The harmonic drive gear consists of the

3 HYSTERESIS MODEL 3 FIGURE 1 View of the harmonic drive test apparatus. mechanical assembly of three components: a rigid Circular Spline, an elliptical Wave Generator, and a nonrigid flexible spline or Flexspline, which form together a compact high-torque, high-ratio, in-line gear mechanism. A harmonic drive test apparatus was designed and built at Rice University as a platform to perform various types of experiments on the harmonic drive and to characterize the different errors inherent in its operation 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 to avoid the radial loading problem. A special design of vertical support plates and circular steel pipe sections with a highly stable platform was also used to maintain torsional integrity 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 objective was to avoid any torsion in the system produced by elements other than the flexpline and the harmonic drive as a hole. The harmonic drive system is driven by an AC servo motor with a dedicated power supply and controller. The total system is controlled with an IBM PC to which it is interfaced through a DSP board made by dspace [16]. Position feedback from the motor and the load are provided with high resolution optical encoders. The torque sensor used is a DC-operated noncontact torque sensor with a large capacity matching that of the harmonic drive. The signals and feedback are processed by the dspace board and may be displayed at the terminal in 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 programs necessary for the operation and control of the system were developed so that the user could 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 representation relating two variables x(t) andy(t) [3]: dy ¼ hðyþg½xðtþ fðyðtþþš, dt ð1þ

4 4 R. DHAOUADI AND F. GHORBEL where f(), g(), and h() are real-valued continuous and differentiable functions with continuous first-order derivatives and satisfying f 0 > 0, g 0 > 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 the desired nonlinear phenomena of hysteresis. The conditions imposed on f, g, and h will insure the existence and uniquences of a solution of (1) when x(t) is a continuous variable. This property is very important, since our objective is to obtain a reliable hysteresis model that can be integrated easily in the global harmonic drive model to yield 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 angular displacement across the flexpline, relative to a fixed reference position ¼ 1 N 2, ð2þ where 1 is the wave-generator position, 2 is the load position at the end side of the flexpline and N is the reduction ratio. If we replace the variable x by the torsional torque and the variable y by the angular displacement, Eq. (1) becomes: dðþ t ¼ hðþg½ðþ f t ððtþþš: dt This equation can be rearranged into the form ðtþ ¼g 1 ðtþ þ f ððtþþ: hððtþþ ð3þ ð4þ Equation (4) can be interpreted as the mechanical dynamic equation across the flexpline describing the parallel combination of a nonlinear torsional spring and a nonlinear viscous damping. The function f() determines the stiffness curve while the function g 1 () represents the nonlinear dynamic friction as shown in Fig. 2. FIGURE 2 Proposed mechanical analog of the hysteresis model.

5 HYSTERESIS MODEL 5 The validity of the nonlinear model (4) will then be established. This consists of first showing that the postulated model exhibits the same significant properties as the actual system and then verifying that the model gives realistic responses to one or more test signals. 4. FORMULATION OF THE EQUATIONS OF MOTION OF THE HARMONIC DRIVE In order to study the dynamic behavior of the complete harmonic drive system, the model of hysteresis will be combined with the wave generator and load dynamic models. The following set of equations represent the complete model of the harmonic drive: J 1 1 þ B 1 _ 1 þ f þ ð _, Þ N ¼ m ð5þ J 2 2 þ B 2 _ 2 þ L ð _, Þ ¼0 _ ð _, Þ ¼g 1 þ f ðþ hðþ ð6þ ð7þ ¼ 1 N 2 ð8þ where J 1 is the total motor and wave generator inertia, J 2 is the total load inertia, 1 is the motor position, 2 is the load position at the end side of the flexpline, N is the reduction gear ratio, B 1 and B 2 are the viscous damping coefficients at the motor side and the load side, is the transmitted torque across the flexpline, m is the driving torque applied by the electric motor, and L is the load torque. f represents a dry frictional torque component at the bearings of the wave generator which is a combination of the necessary torque to initiate motion from rest (static friction) and the 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 Locked Rotor Case When the output of the drive is locked 2 ¼ 0, the motor side can still rotate within a limited angular range allowed by the flexibility of the harmonic drive gear. The model of the setup in this case becomes 2 ¼ 0 ð9þ ¼ 1 N J þ B _ þ ð _, Þ ¼ n ð10þ ð11þ

6 6 R. DHAOUADI AND F. GHORBEL J ¼ N 2 J 1 ð12þ B ¼ N 2 B 1 n ¼ Nð m f Þ ð13þ ð14þ 5. NONLINEAR FUNCTIONS IDENTIFICATION In order to identify the nonlinear functions f, g, andh, a pair of waveforms {(t), (t)} must be measured. If (t) is selected as the excitation signal and d/dt is known, then Eq. (4) is reduced to an algebraic relationship to find the output signal (t). To be able to perform the proposed experiments, the experimental setup should be configured in a way to allow the angular displacement to be manipulated as the excitation signal. This consists in having the load side of the harmonic drive locked and then forcing the desired angular displacement through a feedback position control loop. The position control loop gains are adjusted so as to get the desired accuracy of the 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 given frequency 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 a period T, then for each value of, there exist two instants of time t 1 and t 2 such that: ðt 1 Þ¼ðt 2 Þ¼ 0 t 1, t 2 2½0, TŠ ð15þ Then with the function g odd, we have: g 1 _ðt 1 Þ¼ _ðt 2 Þ _ðt 1 Þ _ðt ¼ g 1 2 Þ ¼ X d hððt 1 ÞÞ hððt 2 ÞÞ ð16þ ð17þ In view of Eq. (4), we note that ((t 1 ), (t 1 )) and ((t 2 ), (t 2 )) represent points on the hysteresis loop with the same ordinate: ðt 1 Þ ðt 2 Þ¼2X d ðt 1 Þ¼ðt 2 Þ¼f 1 ðt 1Þþðt 2 Þ ¼ f 1 ðx m Þ 2 ð18þ ð19þ Geometrically, X m represents the midpoint of the two points on the hysteresis loop corresponding to t 1 and t 2 and X d is the horizontal distance from the edge of the hysteresis loop to the midpoint as shown in Fig. 3. Therefore, the locus of the variable X m determines the function f while the locus of the variable X d determines the function g.

7 HYSTERESIS MODEL 7 FIGURE 3 Measurement of stiffness and damping curves. Next, the locus of each of the functions f and g is fitted to an analytical odd polynomial function. Assuming that h is a unity function (h() ¼ 1), the f and g parameters can be estimated through a nonlinear least-square fit. X m ¼ X5 a 2 1 þ " f X d ¼ X5 ¼1 b ¼1 2 1 _ þ "g ð20þ 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 J f ¼ Xn i¼1 J g ¼ Xn i¼1 " 2 f ðiþ ¼Xn i¼1 " 2 g ðþ¼xn i i¼1 " # 2 X m ðiþ X5 a ½ðiÞŠ 2 1 ð22þ ¼1 " # 2 X d ðiþ X5 2 1 _ ðþ i ð23þ b ¼1 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, ^g. f^ðþ¼ X5 ¼1 ^a 2 1 ð24þ ^g 1 ð _Þ ¼ X5 ¼1 ^b ð _Þ 2 1 ð25þ

8 8 R. DHAOUADI AND F. GHORBEL The estimated transmitted torque ^ at the output of the harmonic drive is finally expressed as: ^ð, _Þ ¼ ^g 1 ð _Þþ ^ f ðþ ð26þ 6. EXPERIMENTAL RESULTS Various experiments have been carried out on the harmonic drive. Figure 4 shows the measured motor position which was controlled to follow a sinusoidal reference signal with 10 amplitude and Hz frequency. Figure 5 shows the resulting transmitted torque across the flexpline. It can be seen that the torque is not a pure sine wave, which reflects the nonlinear relationship with the displacement. Figure 6 FIGURE 4 Measured angular displacement. FIGURE 5 Measured transmitted torque.

9 HYSTERESIS MODEL 9 FIGURE 6 Measured steady state hysteresis curve. shows the hysteresis curve obtained when the torque is plotted as a function of the displacement. The results obtained also show that the hysteresis curve depends on the amplitude of the displacement. It follows also that the resulting torque depends on all the previous angular displacements which have been applied on the elastic body 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 locus of the points X m and X d representing respectively the midpoint of the hysteresis loop and the horizontal distance from the edge of the loop to the midpoint. The locus of the variable X m determines the function f while the locus of the variable X d determines the 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 differentiation of the angular position measured from the position sensor. Because of the high frequency noise, the accuracy of the parameters estimates of the function g will be affected. On the other hand, filtering the speed signal will introduce a phase shift which will also affect the accuracy. Therefore the motor velocity is replaced with an estimated signal obtained from the angular position. Since is assumed to follow a pure sine wave, its derivative will also be a sine signal displaced by 90 or equivalently a cosine function as shown in Fig. 7(b). The experimental values of the f and g functions are shown in Figs. 8 and 9. The estimated parameters of the analytical odd polynomial function are listed in Table I. 7. SIMULATION RESULTS AND MODEL VALIDATION To validate the proposed hysteresis model, a simulation of the harmonic drive system is performed with the locked output shaft. The data of the identified parameters is used to represent the stiffness and viscous damping nonlinear functions of the flexspline. The transmitted torque across the flexspline is computed using the actual displacement angle and the estimated stiffness and damping functions as given by Eq. (26). Figure 7(b)

10 10 R. DHAOUADI AND F. GHORBEL FIGURE 7 Measured and estimated angular velocity. FIGURE 8 Measured and stiffness function.

11 HYSTERESIS MODEL 11 shows the results of simulation in comparison with experiments. We clearly observe a very good match of experimental results with those of simulation. This proves that the proposed model is very well suited for the purpose. The proposed model is shown to be useful because the functions and parameters can be fine tuned to match experimental results in a given situation. The nonlinear ordinary differential equation has guaranteed existence and uniqueness of solution. The nonlinear functions are also strictly monotonically increasing and differentiable functions. Thus, the resulting FIGURE 9 Measured viscous damping function. TABLE I Identified parameters Parameter Value Unit ^a N m/(deg) ^a N m/(deg) 3 ^a N m/(deg) 5 ^a N m/(deg) 7 ^a N m/(deg) 9 ^b N m/(deg/s) ^b N m/(deg/s) 3 ^b N m/(deg/s) 5 ^b N m/(deg/s) 7 ^b N m/(deg/s) 9 TABLE II A characteristics of the harmonic drive system Characteristic Value Gear rated output torque 226 N m Gear maximum input speed 2800 rpm Gear reduction ratio 50 Wave generator inertia Motor rated output torque 3.8 N m Motor rated speed 4000 rpm Motor inertia kg m 2

12 12 R. DHAOUADI AND F. GHORBEL FIGURE 10 Measured and estimated stiffness function ( ): measured; (- - -): estimated. FIGURE 11 Measured and estimated viscous damping function ( ): measured; (- - -): estimated.

13 HYSTERESIS MODEL 13 FIGURE 12 Measured and estimated hysteresis loop ( ): measured; (- - -): estimated. differential equation will be easily analyzed by standard nonlinear systems stability analysis tools and control methodologies. see also Table II, Figs CONCLUSION A mathematical model for the hysteresis phenomenon in harmonic drives has been presented. The proposed model is described by a nonlinear differential equation representing the torque displacement relationship across the flexpline of the harmonic drive. A mechanical analogy obtained through the proposed methodology amounts to having 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), [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), ,. [3] L.O. Chua and K.A. Stromsmoe (1971). Mathematical models for dynamic hysteresis loops. Int. Journal of Eng. Science, [4] L.O. Chua and S.C. Bass (1972 Jan). A generalized hysteresis model. IEEE Trans. on Circuit Theory, CT-19(1), [5] B.D. Coleman and M. Hodgdon (1986). A constitutive relation for rate-independent hysteresis in ferromagnetically soft materials. Int. Journal of Eng. Science, 24(6), [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), [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 of Mechanical Engineering.

14 14 R. DHAOUADI AND F. GHORBEL [8] M. Hodgdon (1988 Jan). Application of a theory of ferromagnetic hysteresis. IEEE Trans. on Magnetics, 24(1), [9] M. Hodgdon (1988 Nov). Mathematical theory and calculations of magnetic hysteresis curves. IEEE Trans. on Magnetics, 24(6), [10] N. Kircanski, A.A. Goldenberg and S. Jia (1993). An experimental study of nonlinear stiffness, hysteresis and friction effects in robot joints with harmonic drives and torque sensors. Third International Symposium on Experimental Robotics, pp Oct , 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), ,. [12] J.W. Macki, P. Nistri and P. Zecca (1993 March). Mathematical models for hysteresis. SIAM Review, 35(1), [13] T. Marilier and J.A. Richard (1989). Nonlinear mechanic and electric behaviour of a robotic axis with a harmonic drive gear. Robotics and Computer Integrated Manufacturing, 5(2 3), [14] W. Seyfferth, A.J. Maghazal and J. Angeles (1995). Nonlinear modeling and parameter identification of harmonic drive robot transmissions. IEEE International Conference on Robotics and Automation, [15] T. Tuttle and W. Seering (1993). Modeling a harmonic drive gear transmission. Proc International Conf. on Robotics and Automation, [16] dspace (1993). Digital Signal Processing And Control Engineering GmbH. DSP-CITeco LD31/ LD31NET User s Guide.

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